CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION GROUPS OF EXCEPTIONAL TYPE — THE DETAILS

Christian Krattenthaler^{†} and Thomas W. Müller^{‡}

† Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria.

                      WWW: http://www.mat.univie.ac.at/~kratt

‡ School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom.

                       WWW: http://www.maths.qmw.ac.uk/~twm/

Dedicated to the memory of Herb Wilf

Abstract. We prove that the generalised non-crossing partitions associated with well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, and by Bessis and Reiner. This manuscript accompanies the paper “Cyclic sieving for generalised non-crossing partitions associated with complex reflection groups of exceptional type” [arχiv:1001.0028], for which it provides the computational details.

1. Introduction

In his memoir [3], Armstrong introduced generalised non-crossing partitions associated with finite (real) reflection groups, thereby embedding Kreweras’ non-crossing partitions [22], Edelman’s m-divisible non-crossing partitions [12], the non-crossing partitions associated with reflection groups due to Bessis [7] and Brady and Watt [11] into one uniform framework. Bessis and Reiner [10] observed that Armstrong’s definition can be straightforwardly extended to well-generated complex reflection groups (see Section 2 for the precise definition). These generalised non-crossing partitions possess a wealth of beautiful properties, and they display deep and surprising relations to other combinatorial objects defined for reflection groups (such as the generalised cluster complex of Fomin and Reading [13], or the extended Shi arrangement and the geometric multichains of filters of Athanasiadis [5, 6]); see Armstrong’s memoir [3] and the references given therein.

2000 Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07 20F55.

Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible noncrossing partitions, generalised non-crossing partitions, Fuß–Catalan numbers, cyclic sieving.

†Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory.”

‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.

On the other hand, cyclic sieving is a phenomenon brought to light by Reiner, Stanton and White [29]. It extends the so-called “(−1)-phenomenon” of Stembridge [33, 34]. Cyclic sieving can be defined in three equivalent ways (cf. [29, Prop. 2.1]). The one which gives the name can be described as follows: given a set S of combinatorial objects, an action on S of a cyclic group G = ⟨g⟩ with generator g of order n, and a polynomial P (q) in q with non-negative integer coefficients, we say that the triple (S, P, G) exhibits the cyclic sieving phenomenon, if the number of elements of S fixed by g^{k} equals P (e^{2πik/n}). In [29] it is shown that this phenomenon occurs in surprisingly many contexts, and several further instances have been discovered since then.

In [3, Conj. 5.4.7] (also appearing in [10, Conj. 6.4]) and [10, Conj. 6.5], Armstrong, respectively Bessis and Reiner, conjecture that generalised non-crossing partitions for irreducible well-generated complex reflection groups exhibit two different cyclic sieving phenomena (see Sections 3 and 7 for the precise statements).

According to the classification of these groups due to Shephard and Todd [31], there are two infinite families of irreducible well-generated complex reflection groups, namely the groups G(d, 1, n) and G(e, e, n), where n, d, e are positive integers, and there are 26 exceptional groups. For the infinite families of types G(d, 1, n) and G(e, e, n), the two cyclic sieving conjectures follow from the results in [19].

The purpose of the present article is to prove the cyclic sieving conjectures of Armstrong, and of Bessis and Reiner, for the 26 exceptional types, thus completing the proof of these conjectures. Since the generalised non-crossing partitions feature a parameter m, from the outset this is not a finite problem. Consequently, we first need several auxiliary results to reduce the conjectures for each of the 26 exceptional types to a finite problem. Subsequently, we use Stembridge’s Maple package coxeter [35] and the GAP package CHEVIE [14, 27] to carry out the remaining finite computations. It is interesting to observe that, for the verification of the type E_{8} case, it is essential to use the decomposition numbers in the sense of [17, 18, 20] because, otherwise, the necessary computations would not be feasible in reasonable time with the currently available computer facilities. We point out that, for the special case where the aforementioned parameter m is equal to 1, the first cyclic sieving conjecture has been proved in a uniform fashion by Bessis and Reiner in [10]. (See [4] for a — non-uniform — proof of cyclic sieving for non-crossing partitions associated with real reflection groups under the action of the so-called Kreweras map, a special case of the second cyclic sieving phenomenon discussed in the present paper.) The crucial result on which the proof of Bessis and Reiner is based is (5.5) below, and it plays an important role in our reduction of the conjectures for the 26 exceptional groups to a finite problem.

Our paper is organised as follows. In the next section, we recall the definition of generalised non-crossing partitions for well-generated complex reflection groups and of decomposition numbers in the sense of [17, 18, 20], and we review some basic facts. The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial P in the cyclic sieving phenomena concerning the generalised non-crossing partitions for well-generated complex reflection groups, is always a polynomial with non-negative integer coefficients, as required by the definition of cyclic sieving. (The reader is referred to the first paragraph of Section 4 for comments on other approaches for establishing polynomiality with non-negative coefficients.) Section 5 contains the announced auxiliary results which, for the 26 exceptional types, allow a reduction of the conjecture to a finite problem. The remaining case-by-case verification of the conjecture is then carried out in Section 6. The second cyclic sieving conjecture is stated in Section 7. Section 8 contains the auxiliary results which, for the 26 exceptional types, allow a reduction of the conjecture to a finite problem, while Section 9 contains the remaining case-by-case verification of the conjecture.

2. Preliminaries

A complex reflection group is a group generated by (complex) reflections in ℂ^{n}. (Here, a reflection is a non-trivial element of GL_{n}(ℂ) which fixes a hyperplane pointwise and which has finite order.) We refer to [24] for an in-depth exposition of the theory complex reflection groups.

Shephard and Todd provided a complete classification of all finite complex reflection groups in [31] (see also [24, Ch. 8]). According to this classification, an arbitrary complex reflection group W decomposes into a direct product of irreducible complex reflection groups, acting on mutually orthogonal subspaces of the complex vector space on which W is acting. Moreover, the list of irreducible complex reflection groups consists of the infinite family of groups G(m, p, n), where m, p, n are positive integers, and 34 exceptional groups, denoted G_{4}, G_{5}, …, G_{37} by Shephard and Todd.

In this paper, we are only interested in finite complex reflection groups which are well-generated. A complex reflection group of rank n is called well-generated if it is generated by n reflections.^{1} Well-generation can be equivalently characterised by a duality property due to Orlik and Solomon [28]. Namely, a complex reflection group of rank n has two sets of distinguished integers d_{1} ≤ d_{2} ≤ ⋯ ≤ d_{n} and d_{1}^{∗} ≥ d_{2}^{∗} ≥ ⋯ ≥ d_{n}^{∗}, called its degrees and codegrees, respectively (see [24, p. 51 and Def. 10.27]). Orlik and Solomon observed, using case-by-case checking, that an irreducible complex reflection group W of rank n is well-generated if and only if its degrees and codegrees satisfy

d_{i} + d_{i}^{∗} = d_{n}

for all i = 1, 2, …, n. The reader is referred to [24, App. D.2] for a table of the degrees and codegrees of all irreducible complex reflection groups. Together with the classification of Shephard and Todd [31], this constitutes a classification of well-generated complex reflection groups: the irreducible well-generated complex reflection groups are

— the two infinite families G(d, 1, n) and G(e, e, n), where d, e, n are positive integers,

— the exceptional groups G_{4}, G_{5}, G_{6}, G_{8}, G_{9}, G_{10}, G_{14}, G_{16}, G_{17}, G_{18}, G_{20}, G_{21} of rank 2,

— the exceptional groups G_{23} = H_{3}, G_{24}, G_{25}, G_{26}, G_{27} of rank 3,

— the exceptional groups G_{28} = F_{4}, G_{29}, G_{30} = H_{4}, G_{32} of rank 4,

— the exceptional group G_{33} of rank 5,

— the exceptional groups G_{34}, G_{35} = E_{6} of rank 6,

— the exceptional group G_{36} = E_{7} of rank 7,

1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roughly speaking, the rank of a complex reflection group W is the minimal n such that W can be realized as reflection group on ℂ^{n}.

— and the exceptional group G_{37} = E_{8} of rank 8. In this list, we have made visible the groups H_{3}, F_{4}, H_{4}, E_{6}, E_{7}, E_{8} which appear as exceptional groups in the classification of all irreducible real reflection groups (cf. [16]).

Let W be a well-generated complex reflection group of rank n, and let T ⊆ W denote the set of all (complex) reflections in the group. Let ℓ_{T} : W → ℤ denote the word length in terms of the generators T. This word length is called absolute length or reflection length. Furthermore, we define a partial order ≤_{T} on W by

u ≤_{T} w if and only if ℓ_{T} (w) = ℓ_{T} (u) + ℓ_{T} (u^{−1}w).  (2.1)

This partial order is called absolute order or reflection order. As is well-known and easy to see, the equation in (2.1) is equivalent to the statement that every shortest representation of u by reflections occurs as an initial segment in some shortest product representation of w by reflections.

Now fix a (generalised) Coxeter element^{2} c ∈ W and a positive integer m. The m-divisible non-crossing partitions NC^{m}(W) are defined as the set

NC^{m}(W) = {(w_{0}; w_{1}, …, w_{m}) : w_{0}w_{1} ⋯ w_{m} = c and

ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) = ℓ_{T} (c)}.

A partial order is defined on this set by

(w_{0}; w_{1}, …, w_{m}) ≤ (u_{0}; u_{1}, …, u_{m}) if and only if u_{i} ≤_{T} w_{i} for 1 ≤ i ≤ m.

We have suppressed the dependence on c, since we understand this definition up to isomorphism of posets. To be more precise, it can be shown that any two Coxeter elements are related to each other by conjugation and (possibly) an automorphism on the field of complex numbers (see [32, Theorem 4.2] or [24, Cor. 11.25]), and hence the resulting posets NC^{m}(W) are isomorphic to each other. If m = 1, then NC^{1}(W) can be identified with the set NC(W) of non-crossing partitions for the (complex) reflection group W as defined by Bessis and Corran (cf. [9] and [8, Sec. 13]; their definition extends the earlier definition by Bessis [7] and Brady and Watt [11] for real reflection groups).

The following result has been proved by a collaborative effort of several authors (see [8, Prop. 13.1]).

Theorem 1. Let W be an irreducible well-generated complex reflection group, and let d_{1} ≤ d_{2} ≤ ⋯ ≤ d_{n} be its degrees and h := d_{n} its Coxeter number. Then

|NC^{m}(W)| = ∏_{i=1}^{n} \frac{mh + d_{i}}{d_{i}} .  (2.2)

2An element of an irreducible well-generated complex reflection group W of rank n is called a Coxeter element if it is regular in the sense of Springer [32] (see also [24, Def. 11.21]) and of order d_{n}.

An element of W is called regular if it has an eigenvector which lies in no reflecting hyperplane of a reflection of W. It follows from an observation of Lehrer and Springer, proved uniformly by Lehrer and Michel [23] (see [24, Theorem 11.28]), that there is always a regular element of order d_{n} in an irreducible well-generated complex reflection group W of rank n. More generally, if a well-generated complex reflection group W decomposes as W ≅ W_{1} × W_{2} × ⋯ × W_{k}, where the W_{i}’s are irreducible, then a Coxeter element of W is an element of the form c = c_{1}c_{2} ⋯ c_{k}, where c_{i} is a Coxeter element of W_{i}, i = 1, 2, …, k. If W is a real reflection group, that is, if all generators in T have order 2, then the notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical sense (cf. [16, Sec. 3.16]).

Remark 1. (1) The number in (2.2) is called the Fuß–Catalan number for the reflection group W.

(2) If c is a Coxeter element of a well-generated complex reflection group W of rank n, then ℓ_{T} (c) = n. (This follows from [8, Sec. 7].)

We conclude this section by recalling the definition of decomposition numbers from [17, 18, 20]. Although we need them here only for (very small) real reflection groups, and although, strictly speaking, they have been only defined for real reflection groups in [17, 18, 20], this definition can be extended to well-generated complex reflection groups without any extra effort, which we do now.

Given a well-generated complex reflection group W of rank n, types T_{1}, T_{2}, …, T_{d} (in the sense of the classification of well-generated complex reflection groups) such that the sum of the ranks of the T_{i}’s equals n, and a Coxeter element c, the decomposition number N_{W} (T_{1}, T_{2}, …, T_{d}) is defined as the number of “minimal” factorisations c = c_{1}c_{2} ⋯ c_{d}, “minimal” meaning that ℓ_{T} (c_{1}) + ℓ_{T} (c_{2}) + ⋯ + ℓ_{T} (c_{d}) = ℓ_{T} (c) = n, such that, for i = 1, 2, …, d, the type of c_{i} as a parabolic Coxeter element is T_{i}. (Here, the term “parabolic Coxeter element” means a Coxeter element in some parabolic subgroup. It follows from [30, Prop.6.3] that any element c_{i} is indeed a Coxeter element in a unique parabolic subgroup of W.^{3} By definition, the type of c_{i} is the type of this parabolic subgroup.) Since any two Coxeter elements are related to each other by conjugation plus field automorphism, the decomposition numbers are independent of the choice of the Coxeter element c.

The decomposition numbers for real reflection groups have been computed in [17, 18, 20]. To compute the decomposition numbers for well-generated complex reflection groups is a task that remains to be done.

3. Cyclic sieving I

In this section we present the first cyclic sieving conjecture due to Armstrong [3, Conj. 5.4.7], and to Bessis and Reiner [10, Conj. 6.4].

Let ϕ : NC^{m}(W) → NC^{m}(W) be the map defined by

(w_{0}; w_{1}, …, w_{m}) ↦ ((cw_{m}c^{−1})w_{0}(cw_{m}c^{−1})^{−1}; cw_{m}c^{−1}, w_{1}, w_{2}, …, w_{m−1}).  (3.1)

It is indeed not difficult to see that, if the (m + 1)-tuple on the left-hand side is an element of NC^{m}(W), then so is the (m + 1)-tuple on the right-hand side. For m = 1, this action reduces to conjugation by the Coxeter element c (applied to w_{1}). Cyclic sieving arising from conjugation by c has been the subject of [10].

It is easy to see that ϕ^{mh} acts as the identity, where h is the Coxeter number of W (see (5.1) and Lemma 29 below). By slight abuse of notation, let C_{1} be the cyclic group of order mh generated by ϕ. (The slight abuse consists in the fact that we insist on C_{1} to be a cyclic group of order mh, while it may happen that the order of the action of ϕ given in (3.1) is actually a proper divisor of mh.)

3The uniqueness can be argued as follows: suppose that c_{i} were a Coxeter element in two parabolic subgroups of W, say U_{1} and U_{2}. Then it must also be a Coxeter element in the intersection U_{1} ∩ U_{2}.

On the other hand, the absolute length of a Coxeter element of a complex reflection group U is always equal to rk(U), the rank of U. (This follows from the fact that, for each element u of U, we have ℓ_{T} (u) = codim(ker(u − id)), with id denoting the identity element in U; see e.g. [30, Prop. 1.3]). We conclude that ℓ_{T} (c_{i}) = rk(U_{1}) = rk(U_{2}) = rk(U_{1} ∩ U_{2}), This implies that U_{1} = U_{2}.

Given these definitions, we are now in the position to state the first cyclic sieving conjecture of Armstrong, respectively of Bessis and Reiner. By the results of [19] and of this paper, it becomes the following theorem.

Theorem 2. For an irreducible well-generated complex reflection group W and any m ≥ 1, the triple (NC^{m}(W), Cat^{m}(W; q), C_{1}), where Cat^{m}(W; q) is the q-analogue of the Fuß–Catalan number defined by

Cat^{m}(W; q) := ∏_{i=1}^{n} \frac{[mh + d_{i}]_{q}}{[d_{i}]_{q}} ,  (3.2)

exhibits the cyclic sieving phenomenon in the sense of Reiner, Stanton and White [29]. Here, n is the rank of W, d_{1}, d_{2}, …, d_{n} are the degrees of W, h is the Coxeter number of W, and [α]_{q} := (1 − q^{α})/(1 − q).

Remark 2. We write Cat^{m}(W) for Cat^{m}(W; 1).

By definition of the cyclic sieving phenomenon, we have to prove that Cat^{m}(W; q) is a polynomial in q with non-negative integer coefficients, and that

| Fix_{NC^{m}(W)}(ϕ^{p})| = Cat^{m}(W; q)|_{q=e^{2πip/mh}},  (3.3)

for all p in the range 0 ≤ p < mh. The first fact is established in the next section, while the proof of the second is achieved by making use of several auxiliary results, given in Section 5, to reduce the proof to a finite problem, and a subsequent case-by-case analysis, which occupies Section 6.

4. The q-Fusz–Catalan numbers Cat^{m}(W; q)

The purpose of this section is to provide an elementary, self-contained proof of the fact that, for all irreducible complex reflection groups W, the q-Fuß–Catalan number Cat^{m}(W; q) is a polynomial in q with non-negative integer coefficients. For most of the groups, this is a known property. However, aside from the fact that, for many of the known cases, the proof is very indirect and uses deep algebraic results on rational Cherednik algebras, there still remained some cases where this property had not been formally established. The reader is referred to the Theorem in Section 1.6 of [15], which says that, under the assumption of a certain rank condition ([15, Hypothesis 2.4]), the q-Fuß–Catalan number Cat^{m}(W; q) is a Hilbert series of a finite-dimensional quotient of the ring of invariants of W and also the graded character of a finite-dimensional irreducible representation of a spherical rational Cherednik algebra associated with W. At present, this rank condition has been proven for all irreducible well-generated complex reflection groups apart from G_{17}, G_{18}, G_{29}, G_{33}, G_{34}; see [25, Tables 8 and 9, column “rank”], and the recent paper [26], which establishes the result in the case of G_{32}.

In the sequel, aside from the standard notation [α]_{q} = (1 − q^{α})/(1 − q) for q-integers, we shall also use the q-binomial coefficient, which is defined by

[\begin{matrix}n \\ k \end{matrix}]_{q} := \begin{cases}1, & if k = 0, \\ \frac{[n]_{q} [n−1]_{q}···[n−k+1]_{q}}{[k]_{q} [k−1]_{q}···[1]_{q}} , & if k > 0. \end{cases}

We begin with several auxiliary results.

Proposition 3. For all non-negative integers n and k, the q-binomial coefficient [ \begin{matrix}n \\ k \end{matrix}]_{q} is a polynomial in q with non-negative integer coefficients.

Proof. This is a well-known fact, which can be derived either from the recurrence relation(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from the fact that the qbinomial coefficient [ \begin{matrix}n \\ k \end{matrix}]_{q} is the generating function for (integer) partitions with at most k parts all of which are at most n − k (cf. [1, Theorem 3.1]). □

Proposition 4. For all non-negative integers m and n, the q-Fuß–Catalan number of type A_{n},

\frac{1}{[(m + 1)n + 1]_{q}} [\begin{matrix}(m + 1)n + 1 \\ n \end{matrix}]_{q} ,

is a polynomial in q with non-negative integer coefficients.

Proof. This follows by replacing n by mn + 1 and j by n in Theorem 2 of [2]. □

Lemma 5. If a and b are coprime positive integers, then

\frac{[ab]_{q}}{[a]_{q} [b]_{q}}  (4.1)

is a polynomial in q of degree (a − 1)(b − 1), all of whose coefficients are in {0, 1, −1}. Moreover, if one disregards the coefficients which are 0, then +1’s and (−1)’s alternate, and the constant coefficient as well as the leading coefficient of the polynomial equal +1.

Proof. Let Φ_{n}(q) denote the n-th cyclotomic polynomial in q. Using the classical formula

1 − q^{n} = ∏_{d|n} Φ_{d}(q),

we see that

\frac{(1 − q)(1 − q^{ab})}{(1 − q^{a})(1 − q^{b})} = _{∏_{d_{1}|a, d_{1}≠1} ; d_{2}|a, d_{2}≠1}Φ_{d_{1}d_{2}}(q),

so that, manifestly, the expression in (4.1) is a polynomial in q. The claim concerning the degree of this polynomial is obvious.

In order to establish the claim on the coefficients, we start with a sub-expression of (4.1),

\frac{(1 − q^{ab})}{(1 − q^{a})(1 − q^{b})} = ( ∑_{i=0}^{b−1} q^{ia})( ∑_{j=0}^{∞} q^{jb}) = ∑_{k=0}^{∞} C_{k}q^{k},  (4.2)

say. The assumption that a and b are coprime implies that 0 ≤ C_{k} ≤ 1 for k ≤ (a − 1)(b − 1). Multiplying both sides of (4.2) by 1 − q, we obtain the equation

\frac{[ab]_{q}}{[a]_{q} [b]_{q}} = (1 − q) ∑_{k=0}^{(a−1)(b−1)} C_{k}q^{k} + (1 − q) ∑_{k=(a−1)(b−1)+1}^{∞} C_{k}q^{k}.  (4.3)

By our previous observation on the coefficients C_{k} with k ≤ (a − 1)(b − 1), it is obvious that the coefficients of the first expression on the right-hand side of (4.3) are alternately +1 and −1, when 0’s are disregarded. Since we already know that the left-hand side is a polynomial in q of degree (a − 1)(b − 1), we may ignore the second expression.

The proof is concluded by observing that the claims on the constant and leading coefficients are obvious. □

Corollary 6. Let a and b be coprime positive integers, and let γ be an integer with γ ≥ (a − 1)(b − 1). Then the expression

\frac{[γ]_{q} [ab]_{q}}{[a]_{q} [b]_{q}}

is a polynomial in q with non-negative integer coefficients.

Proof. Let

\frac{[ab]_{q}}{[a]_{q} [b]_{q}} = ∑_{k=0}^{(a−1)(b−1)} D_{k}q^{k}.

We then have

\frac{[γ]_{q} [ab]_{q}}{[a]_{q} [b]_{q}} = ∑_{N=0}^{(a−1)(b−1)+γ−1} q^{N} ∑_{k=max\{0,N−γ+1\}}^{N} D_{k}.  (4.4)

If N ≤ γ − 1, then, by Lemma 5, the sum over k on the right-hand side of (4.4) equals 1 − 1 + 1 − 1 + ⋯, which is manifestly non-negative. On the other hand, if N > γ − 1, then we may rewrite the sum over k on the right-hand side of (4.4) as

∑_{k=max{0,N−γ+1}}^{N} D_{k} = ∑_{k=N−γ+1}^{(a−1)(b−1)} D_{k} = ∑_{k=0}^{(a−1)(b−1)+γ−1−N} D_{(a−1)(b−1)−k}.

Again, by Lemma 5, this sum equals 1 − 1 + 1 − 1 + ⋯, which is manifestly nonnegative. □

Lemma 7. Let α and β be positive integers with α ≥ 6 and β ≥ 8. Then the expression

[α]_{q^{3}} [β]_{q^{4}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix}

= (1 − q^{3} + q^{9} − q^{15} + q^{18})(1 − q^{4} + q^{8} − q^{12} + q^{16} − q^{20} + q^{24} − q^{28} + q^{32}).

It should be observed that both factors on the right-hand side have the property that coefficients are in {0, 1, −1} and that (+1)’s and (−1)’s alternate, if one disregards the coefficients which are 0. If we now apply the same idea as in the proof of Corollary 6, then we see that [α]_{q^{3}} times the first factor is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{4}} times the second factor. Taken together, this establishes the claim. □

Lemma 8. Let α and β be positive integers with α ≥ 26 and β ≥ 8. Then the expression

[α]_{q} [β]_{q^{4}} \begin{matrix}[15] & q \\ [3] & q & [5] & q \end{matrix} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[15] & q & [72] & q & [4] & q \\ [5] & q & [8] & q & [9] & q & [12] & q \end{matrix}

= (1 − q + q^{5} − q^{6} + q^{9} − q^{11} + q^{12} − q^{13} + q^{14} − q^{15} + q^{17} − q^{20} + q^{21} − q^{25} + q^{26})

× (1 − q^{4} + q^{8} − q^{12} + q^{16} − q^{20} + q^{24} − q^{28} + q^{32}).

Again, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{4}} times the second factor. Taken together, this establishes the claim. □

Lemma 9. Let α and β be positive integers with α ≥ 18 and β ≥ 3. Then the expression

[α]_{q^{3}} [β]_{q^{4}} \begin{matrix}[90] & q & [3] & q & [4] & q \\ [5] & q & [6] & q & [9] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[90] & q & [3] & q & [4] & q \\ [5] & q & [6] & q & [9] & q \end{matrix}

= (1 − q^{3} + q^{9} − q^{12} + q^{18} − q^{21} + q^{27} − q^{33} + q^{36} − q^{42} + q^{45} − q^{51} + q^{54})
× (1 − q^{4} + q^{5} + q^{6} − q^{9} + q^{11} + q^{12} − q^{14} + q^{17} + q^{18} − q^{19} + q^{23})
= (1 − q^{3} + q^{9} − q^{12} + q^{18} − q^{21} + q^{27} − q^{33} + q^{36} − q^{42} + q^{45} − q^{51} + q^{54})

× (1 − q^{4} + q^{12} + q^{5}(1 − q^{4} + q^{12}) + q^{6}(1 − q^{8} + q^{12}) + q^{11}(1 − q^{8} + q^{12})).

Again, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q^{3}} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{4}} times the second factor. Taken together, this establishes the claim. □

Lemma 10. Let α and β be positive integers with α ≥ 20 and β ≥ 18. Then the expression

[α]_{q} [β]_{q^{3}} \begin{matrix}[90] & q & [3] & q \\ [5] & q & [6] & q & [9] & q \end{matrix}

Proof. We have

\begin{matrix}[90] & q & [3] & q \\ [5] & q & [6] & q & [9] & q \end{matrix}

= (1 − q^{3} + q^{9} − q^{12} + q^{18} − q^{21} + q^{27} − q^{33} + q^{36} − q^{42} + q^{45} − q^{51} + q^{54})
× (1 − q + q^{5} − q^{7} + q^{10} − q^{13} + q^{15} − q^{19} + q^{20}).

Again, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q^{3}} times the second factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{3}} times the first factor. Taken together, this establishes the claim. □

Lemma 11. Let α be a positive integer with α ≥ 26. Then the expression

[α]_{q} \begin{matrix}[15] & q \\ [3] & q & [5] & q \end{matrix} \begin{matrix}[12] & q_{3} \\ [3] & q_{3} & [4] & q_{3} \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[15] & q \\ [3] & q & [5] & q \end{matrix} \begin{matrix}[12] & q_{3} \\ [3] & q_{3} & [4] & q_{3} \end{matrix}

= 1 − q + q^{5} − q^{6} + q^{9} − q^{11} + q^{12} − q^{13} + q^{14} − q^{15} + q^{17} − q^{20} + q^{21} − q^{25} + q^{26}.

Once again, the coefficients of the polynomial on the right-hand side are in {0, 1, −1} and (+1)’s and (−1)’s alternate, if one disregards the coefficients which are 0. The argument from the proof of Corollary 6 then implies that this is a polynomial in q with non-negative integer coefficients. □

Lemma 12. Let α be a positive integer with α ≥ 14. Then the expression

[α]_{q} \begin{matrix}[15] & q \\ [3] & q & [5] & q \end{matrix} \begin{matrix}[6] & q_{3} \\ [2] & q_{3} & [3] & q_{3} \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[15] & q \\ [3] & q & [5] & q \end{matrix} \begin{matrix}[6] & q_{3} \\ [2] & q_{3} & [3] & q_{3} \end{matrix} = 1 − q + q^{5} − q^{7} + q^{9} − q^{13} + q^{14}.

Repeating the arguments of the previous proof, this establishes the claim. □

Lemma 13. Let α and β be positive integers with α ≥ 30 and β ≥ 20. Then the expression

[α]_{q} [β]_{q^{2}} \begin{matrix}[84] & q & [2] & q \\ [4] & q & [6] & q & [7] & q \end{matrix}

Proof. We have

\begin{matrix}[84] & q & [2] & q \\ [4] & q & [6] & q & [7] & q \end{matrix}

= (1 − q + q^{6} − q^{8} + q^{12} − q^{15} + q^{18} − q^{22} + q^{24} − q^{29} + q^{30})
× (1 − q^{2} + q^{4} − q^{6} + q^{8} − q^{10} + q^{12} − q^{14} + q^{16} − q^{18} + q^{20} − q^{22}
+ q^{24} − q^{26} + q^{28} − q^{30} + q^{32} − q^{34} + q^{36} − q^{38} + q^{40}).

Once again, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{2}} times the second factor. Taken together, this establishes the claim. □

Lemma 14. Let α and β be positive integers with α ≥ 24 and β ≥ 68. Then the expression

[α]_{q} [β]_{q} \begin{matrix}[105] & q \\ [3] & q & [5] & q & [7] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[105] & q \\ [3] & q & [5] & q & [7] & q \end{matrix}

= (1 − q + q^{5} − q^{6} + q^{7} − q^{8} + q^{10} − q^{11} + q^{12} − q^{13} + q^{14} − q^{16}
+ q^{17} − q^{18} + q^{19} − q^{23} + q^{24})

× (1 − q + q^{3} − q^{4} + q^{6} − q^{7} + q^{9} − q^{10} + q^{12} − q^{13} + q^{15} − q^{16} + q^{18} − q^{19}

+ q^{21} − q^{22} + q^{24} − q^{25} + q^{27} − q^{28} + q^{30} − q^{31} + q^{33} − q^{34} + q^{35} − q^{37}
+ q^{38} − q^{40} + q^{41} − q^{43} + q^{44} − q^{46} + q^{47} − q^{49} + q^{50} − q^{52} + q^{53} − q^{55}
+ q^{56} − q^{58} + q^{59} − q^{61} + q^{62} − q^{64} + q^{65} − q^{67} + q^{68}).

Once again, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q} times the second factor. Taken together, this establishes the claim. □

Lemma 15. Let α and β be positive integers with α ≥ 24 and β ≥ 34. Then the expression

[α]_{q} [β]_{q} \begin{matrix}[70] & q \\ [2] & q & [5] & q & [7] & q \end{matrix}

Proof. We have

\begin{matrix}[70] & q \\ [2] & q & [5] & q & [7] & q \end{matrix}

= (1 − q + q^{5} − q^{6} + q^{7} − q^{8} + q^{10} − q^{11} + q^{12} − q^{13}
+ q^{14} − q^{16} + q^{17} − q^{18} + q^{19} − q^{23} + q^{24})

× (1 − q + q^{2} − q^{3} + q^{4} − q^{5} + q^{6} − q^{7} + q^{8} − q^{9} + q^{10} − q^{11} + q^{12} − q^{13}

+ q^{14} − q^{15} + q^{16} − q^{17} + q^{18} − q^{19} + q^{20} − q^{21} + q^{22} − q^{23} + q^{24} − q^{25}
+ q^{26} − q^{27} + q^{28} − q^{29} + q^{30} − q^{31} + q^{32} − q^{33} + q^{34}).

Also here, if we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q} times the second factor. Taken together, this establishes the claim. □

Lemma 16. Let α and β be positive integers with α ≥ 4 and β ≥ 2. Then the expression

[α]_{q^{2}} [β]_{q^{5}} \begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix} = 1 + q − q^{3} − q^{4} − q^{5} + q^{7} + q^{8}.

If we multiply this expression by [α]_{q^{2}}, then, for α = 4 we obtain

1 + q + q^{2} − q^{5} − q^{9} + q^{12} + q^{13} + q^{14},

for α = 5 we obtain

1 + q + q^{2} − q^{5} + q^{8} − q^{11} + q^{14} + q^{15} + q^{16},

and, for α ≥ 6, we obtain

1 + q + q^{2} − q^{5} + q^{8} + q^{10} + p_{1}(q) + q^{2α−4} + q^{2α−2} − q^{2α+1} + q^{2α+4} + q^{2α+5} + q^{2α+6},

where p_{1}(q) is a polynomial in q with non-negative coefficients of order at least 11 and degree at most 2α − 5. In all cases it is obvious that the product of the result and [β]_{q^{5}}, with β ≥ 2, is a polynomial in q with non-negative coefficients. □

Lemma 17. Let α and β be positive integers with α ≥ 14 and β ≥ 2. Then the expression

[α]_{q} [β]_{q^{5}} \begin{matrix}[14] & q \\ [2] & q & [7] & q \end{matrix} \begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[14] & q \\ [2] & q & [7] & q \end{matrix} \begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix} = 1 − q^{3} − q^{5} + q^{6} + q^{7} + q^{8} − q^{9} − q^{11} + q^{14}.

If we multiply this expression by [α]_{q}, then, for α ≥ 14, we obtain

1 + q + q^{2} − q^{5} + q^{7} + 2q^{8} + q^{9} + q^{10} + p_{2}(q)

+ q^{α+3} + q^{α+4} + 2q^{α+5} + q^{α+6} − q^{α+8} + q^{α+11} + q^{α+12} + q^{α+13},

where p_{2}(q) is a polynomial in q with non-negative coefficients of order at least 11 and degree at most α + 2. It is obvious that the product of the result and [β]_{q^{5}}, with β ≥ 2, is a polynomial in q with non-negative coefficients. □

Lemma 18. Let α and β be positive integers with α ≥ 32 and β ≥ 12. Then the expression

[α]_{q} [β]_{q^{2}} \begin{matrix}[35] & q \\ [5] & q & [7] & q \end{matrix} \begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[35] & q \\ [5] & q & [7] & q \end{matrix} \begin{matrix}[30] & q & [2] & q & [3] & q & [5] & q \\ [6] & q & [10] & q & [15] & q \end{matrix} = 1−q^{2} −q^{3} +q^{5} +q^{6} +q^{7} −q^{8} −2q^{9} +q^{11} +q^{12} −q^{16} +q^{20} +q^{21}

− 2q^{23} − q^{24} + q^{25} + q^{26} + q^{27} − q^{29} − q^{30} + q^{32}.

If we multiply this expression by [α]_{q}, then, for α ≥ 32, we obtain

1 + q − q^{3} − q^{4} + q^{6} + 2q^{7} + q^{8} − q^{9} − q^{10} + q^{12} + q^{13} + q^{14} + q^{15} + q^{20} + 2q^{21} + 2q^{22}

− q^{24} + q^{26} + 2q^{27} + 2q^{28} + q^{29} + p_{3}(q) + q^{α+2} + 2q^{α+4} + 2q^{α+5} + q^{α+6} − q^{α+8}

+ 2q^{α+9} + 2q^{α+10} + q^{α+11} + q^{α+16} + q^{α+17} + q^{α+18} + q^{α+19} − q^{α+21} − q^{α+22} + q^{α+23}

+ 2q^{α+24} + q^{α+25} − q^{α+27} − q^{α+28} + q^{α+30} + q^{α+31},

where p_{3}(q) is a polynomial in q with non-negative coefficients of order at least 30 and degree at most α+1. It is obvious that the product of the result and [β]_{q^{2}}, with β ≥ 12, is a polynomial in q with non-negative coefficients. □

Lemma 19. Let α and β be positive integers with α ≥ 16 and β ≥ 2. Then the expression

[α]_{q^{2}} [β]_{q^{5}} \begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix} = 1 + q − q^{3} − q^{4} − q^{5} − q^{6} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} − q^{14} − q^{15} − q^{16}

− q^{17} − q^{18} + q^{20} + q^{21} + q^{22} + q^{23} + q^{24} − q^{26} − q^{27} − q^{28} − q^{29} + q^{31} + q^{32}.

If we multiply this expression by [α]_{q^{2}}, then, for α ≥ 16, we obtain

1 + q + q^{2} − q^{5} − q^{6} − q^{7} + q^{10} + q^{11} + 2q^{12} + q^{13} + q^{14}

− q^{17} − q^{18} − q^{19} + q^{22} + q^{23} + 2q^{24} + p_{4}(q) + s_{4}(q), where p_{4}(q) is a polynomial in q with non-negative coefficients of order at least 25 and degree at most 2α + 5 and s_{4}(q) completes the above expression to a symmetric polynomial. It is obvious that the product of the result and [β]_{q^{5}}, with β ≥ 2, is a polynomial in q with non-negative coefficients. □

Lemma 20. Let α and β be positive integers with α ≥ 56 and β ≥ 4. Then the expression

[α]_{q} [β]_{q^{2}} \begin{matrix}[35] & q \\ [5] & q & [7] & q \end{matrix} \begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[35] & q \\ [5] & q & [7] & q \end{matrix} \begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix} = 1 − q^{2} − q^{3} + q^{5} + q^{7} − q^{9} + q^{12} − q^{13} + q^{15} − q^{16} − q^{18}

+ q^{19} + q^{20} + q^{22} − 2q^{23} + q^{27} − q^{28} + q^{29} − 2q^{33} + q^{34} + q^{36} + q^{37} − q^{38}

− q^{40} + q^{41} − q^{43} + q^{44} − q^{47} + q^{49} + q^{51} − q^{53} − q^{54} + q^{56}.

If we multiply this expression by [α]_{q}, then, for α ≥ 56, we obtain

1 + q − q^{3} − q^{4} + q^{7} + q^{8} + q^{12} + q^{15} − q^{18} + q^{20} + q^{21} + 2q^{22} + q^{27}

+ q^{29} + q^{30} + q^{31} + q^{32} − q^{33} + q^{36} + 2q^{37} + q^{38} + p_{5}(q) + s_{5}(q),

where p_{5}(q) is a polynomial in q with non-negative coefficients of order at least 39 and degree at most α + 16 and s_{5}(q) completes the above expression to a symmetric polynomial. It is obvious that the product of the result and [β]_{q^{2}}, with β ≥ 4, is a polynomial in q with non-negative coefficients. □

Lemma 21. Let α and β be positive integers with α ≥ 38 and β ≥ 2. Then the expression

[α]_{q} [β]_{q^{5}} \begin{matrix}[14] & q \\ [2] & q & [7] & q \end{matrix} \begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[14] & q \\ [2] & q & [7] & q \end{matrix} \begin{matrix}[60] & q & [2] & q & [3] & q & [5] & q \\ [10] & q & [12] & q & [15] & q \end{matrix} = 1−q^{3} −q^{5} +q^{7} +q^{8} −q^{13} +q^{19} −q^{25} +q^{30} +q^{31} −q^{33} −q^{35} +q^{38}.

If we multiply this expression by [α]_{q}, then, for α ≥ 38, we obtain

1 + q + q^{2} − q^{5} − q^{6} + q^{8} + q^{9} + q^{10} + q^{11} + p_{6}(q) + s_{6}(q),

where p_{6}(q) is a polynomial in q with non-negative coefficients of order at least 12 and degree at most α + 25 and s_{6}(q) completes the above expression to a symmetric polynomial. It is obvious that the product of the result and [β]_{q^{2}}, with β ≥ 2, is a polynomial in q with non-negative coefficients. □

Lemma 22. Let α and β be positive integers with α ≥ 30 and β ≥ 26. Then the expression

[α]_{q} [β]_{q^{3}} \begin{matrix}[126] & q & [3] & q \\ [6] & q & [7] & q & [9] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[126] & q & [3] & q \\ [6] & q & [7] & q & [9] & q \end{matrix} = (1 − q + q^{6} − q^{8} + q^{12} − q^{15} + q^{18} − q^{22} + q^{24} − q^{29} + q^{30})

× (1 − q^{3} + q^{9} − q^{12} + q^{18} − q^{21} + q^{27} − q^{30} + q^{36} − q^{39}
+ q^{42} − q^{48} + q^{51} − q^{57} + q^{60} − q^{66} + q^{69} − q^{75} + q^{78}).

If we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{3}} times the second factor. Taken together, this establishes the claim. □

Lemma 23. Let α and β be positive integers with α ≥ 66 and β ≥ 54. Then the expression

[α]_{q} [β]_{q^{3}} \begin{matrix}[252] & q & [3] & q \\ [7] & q & [9] & q & [12] & q \end{matrix}

is a polynomial in q with non-negative integer coefficients.

Proof. We have

\begin{matrix}[252] & q & [3] & q \\ [7] & q & [9] & q & [12] & q \end{matrix}

= (1 − q + q^{7} − q^{8} + q^{12} − q^{13} + q^{14} − q^{15} + q^{19} − q^{20} + q^{21} − q^{22} + q^{24} − q^{25}

+ q^{26} − q^{27} + q^{28} − q^{29} + q^{31} − q^{32} + q^{33} − q^{34} + q^{35} − q^{37} + q^{38}
− q^{39} + q^{40} − q^{41} + q^{42} − q^{44} + q^{45} − q^{46} + q^{47} − q^{51} + q^{52} − q^{53}
+ q^{54} − q^{58} + q^{59} − q^{65} + q^{66})
× (1 − q^{3} + q^{9} − q^{12} + q^{18} − q^{21} + q^{27} − q^{30} + q^{36} − q^{39} + q^{45} − q^{48}
+ q^{54} − q^{57} + q^{63} − q^{66} + q^{72} − q^{75} + q^{81} − q^{87} + q^{90} − q^{96} + q^{99}
− q^{105} + q^{108} − q^{114} + q^{117} − q^{123} + q^{126} − q^{132} + q^{135} − q^{141}
+ q^{144} − q^{150} + q^{153} − q^{159} + q^{162}).

If we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{3}} times the second factor. Taken together, this establishes the claim. □

Lemma 24. Let α and β be positive integers with α ≥ 54 and β ≥ 34. Then the expression

[α]_{q} [β]_{q^{2}} \begin{matrix}[140] & q & [2] & q \\ [4] & q & [7] & q & [10] & q \end{matrix}

Proof. We have

\begin{matrix}[140] & q & [2] & q \\ [4] & q & [7] & q & [10] & q \end{matrix}

= (1 − q + q^{7} − q^{8} + q^{10} − q^{11} + q^{14} − q^{15} + q^{17} − q^{18} + q^{20} − q^{22} + q^{24} − q^{25}
+ q^{27} − q^{29} + q^{30} − q^{32} + q^{34} − q^{36} + q^{37} − q^{39} + q^{40} − q^{43} + q^{44} − q^{46}
+ q^{47} − q^{53} + q^{54})
× (1 − q^{2} + q^{4} − q^{6} + q^{8} − q^{10} + q^{12} − q^{14} + q^{16} − q^{18} + q^{20} − q^{22} + q^{24}
− q^{26} + q^{28} − q^{30} + q^{32} − q^{34} + q^{36} − q^{38} + q^{40} − q^{42} + q^{44} − q^{46}
+ q^{48} − q^{50} + q^{52} − q^{54} + q^{56} − q^{58} + q^{60} − q^{62} + q^{64} − q^{66} + q^{68}),

If we apply the same idea as in the proof of Corollary 6, then we see that [α]_{q} times the first factor on the the right-hand side of the above equation is a polynomial in q with non-negative integer coefficients, as is [β]_{q^{2}} times the second factor. Taken together, this establishes the claim. □

We are now ready for the proof of the main result of this section.

Theorem 25. For all irreducible well-generated complex reflection groups and positive integers m, the q-Fuß–Catalan number Cat^{m}(W; q) is a polynomial in q with nonnegative integer coefficients.

Proof. First, let W = A_{n}. In this case, the degrees are 2, 3, …, n + 1, and hence

Cat^{m}(A_{n}; q) = \frac{1}{[(m + 1)n + 1]_{q}} [\begin{matrix}(m + 1)n + 1 \\ n \end{matrix}]_{q} ,

which, by Proposition 4, is a polynomial in q with non-negative integer coefficients. Next, let W = G(d, 1, n). In this case, the degrees are d, 2d, …, nd, and hence

Cat^{m}(G(d, 1, n); q) = [\begin{matrix}(m + 1)n \\ n \end{matrix}]_{q^{d}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. Now, let W = G(e, e, n). In this case, the degrees are e, 2e, …, (n − 1)e, n, and hence

Cat^{m}(G(e, e, n); q) = \frac{[m(n − 1)e + n]_{q}}{[n]_{q}} ∏_{i=1}^{n−1} \frac{[m(n − 1)e + ie]_{q}}{[ie]_{q}}

= [\begin{matrix}(m + 1)(n − 1) \\ n − 1 \end{matrix}]_{q^{e}} + q^{n}[e]_{q^{n}} [\begin{matrix}(m + 1)(n − 1) \\ n \end{matrix}]_{q^{e}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. It remains to verify the claim for the exceptional groups. For W = G_{4}, the degrees are 4, 6, and hence

Cat^{m}(G_{4}; q) = \frac{[6m + 4]_{q} [6m + 6]_{q}}{[4]_{q} [6]_{q}} = \frac{1}{[3m + 4]_{q^{2}}} [\begin{matrix}3m + 4 \\ 3 \end{matrix}]_{q^{2}} ,

For W = G_{5}, the degrees are 6, 12, and hence

Cat^{m}(G_{5}; q) = \frac{[12m + 6]_{q} [12m + 12]_{q}}{[6]_{q} [12]_{q}} = [\begin{matrix}2m + 2 \\ 2 \end{matrix}]_{q^{6}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{6}, the degrees are 4, 12, and hence

Cat^{m}(G_{6}; q) = \frac{[12m + 4]_{q} [12m + 12]_{q}}{[4]_{q} [12]_{q}} = [3m + 1]_{q^{4}} [m + 1]_{q^{12}},

which is manifestly a polynomial in q with non-negative integer coefficients. For W = G_{8}, the degrees are 8, 12, and hence

Cat^{m}(G_{8}; q) = \frac{[12m + 8]_{q} [12m + 12]_{q}}{[8]_{q} [12]_{q}} = \frac{1}{[3m + 4]_{q^{4}}} [\begin{matrix}3m + 4 \\ 3 \end{matrix}]_{q^{4}} ,

which, by Proposition 4, is a polynomial in q with non-negative integer coefficients. For W = G_{9}, the degrees are 8, 24, and hence

Cat^{m}(G_{9}; q) = \frac{[24m + 8]_{q} [24m + 24]_{q}}{[8]_{q} [24]_{q}} = [3m + 1]_{q^{8}} [m + 1]_{q^{24}},

which is manifestly a polynomial in q with non-negative integer coefficients. For W = G_{10}, the degrees are 12, 24, and hence

Cat^{m}(G_{10}; q) = \frac{[24m + 12]_{q} [24m + 24]_{q}}{[12]_{q} [24]_{q}} = [\begin{matrix}2m + 2 \\ 2 \end{matrix}]_{q^{12}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{14}, the degrees are 6, 24, and hence

Cat^{m}(G_{14}; q) = \frac{[24m + 6]_{q} [24m + 24]_{q}}{[6]_{q} [24]_{q}} = [4m + 1]_{q^{6}} [m + 1]_{q^{24}},

which is manifestly a polynomial in q with non-negative integer coefficients. For W = G_{16}, the degrees are 20, 30, and hence

Cat^{m}(G_{16}; q) = \frac{[30m + 20]_{q} [30m + 30]_{q}}{[20]_{q} [30]_{q}} = \frac{1}{[3m + 4]_{q^{10}}} [\begin{matrix}3m + 4 \\ 3 \end{matrix}]_{q^{10}} ,

which, by Proposition 4, is a polynomial in q with non-negative integer coefficients. For W = G_{17}, the degrees are 20, 60, and hence

Cat^{m}(G_{17}; q) = \frac{[60m + 20]_{q} [60m + 60]_{q}}{[20]_{q} [60]_{q}} = [3m + 1]_{q^{20}} [m + 1]_{q^{60}},

which is manifestly a polynomial in q with non-negative integer coefficients. For W = G_{18}, the degrees are 30, 60, and hence

Cat^{m}(G_{18}; q) = \frac{[60m + 30]_{q} [60m + 60]_{q}}{[30]_{q} [60]_{q}} = [\begin{matrix}2m + 2 \\ 2 \end{matrix}]_{q^{30}} ,

For W = G_{20}, the degrees are 12, 30, and hence

Cat^{m}(G_{20}; q) = \frac{[30m + 12]_{q} [30m + 30]_{q}}{[12]_{q} [30]_{q}}

= \begin{cases}[\frac{5m+2}{2}]_{q^{12}} [m + 1]_{q^{30}} , & if m is even, \\ [5m + 2]_{q^{6}} [\frac{m+1}{2}]_{q^{60}} \frac{[10]_{q^{6}}}{[2]_{q^{6}}[5]_{q^{6}}} , & if m is odd, \end{cases}

which, by Corollary 6, are polynomials in q with non-negative integer coefficients in both cases.

For W = G_{21}, the degrees are 12, 60, and hence

Cat^{m}(G_{21}; q) = \frac{[60m + 12]_{q} [60m + 60]_{q}}{[12]_{q} [60]_{q}} = [5m + 1]_{q^{12}} [m + 1]_{q^{60}},

which is manifestly a polynomial in q with non-negative integer coefficients. For W = G_{23} = H_{3}, the degrees are 2, 6, 10, and hence

Cat^{m}(H_{3}; q) = \frac{[10m + 2]_{q} [10m + 6]_{q} [10m + 10]_{q}}{[2]_{q} [6]_{q} [10]_{q}}

= \begin{cases}[5m + 2]_{q^{2}} [\frac{5m+3}{3}]_{q^{6}} [m + 1]_{q^{10}} , & if m ≡ 0 (mod 3), \\ [\frac{5m+1}{3}]_{q^{6}} [5m + 3]_{q^{2}} [m + 1]_{q^{10}} , & if m ≡ 1 (mod 3), \\ [5m + 2]_{q^{2}} [5m + 3]_{q^{2}} [\frac{m+1}{3}]_{q^{30}} \frac{[15]_{q^{2}}}{[3]_{q^{2}}[5]_{q^{2}}} , & if m ≡ 2 (mod 3), \end{cases}

which, by Corollary 6, are polynomials in q with non-negative integer coefficients in all cases.

For W = G_{24}, the degrees are 4, 6, 14, and hence

Cat^{m}(G_{24}; q) = \frac{[14m + 4]_{q} [14m + 6]_{q} [14m + 14]_{q}}{[4]_{q} [6]_{q} [14]_{q}} .

We have

Cat^{m}(G_{24}; q) = \begin{cases}[\frac{7m}{2} + 1]_{q^{4}} [\frac{14m}{6} + 1]_{q^{6}} [m + 1]_{q^{14}} , & if m ≡ 0 (mod 6), \\ [\frac{7m+2}{3}]_{q^{6}} [\frac{7m+3}{2}]_{q^{4}} [m + 1]_{q^{14}} , & if m ≡ 1 (mod 6), \\ [\frac{7m}{2} + 1]_{q^{4}} [7m + 3]_{q^{2}} [\frac{m+1}{3}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}}[7]_{q^{2}}} , & if m ≡ 2 (mod 6), \\ [7m + 2]_{q^{2}} [\frac{7m}{3} + 1]_{q^{6}} [\frac{m+1}{2}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}}[7]_{q^{2}}} , & if m ≡ 3 (mod 6), \\ [\frac{7m+2}{6}]_{q^{12}} \frac{[6]_{q^{2}}}{[2]_{q^{2}}[3]_{q^{2}}} [7m + 3]_{q^{2}} [m + 1]_{q^{14}} , & if m ≡ 4 (mod 6), \\ [7m + 2]_{q^{2}} [\frac{7m+3}{2}]_{q^{4}} [\frac{m+1}{3}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}}[7]_{q^{2}}} , & if m ≡ 5 (mod 6), \end{cases}

which, by Corollary 6, are polynomials in q with non-negative integer coefficients in all cases.

For W = G_{25}, the degrees are 6, 9, 12, and hence

Cat^{m}(G_{25}; q) = \frac{[12m + 6]_{q} [12m + 9]_{q} [12m + 12]_{q}}{[6]_{q} [9]_{q} [12]_{q}} = \frac{1}{[4m + 5]_{q^{3}}} [\begin{matrix}4m + 5 \\ 4 \end{matrix}]_{q^{3}} ,

For W = G_{26}, the degrees are 6, 12, 18, and hence

Cat^{m}(G_{26}; q) = \frac{[18m + 6]_{q} [18m + 12]_{q} [18m + 18]_{q}}{[6]_{q} [12]_{q} [18]_{q}} = [\begin{matrix}3m + 3 \\ 3 \end{matrix}]_{q^{6}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{27}, the degrees are 6, 12, 30, and hence

Cat^{m}(G_{27}; q) = \frac{[30m + 6]_{q} [30m + 12]_{q} [30m + 30]_{q}}{[6]_{q} [12]_{q} [30]_{q}} = [m + 1]_{q^{30}} [\begin{matrix}5m + 2 \\ 2 \end{matrix}]_{q^{6}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{28} = F_{4}, the degrees are 2, 6, 8, 12, and hence

Cat^{m}(F_{4}; q) = \frac{[12m + 2]_{q} [12m + 6]_{q} [12m + 8]_{q} [12m + 12]_{q}}{[2]_{q} [6]_{q} [8]_{q} [12]_{q}}

= \begin{cases}[6m + 1]_{q^{2}} [2m + 1]_{q^{6}} [\frac{3}{2}m + 1]_{q^{8}} [m + 1]_{q^{12}} , & if m is even, \\ [6m + 1]_{q^{2}} [2m + 1]_{q^{6}} [\frac{m+1}{2}]_{q^{24}} [3m + 2]_{q^{4}} \frac{[6]_{q^{4}}}{[2]_{q^{4}} [3]_{q^{4}}} , & if m is odd, \end{cases}

which in both cases is a polynomial in q with non-negative integer coefficients; in the second case this is due to Corollary 6.

For W = G_{29}, the degrees are 4, 8, 12, 20, and hence

Cat^{m}(G_{29}; q) = \frac{[20m + 4]_{q} [20m + 8]_{q} [20m + 12]_{q} [20m + 20]_{q}}{[4]_{q} [8]_{q} [12]_{q} [20]_{q}}

= [m + 1]_{q^{20}} [\begin{matrix}5m + 3 \\ 3 \end{matrix}]_{q^{4}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{30} = H_{4}, the degrees are 2, 12, 20, 30, and hence

Cat^{m}(H_{4}; q) = \frac{[30m + 2]_{q} [30m + 12]_{q} [30m + 20]_{q} [30m + 30]_{q}}{[2]_{q} [12]_{q} [20]_{q} [30]_{q}} .

If m is even, then we have

Cat^{m}(H_{4}; q) = [15m + 1]_{q^{2}} [\frac{5}{2}m + 1]_{q^{12}} [\frac{3}{2}m + 1]_{q^{20}} [m + 1]_{q^{30}} ,

which is manifestly a polynomial in q with non-negative integer coefficients. On the other hand, if m is odd, then we may write

Cat^{m}(H_{4}; q) = [\frac{15m+1}{2}]_{q^{4}} [5m + 2]_{q^{6}} [3m + 2]_{q^{10}} [\frac{m+1}{2}]_{q^{60}} \frac{[30]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[6]_{q^{6}} [10]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemma 16, is a polynomial in q with non-negative integer coefficients. For W = G_{32}, the degrees are 12, 18, 24, 30, and hence

Cat^{m}(G_{32}; q) = \frac{[30m + 12]_{q} [30m + 18]_{q} [30m + 24]_{q} [30m + 30]_{q}}{[12]_{q} [18]_{q} [24]_{q} [30]_{q}}

= \frac{1}{[5m + 6]_{q^{6}}} [\begin{matrix}5m + 6 \\ 5 \end{matrix}]_{q^{6}} ,

For W = G_{33}, the degrees are 4, 6, 10, 12, 18, and hence

Cat^{m}(G_{33}; q) = \frac{[18m + 4]_{q} [18m + 6]_{q} [18m + 10]_{q} [18m + 12]_{q} [18m + 18]_{q}}{[4]_{q} [6]_{q} [10]_{q} [12]_{q} [18]_{q}} .

If m ≡ 0 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9}{2}m + 1]_{q^{4}} [3m + 1]_{q^{6}} [\frac{9}{5}m + 1]_{q^{10}} [\frac{3}{2}m + 1]_{q^{12}} [m + 1]_{q^{18}} ,

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 1 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{3m+1}{2}]_{q^{12}} [\frac{9m+5}{2}]_{q^{4}} [\frac{3m+2}{5}]_{q^{30}} [m + 1]_{q^{18}} [9m + 2]_{q^{2}} \frac{[15]_{q^{2}}}{[3]_{q^{2}} [5]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 2 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9m+2}{10}]_{q^{20}} [3m + 1]_{q^{6}} [\frac{3}{2}m + 1]_{q^{12}} [m + 1]_{q^{18}} [9m + 5]_{q^{2}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 3 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{3m+1}{10}]_{q^{60}} [\frac{9m+5}{2}]_{q^{4}} [3m + 2]_{q^{6}} [m + 1]_{q^{18}} [9m + 2]_{q^{2}} \frac{[30]_{q^{2}}}{[5]_{q^{2}} [6]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 4 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9}{2}m + 1]_{q^{4}} [3m + 1]_{q^{6}} [\frac{3}{2}m + 1]_{q^{12}} [\frac{m+1}{5}]_{q^{90}} [9m + 5]_{q^{2}} \frac{[45]_{q^{2}}}{[5]_{q^{2}} [9]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 5 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{3m+1}{2}]_{q^{12}} [\frac{9m+5}{10}]_{q^{20}} [3m + 2]_{q^{6}} [m + 1]_{q^{18}} [9m + 2]_{q^{2}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 6 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9}{2}m + 1]_{q^{4}} [3m + 1]_{q^{6}} [\frac{3m+2}{10}]_{q^{60}} [m + 1]_{q^{18}} [9m + 5]_{q^{2}} \frac{[30]_{q^{2}}}{[5]_{q^{2}} [6]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 7 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9m+2}{5}]_{q^{10}} [\frac{3m+1}{2}]_{q^{12}} [\frac{9m+5}{2}]_{q^{4}} [3m + 2]_{q^{6}} [m + 1]_{q^{18}} ,

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 8 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{9}{2}m + 1]_{q^{4}} [\frac{3m+1}{5}]_{q^{30}} [\frac{3}{2}m + 1]_{q^{12}} [m + 1]_{q^{18}} [9m + 5]_{q^{2}} \frac{[15]_{q^{2}}}{[3]_{q^{2}} [5]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. Finally, if m ≡ 9 (mod 10), then we have

Cat^{m}(G_{33}; q) = [\frac{3m+1}{2}]_{q^{12}} [\frac{9m+5}{2}]_{q^{4}} [3m + 2]_{q^{6}} [\frac{m+1}{5}]_{q^{90}} [9m + 2]_{q^{2}} \frac{[45]_{q^{2}}}{[5]_{q^{2}} [9]_{q^{2}}} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. For W = G_{34}, the degrees are 6, 12, 18, 24, 30, 42, and hence

Cat^{m}(G_{34}; q) = \frac{[42m + 6]_{q} [42m + 12]_{q} [42m + 18]_{q} [42m + 24]_{q} [42m + 30]_{q} [42m + 42]_{q}}{[6]_{q} [12]_{q} [18]_{q} [24]_{q} [30]_{q} [42]_{q}}

= [m + 1]_{q^{42}} [\begin{matrix}7m + 5 \\ 5 \end{matrix}]_{q^{6}} ,

which, by Proposition 3, is a polynomial in q with non-negative integer coefficients. For W = G_{35} = E_{6}, the degrees are 2, 5, 6, 8, 9, 12, and hence

Cat^{m}(E_{6}; q) = \frac{[12m + 2]_{q} [12m + 5]_{q} [12m + 6]_{q} [12m + 8]_{q} [12m + 9]_{q} [12m + 12]_{q}}{[2]_{q} [5]_{q} [6]_{q} [8]_{q} [9]_{q} [12]_{q}} .

If m ≡ 0 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}[\frac{4m+3}{3}]_{q^{9}}[m + 1]_{q^{12}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 1 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{2m+1}{3}]_{q^{18}}[4m + 3]_{q^{3}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [\frac{3m+2}{5}]_{q^{20}}[12m + 5]_{q} \frac{[20]_{q}}{[4]_{q} [5]_{q}} [\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} .

If one decomposes [6m+1]_{q^{2}} as [3m+1]_{q^{4}} +q^{2}[3m]_{q^{4}}, then one sees that, by Corollary 6, the above expression is a polynomial in q with non-negative integer coefficients. If m ≡ 2 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{5}]_{q^{30}} \begin{matrix}[30] & q \\ [5]_{q} & [6]_{q} \end{matrix}

× [\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[\frac{m+1}{3}]_{q^{36}} \begin{matrix}[12] & q_{3} \\ [3]_{q^{3}} & [4]_{q^{3}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 3 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}[3m + 2]_{q^{4}}

× [\frac{4m+3}{15}]_{q^{45}} \begin{matrix}[45] & q \\ [5]_{q} & [9]_{q} \end{matrix} [\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 4 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{12m+2}{5}]_{q^{5}}[\frac{12m+5}{2}]_{q^{2}}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 5 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[2m + 1]_{q^{6}}

× [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{6}]_{q^{72}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix} ,

which, by Lemma 7, is a polynomial in q with non-negative integer coefficients. If m ≡ 6 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}[\frac{3m+2}{10}]_{q^{40}} \begin{matrix}[40] & q \\ [5]_{q} & [8]_{q} \end{matrix} [\frac{4m+3}{3}]_{q^{9}}[m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 7 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{6m+1}{2}]_{q^{4}}[12m + 5]_{q}[\frac{2m+1}{15}]_{q^{90}}

× \begin{matrix}[90] & q & [3] & q & [4] & q \\ [5] & q & [6] & q & [9] & q \end{matrix} [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{2}]_{q^{24}} \frac{[6]_{q^{4}}}{[2]_{q^{4}} [3]_{q^{4}}} ,

which, by Corollary 6 and Lemma 9, is a polynomial in q with non-negative integer coefficients. If m ≡ 8 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}

× [\frac{4m+3}{5}]_{q^{15}} \frac{[15]_{q}}{[3]_{q} [5]_{q}} [\frac{m+1}{3}]_{q^{36}} \begin{matrix}[12] & q_{3} \\ [3]_{q^{3}} & [4]_{q^{3}} \end{matrix} ,

which, by Lemma 11, is a polynomial in q with non-negative integer coefficients. If m ≡ 9 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{12m+2}{5}]_{q^{5}}[\frac{12m+5}{2}]_{q^{2}}[2m + 1]_{q^{6}}[3m + 2]_{q^{4}}[\frac{4m+3}{3}]_{q^{9}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 10 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix} [\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 11 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}

× [\frac{3m+2}{5}]_{q^{20}} \frac{[20]_{q}}{[4]_{q} [5]_{q}} [4m + 3]_{q^{3}}[\frac{m+1}{6}]_{q^{72}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix} .

If one decomposes [6m+1]_{q^{2}} as [3m+1]_{q^{4}} +q^{2}[3m]_{q^{4}}, then one sees that, by Corollary 6 and Lemma 7, this is a polynomial in q with non-negative integer coefficients. If m ≡ 12 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{5}]_{q^{30}} \begin{matrix}[30] & q \\ [5]_{q} & [6]_{q} \end{matrix} [\frac{3m+2}{2}]_{q^{8}}[\frac{4m+3}{3}]_{q^{9}}[m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 13 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [3m + 2]_{q^{4}}[\frac{4m+3}{5}]_{q^{15}} \frac{[15]_{q}}{[3]_{q} [5]_{q}} [\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Lemma 12, is a polynomial in q with non-negative integer coefficients. If m ≡ 14 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{12m+2}{5}]_{q^{5}}[\frac{12m+5}{2}]_{q^{2}}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[\frac{m+1}{3}]_{q^{36}} \begin{matrix}[12] & q_{3} \\ [3]_{q^{3}} & [4]_{q^{3}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 15 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[2m + 1]_{q^{6}}[3m + 2]_{q^{4}}[\frac{4m+3}{3}]_{q^{9}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 16 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [\frac{3m+2}{10}]_{q^{40}} \begin{matrix}[40] & q \\ [5]_{q} & [8]_{q} \end{matrix} [4m + 3]_{q^{3}}[m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 17 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{5}]_{q^{30}} \frac{[30]_{q}}{[5]_{q} [6]_{q}}

× [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{6}]_{q^{72}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix} ,

which, by Corollary 6 and Lemma 7, is a polynomial in q with non-negative integer coefficients. If m ≡ 18 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}[\frac{4m+3}{15}]_{q^{45}} \begin{matrix}[45] & q \\ [5]_{q} & [9]_{q} \end{matrix} [m + 1]_{q^{12}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 19 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{12m+2}{5}]_{q^{5}}[\frac{12m+5}{2}]_{q^{2}}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 20 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[\frac{m+1}{3}]_{q^{36}} \begin{matrix}[12] & q_{3} \\ [3]_{q^{3}} & [4]_{q^{3}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 21 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}

× [\frac{3m+2}{5}]_{q^{20}} \frac{[20]_{q}}{[4]_{q} [5]_{q}} [\frac{4m+3}{3}]_{q^{9}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 22 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{15}]_{q^{90}} \begin{matrix}[90]_{q} & [3]_{q} \\ [5]_{q} & [6]_{q} & [9]_{q} \end{matrix}

× [\frac{3m+2}{2}]_{q^{8}}[4m + 3]_{q^{3}}[m + 1]_{q^{12}},

which, by Lemma 10, is a polynomial in q with non-negative integer coefficients. If m ≡ 23 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}

× [3m + 2]_{q^{4}}[\frac{4m+3}{5}]_{q^{15}} \frac{[15]_{q}}{[3]_{q} [5]_{q}} [\frac{m+1}{6}]_{q^{72}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix} ,

which, by Lemma 8, is a polynomial in q with non-negative integer coefficients. If m ≡ 24 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{12m+2}{5}]_{q^{5}}[\frac{12m+5}{2}]_{q^{2}}[2m + 1]_{q^{6}}[\frac{3m+2}{2}]_{q^{8}}[\frac{4m+3}{3}]_{q^{9}}[m + 1]_{q^{12}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 25 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[\frac{12m+5}{5}]_{q^{5}}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 26 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[2m + 1]_{q^{6}}

× [\frac{3m+2}{10}]_{q^{40}} \begin{matrix}[40] & q \\ [5]_{q} & [8]_{q} \end{matrix} [4m + 3]_{q^{3}}[\frac{m+1}{3}]_{q^{36}} \begin{matrix}[12] & q_{3} \\ [3]_{q^{3}} & [4]_{q^{3}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 27 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{5}]_{q^{30}} \begin{matrix}[30] & q \\ [5]_{q} & [6]_{q} \end{matrix}

× [3m + 2]_{q^{4}}[\frac{4m+3}{3}]_{q^{9}}[\frac{m+1}{2}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 28 (mod 30), then we have

Cat^{m}(E_{6}; q) = [6m + 1]_{q^{2}}[12m + 5]_{q}[\frac{2m+1}{3}]_{q^{18}} \begin{matrix}[6] & q_{3} \\ [2]_{q^{3}} & [3]_{q^{3}} \end{matrix}

× [\frac{3m+2}{2}]_{q^{8}}[\frac{4m+3}{5}]_{q^{15}} \frac{[15]_{q}}{[3]_{q} [5]_{q}} [m + 1]_{q^{12}},

which, by Lemma 12, is a polynomial in q with non-negative integer coefficients. If m ≡ 29 (mod 30), then we have

Cat^{m}(E_{6}; q) = [\frac{6m+1}{5}]_{q^{10}} \frac{[10]_{q}}{[2]_{q} [5]_{q}} [12m + 5]_{q}[2m + 1]_{q^{6}}

× [3m + 2]_{q^{4}}[4m + 3]_{q^{3}}[\frac{m+1}{6}]_{q^{72}} \begin{matrix}[72] & q & [3] & q & [4] & q \\ [8] & q & [9] & q & [12] & q \end{matrix} ,

which, by Corollary 6 and Lemma 7, is a polynomial in q with non-negative integer coefficients.

For W = G_{36} = E_{7}, the degrees are 2, 6, 8, 10, 12, 14, 18, and hence

Cat^{m}(E_{7}; q) = \frac{[18m + 2]_{q} [18m + 6]_{q} [18m + 8]_{q} [18m + 10]_{q}}{[2]_{q} [6]_{q} [8]_{q} [10]_{q}}

× \frac{[18m + 12]_{q} [18m + 14]_{q} [18m + 18]_{q}}{[12]_{q} [14]_{q} [18]_{q}} .

If m ≡ 0 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 1 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 2 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 3 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 4 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 5 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{7}]_{q^{14}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 6 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 7 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 8 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}

× [\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 9 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{5}]_{q^{10}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 10 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{5}]_{q^{10}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 11 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{4}]_{q^{8}}

× [\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 12 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 13 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 14 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}}.

If one decomposes [9m+5]_{q^{2}} as [\frac{9m}{2}+3]_{q^{4}}+q^{2}[\frac{9m}{2}+2]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 15 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [\frac{9m+5}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 16 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 17 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 4]_{q^{2}}[\frac{9m+5}{2}]_{q^{4}}

× [3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 18 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \frac{[15]_{q^{2}}}{[3]_{q^{2}} [5]_{q^{2}}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{28}]_{q^{168}} \frac{[84]_{q^{2}} [2]_{q^{2}}}{[4]_{q^{2}} [6]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6 and Lemma 13, is a polynomial in q with non-negative integer coefficients. If m ≡ 19 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}}, which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 20 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{2}]_{q^{12}}

× [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 21 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 4]_{q^{2}}[\frac{9m+5}{2}]_{q^{4}}

× [3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 22 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 23 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{35}]_{q^{210}} \begin{matrix}[105] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}[9m + 5]_{q^{2}}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[\frac{m+1}{2}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} ,

which, by Corollary 6 and Lemma 14, is a polynomial in q with non-negative integer coefficients. If m ≡ 24 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}[\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 25 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}

× [9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 26 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 27 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 28 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 29 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 30 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{5}]_{q^{10}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 31 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}}, which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 32 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 33 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [\frac{9m+4}{7}]_{q^{14}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 34 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} .

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 35 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 36 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 37 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 38 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 39 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 40 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix}

× [\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 41 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 42 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 43 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 44 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 45 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}}, which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 46 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{28}]_{q^{168}} \frac{[84]_{q^{2}} [2]_{q^{2}}}{[4]_{q^{2}} [6]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Lemma 13, is a polynomial in q with non-negative integer coefficients. If m ≡ 47 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{7}]_{q^{14}}[9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 48 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 49 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+4}{5}]_{q^{10}}[\frac{9m+5}{2}]_{q^{4}}

× [3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 50 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}

× [\frac{9m+5}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 51 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 52 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}}, which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 53 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 54 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{70}]_{q^{140}} \begin{matrix}[70] & q_{2} \\ [2]_{q^{2}} & [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2} +4]_{q^{4}}+q^{2}[\frac{9m}{2} +3]_{q^{4}}, then one sees that, by Corollary 6 and Lemma 15, this is a polynomial in q with non-negative integer coefficients. If m ≡ 55 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}

× [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 56 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 57 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{2}]_{q^{4}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [9m + 4]_{q^{2}}[\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 58 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{35}]_{q^{210}} \begin{matrix}[105] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6 and Lemma 14, is a polynomial in q with non-negative integer coefficients. If m ≡ 59 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}}, which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 60 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{5}]_{q^{10}}

× [\frac{3m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 61 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{7}]_{q^{14}}[9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 62 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 63 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 64 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 65 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 66 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 67 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 68 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [\frac{9m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 69 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 70 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 71 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 72 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}}, which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 73 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 74 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{28}]_{q^{168}} \frac{[84]_{q^{2}} [2]_{q^{2}}}{[4]_{q^{2}} [6]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2} +4]_{q^{4}}+q^{2}[\frac{9m}{2} +3]_{q^{4}}, then one sees that, by Corollary 6 and Lemma 13, this is a polynomial in q with non-negative integer coefficients. If m ≡ 75 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{7}]_{q^{14}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 76 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 77 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{2}]_{q^{4}}[3m + 2]_{q^{6}}[\frac{9m+7}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 78 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 79 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{5}]_{q^{10}}[9m + 5]_{q^{2}}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 80 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 81 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 82 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}}.

If one decomposes [9m+5]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+2]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 83 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}[9m + 5]_{q}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 84 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 85 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}

× [9m + 4]_{q^{2}}[\frac{9m+5}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 86 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 87 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 88 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 89 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 90 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 91 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 92 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}}, which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 93 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{35}]_{q^{210}} \begin{matrix}[105] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[\frac{m+1}{2}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} ,

which, by Corollary 6 and Lemma 14, is a polynomial in q with non-negative integer coefficients. If m ≡ 94 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}[\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 95 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}

× [\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 96 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 97 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 4]_{q^{2}}[\frac{9m+5}{2}]_{q^{4}}

× [3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 98 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 99 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 100 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}

× [\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 101 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 102 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{28}]_{q^{168}} \frac{[84]_{q^{2}} [2]_{q^{2}}}{[4]_{q^{2}} [6]_{q^{2}} [7]_{q^{2}}} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Lemma 13, is a polynomial in q with non-negative integer coefficients. If m ≡ 103 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [\frac{9m+4}{7}]_{q^{14}}[9m + 5]_{q^{2}}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 104 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{20}]_{q^{40}} \begin{matrix}[20] & q_{2} \\ [4]_{q^{2}} & [5]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 105 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [9m + 4]_{q^{2}}[\frac{9m+5}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} [3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}}.

If one decomposes [9m+1]_{q^{2}} as [\frac{9m+1}{2}]_{q^{4}} +q^{2}[\frac{9m+1}{2}]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 106 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix}

× [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 107 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 108 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 109 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 110 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}}

× [\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 111 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}

× [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 112 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 113 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 114 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 115 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}

× [\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[9m + 7]_{q}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 116 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 117 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+4}{7}]_{q^{14}}[\frac{9m+5}{2}]_{q^{4}}

× [3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}}, which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 118 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 119 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 120 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}

× [\frac{9m+5}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 121 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 122 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 123 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 124 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[\frac{m+1}{5}]_{q^{90}} \begin{matrix}[45] & q_{2} \\ [5]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 125 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{2}]_{q^{12}}

× [9m + 4]_{q^{2}}[\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 126 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[9m + 5]_{q^{2}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [\frac{9m+7}{7}]_{q^{14}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 127 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[9m + 4]_{q^{2}}[\frac{9m+5}{7}]_{q^{14}}[3m + 2]_{q^{6}}[\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 128 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{35}]_{q^{210}} \begin{matrix}[105] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Lemma 14, is a polynomial in q with non-negative integer coefficients. If m ≡ 129 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{7}]_{q^{14}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{4}]_{q^{8}}[m + 1]_{q^{18}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 130 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}

× [\frac{9m+4}{2}]_{q^{4}}[\frac{9m+5}{5}]_{q^{10}}[\frac{3m+2}{28}]_{q^{168}} \frac{[84]_{q^{2}} [2]_{q^{2}}}{[4]_{q^{2}} [6]_{q^{2}} [7]_{q^{2}}} [9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Lemma 13, is a polynomial in q with non-negative integer coefficients. If m ≡ 131 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{5}]_{q^{10}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{7}]_{q^{14}}[\frac{9m+5}{4}]_{q^{8}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}}, which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 132 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}

× [9m + 5]_{q^{2}}[\frac{3m+2}{2}]_{q^{12}}[\frac{9m+7}{5}]_{q^{10}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 133 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{10}]_{q^{60}} \begin{matrix}[30] & q_{2} \\ [5]_{q^{2}} & [6]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [9m + 5]_{q^{2}}[3m + 2]_{q^{6}}[\frac{9m+7}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 134 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[3m + 1]_{q^{6}}[\frac{9m+4}{10}]_{q^{20}} \frac{[10]_{q^{2}}}{[2]_{q^{2}} [5]_{q^{2}}}

× [\frac{9m+5}{7}]_{q^{14}}[\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 135 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [9m + 4]_{q^{2}}

× [\frac{9m+5}{5}]_{q^{10}}[3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 136 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{35}]_{q^{70}} \begin{matrix}[35] & q_{2} \\ [5]_{q^{2}} & [7]_{q^{2}} \end{matrix} [3m + 1]_{q^{6}}[\frac{9m+4}{4}]_{q^{8}}[9m + 5]_{q^{2}}

× [\frac{3m+2}{2}]_{q^{12}}[9m + 7]_{q^{2}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 137 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 4]_{q^{2}}[\frac{9m+5}{2}]_{q^{4}}

× [\frac{3m+2}{7}]_{q^{42}} \begin{matrix}[21] & q_{2} \\ [3]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{9m+7}{5}]_{q^{10}}[m + 1]_{q^{18}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 138 (mod 140), then we have

Cat^{m}(E_{7}; q) = [9m + 1]_{q^{2}}[\frac{3m+1}{5}]_{q^{30}} \begin{matrix}[15] & q_{2} \\ [3]_{q^{2}} & [5]_{q^{2}} \end{matrix} [\frac{9m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [9m + 5]_{q^{2}}

× [\frac{3m+2}{4}]_{q^{24}} \begin{matrix}[6] & q_{4} \\ [2]_{q^{4}} & [3]_{q^{4}} \end{matrix} [9m + 7]_{q^{2}}[m + 1]_{q^{18}}.

If one decomposes [9m+7]_{q^{2}} as [\frac{9m}{2}+4]_{q^{4}}+q^{2}[\frac{9m}{2}+3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 139 (mod 140), then we have

Cat^{m}(E_{7}; q) = [\frac{9m+1}{4}]_{q^{8}}[\frac{3m+1}{2}]_{q^{12}}[\frac{9m+4}{5}]_{q^{10}}[9m + 5]_{q^{2}}

× [3m + 2]_{q^{6}}[9m + 7]_{q^{2}}[\frac{m+1}{7}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. For W = G_{37} = E_{8}, the degrees are 2, 8, 12, 14, 18, 20, 24, 30, and hence

Cat^{m}(E_{7}; q) = \frac{[30m + 2]_{q} [30m + 8]_{q} [30m + 12]_{q} [30m + 14]_{q}}{[2]_{q} [8]_{q} [12]_{q} [14]_{q}}

× \frac{[30m + 18]_{q} [30m + 20]_{q} [30m + 24]_{q} [30m + 30]_{q}}{[18]_{q} [20]_{q} [24]_{q} [30]_{q}} .

If m ≡ 0 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[\frac{15m+7}{7}]_{q^{14}}

× [\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 1 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 2 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 3 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[\frac{15m+4}{7}]_{q^{14}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 4 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}

× [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}[\frac{3m+2}{14}]_{q^{140}} \begin{matrix}[70] & q_{2} \\ [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 5 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{28}]_{q^{168}} \frac{[84]_{q^{2}}}{[7]_{q^{2}} [12]_{q^{2}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 6 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}

× [\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{4}]_{q^{40}} \frac{[10]_{q^{4}}}{[2]_{q^{4}} [5]_{q^{4}}} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 7 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}8]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 8 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{42}]_{q^{252}} \begin{matrix}[126]_{q^{2}} & [3]_{q^{2}} \\ [6]_{q^{2}} & [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

which, by Lemma 22, is a polynomial in q with non-negative integer coefficients. If m ≡ 9 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 10 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+2}{4}]_{q^{24}}

× [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}} [m + 1]_{q^{30}}.

If one decomposes [15m + 7]_{q^{2}} as [\frac{15m}{2} + 4]_{q^{4}} + q^{2}[\frac{15m}{2} + 3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 11 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 20, is a polynomial in q with non-negative integer coefficients. If m ≡ 12 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}

× [15m + 7]_{q^{2}}[\frac{5m+3}{21}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 13 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{28}]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 14 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [\frac{15m+7}{7}]_{q^{14}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 15 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 16 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{84}]_{q^{504}} \frac{[252]_{q^{2}} [3]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}} [12]_{q^{2}}} [m + 1]_{q^{30}},

which, by Lemma 23, is a polynomial in q with non-negative integer coefficients. If m ≡ 17 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[\frac{15m+4}{7}]_{q^{14}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 18 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}

[\frac{3m+2}{28}]_{q^{280}} \begin{matrix}[140]_{q^{2}} & [2]_{q^{2}} \\ [4]_{q^{2}} & [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Lemma 24, is a polynomial in q with non-negative integer coefficients. If m ≡ 19 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{1}4]_{q^{84}} \frac{[42]_{q^{2}}}{[6]_{q^{2}} [7]_{q^{2}}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 20 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 21 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 17, is a polynomial in q with non-negative integer coefficients. If m ≡ 22 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{28}]_{q^{168}} \begin{matrix}[84] & q_{2} \\ [7]_{q^{2}} & [12]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 23 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 24 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}

× [\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 25 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Lemma 18, is a polynomial in q with non-negative integer coefficients. If m ≡ 26 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[\frac{5m+3}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}}

× [\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}}.

If one decomposes [15m + 1]_{q^{2}} as [5m + 1]_{q^{6}} + q^{2}[5m]_{q^{6}} + q^{4}[5m]_{q^{6}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 27 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 21, is a polynomial in q with non-negative integer coefficients. If m ≡ 28 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[\frac{15m+7}{7}]_{q^{14}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 29 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{21}]_{q^{126}} \frac{[63]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}}} [\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 30 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}

[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{14}]_{q^{84}} \frac{[42]_{q^{2}}}{[6]_{q^{2}} [7]_{q^{2}}} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 31 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[\frac{15m+4}{7}]_{q^{14}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 32 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{14}]_{q^{140}} \begin{matrix}[70] & q_{2} \\ [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 33 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{84}]_{q^{504}} \frac{[252]_{q^{2}} [3]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}} [12]_{q^{2}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Lemmas 16 and 23, is a polynomial in q with non-negative integer coefficients. If m ≡ 34 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}

[5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 35 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{28}]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 36 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}

× [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 37 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m + 4}{21}]_{q^{126}} \frac{[63]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}}} [\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 38 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+2}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix}

× [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}}.

If one decomposes [15m + 7]_{q^{2}} as [\frac{15m}{2} + 4]_{q^{4}} + q^{2}[\frac{15m}{2} + 3]_{q^{4}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 39 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 20, is a polynomial in q with non-negative integer coefficients. If m ≡ 40 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}[\frac{5m+3}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}}

× [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}}.

If one decomposes [15m + 7]_{q^{2}} as [5m + 1]_{q^{6}} + q^{2}[5m]_{q^{6}} + q^{4}[5m]_{q^{6}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 41 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{28}]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 42 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[\frac{15m+7}{7}]_{q^{14}}

× [\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{4}]_{q^{40}} \frac{[10]_{q^{4}}}{[2]_{q^{4}} [5]_{q^{4}}} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 43 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 44 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{28}]_{q^{168}} \begin{matrix}[84] & q_{2} \\ [7]_{q^{2}} & [12]_{q^{2}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 45 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[\frac{15m+4}{7}]_{q^{14}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 46 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}

[5m + 3]_{q^{6}}[\frac{3m+2}{28}]_{q^{280}} \begin{matrix}[140]_{q^{2}} & [2]_{q^{2}} \\ [4]_{q^{2}} & [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6 and Lemma 24, is a polynomial in q with non-negative integer coefficients. If m ≡ 47 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{14}]_{q^{84}} \frac{[42]_{q^{2}}}{[6]_{q^{2}} [7]_{q^{2}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 48 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}

[\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which is manifestly a polynomial in q with non-negative integer coefficients. If m ≡ 49 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 50 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{28}]_{q^{168}} \begin{matrix}[84] & q_{2} \\ [7]_{q^{2}} & [12]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[\frac{m+1}{3}]_{q^{90}} \begin{matrix}[15] & q_{6} \\ [3]_{q^{6}} & [5]_{q^{6}} \end{matrix} ,

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 51 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 52 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 53 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Lemma 18, is a polynomial in q with non-negative integer coefficients. If m ≡ 54 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}

[\frac{5m+3}{21}]_{q^{126}} \begin{matrix}[63] & q_{2} \\ [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} [\frac{3m+2}{4}]_{q^{40}} \frac{[10]_{q^{4}}}{[2]_{q^{4}} [5]_{q^{4}}} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 55 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemma 21, is a polynomial in q with non-negative integer coefficients. If m ≡ 56 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [\frac{15m+7}{7}]_{q^{14}}[5m + 3]_{q^{6}}

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 57 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 58 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{42}]_{q^{252}} \begin{matrix}[126]_{q^{2}} & [3]_{q^{2}} \\ [6]_{q^{2}} & [7]_{q^{2}} & [9]_{q^{2}} \end{matrix} [m + 1]_{q^{30}},

which, by Lemma 22, is a polynomial in q with non-negative integer coefficients. If m ≡ 59 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[\frac{15m+4}{7}]_{q^{14}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 60 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}

× [\frac{3m+2}{14}]_{q^{140}} \begin{matrix}[70] & q_{2} \\ [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 61 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{28}]_{q^{168}} \frac{[84]_{q^{2}}}{[7]_{q^{2}} [12]_{q^{2}}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 62 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 63 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{28}]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 64 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{14}]_{q^{84}} \begin{matrix}[42] & q_{2} \\ [6]_{q^{2}} & [7]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 65 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 66 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}

× [\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{4}]_{q^{40}} \frac{[10]_{q^{4}}}{[2]_{q^{4}} [5]_{q^{4}}} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 67 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemma 20, is a polynomial in q with non-negative integer coefficients. If m ≡ 68 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix}

× [15m + 7]_{q^{2}}[\frac{5m+3}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}} [\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{4}]_{q^{24}}[m + 1]_{q^{30}}.

If one decomposes [15m + 1]_{q^{2}} as [5m + 1]_{q^{6}} + q^{2}[5m]_{q^{6}} + q^{4}[5m]_{q^{6}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 69 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{28}]_{q^{56}} \frac{[28]_{q^{2}}}{[4]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 70 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[\frac{15m+7}{7}]_{q^{14}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 71 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[\frac{5m+2}{21}]_{q^{126}} \frac{[63]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}}} [\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 72 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}[\frac{3m+2}{2}]_{q^{20}}

[\frac{5m+4}{28}]_{q^{168}} \begin{matrix}[84] & q_{2} \\ [7]_{q^{2}} & [12]_{q^{2}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 73 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[\frac{15m+4}{7}]_{q^{14}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{3}]_{q^{18}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 74 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

× [\frac{3m+2}{28}]_{q^{280}} \begin{matrix}[140]_{q^{2}} & [2]_{q^{2}} \\ [4]_{q^{2}} & [7]_{q^{2}} & [10]_{q^{2}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[\frac{m+1}{3}]_{q^{90}} \begin{matrix}[15] & q_{6} \\ [3]_{q^{6}} & [5]_{q^{6}} \end{matrix} ,

which, by Corollary 6 and Lemma 24, is a polynomial in q with non-negative integer coefficients. If m ≡ 75 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{42}]_{q^{252}} \frac{[126]_{q^{2}} [3]_{q^{2}}}{[6]_{q^{2}} [7]_{q^{2}} [9]_{q^{2}}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Lemmas 19 and 22, is a polynomial in q with non-negative integer coefficients. If m ≡ 76 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{7}]_{q^{14}}[\frac{15m+4}{4}]_{q^{8}}[\frac{5m+2}{2}]_{q^{12}}[15m + 7]_{q^{2}}

× [5m + 3]_{q^{6}}[\frac{3m+2}{2}]_{q^{20}}[\frac{5m+4}{12}]_{q^{72}} \begin{matrix}[12] & q_{6} \\ [3]_{q^{6}} & [4]_{q^{6}} \end{matrix} [m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 77 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [\frac{5m+3}{4}]_{q^{24}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 16, is a polynomial in q with non-negative integer coefficients. If m ≡ 78 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{28}]_{q^{168}} \begin{matrix}[84] & q_{2} \\ [7]_{q^{2}} & [12]_{q^{2}} \end{matrix} [15m + 7]_{q^{2}}[\frac{5m+3}{3}]_{q^{18}}

× [\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{2}]_{q^{12}}[m + 1]_{q^{30}},

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 79 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{2}]_{q^{4}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[\frac{5m+4}{21}]_{q^{126}} \frac{[63]_{q^{2}}}{[7]_{q^{2}} [9]_{q^{2}}} [\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 19, is a polynomial in q with non-negative integer coefficients. If m ≡ 80 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{28}]_{q^{56}} \begin{matrix}[28] & q_{2} \\ [4]_{q^{2}} & [7]_{q^{2}} \end{matrix} [\frac{5m+2}{6}]_{q^{36}} \begin{matrix}[6] & q_{6} \\ [2]_{q^{6}} & [3]_{q^{6}} \end{matrix} [15m + 7]_{q^{2}}[5m + 3]_{q^{6}}

which, by Corollary 6, is a polynomial in q with non-negative integer coefficients. If m ≡ 81 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{4}]_{q^{8}}[15m + 4]_{q^{2}}[5m + 2]_{q^{6}}[\frac{15m+7}{2}]_{q^{4}}[\frac{5m+3}{12}]_{q^{72}} \frac{[12]_{q^{6}}}{[3]_{q^{6}} [4]_{q^{6}}}

× [\frac{3m+2}{7}]_{q^{70}} \frac{[35]_{q^{2}}}{[5]_{q^{2}} [7]_{q^{2}}} [5m + 4]_{q^{6}}[\frac{m+1}{2}]_{q^{60}} \begin{matrix}[30]_{q^{2}} & [2]_{q^{2}} & [3]_{q^{2}} & [5]_{q^{2}} \\ [6]_{q^{2}} & [10]_{q^{2}} & [15]_{q^{2}} \end{matrix} ,

which, by Corollary 6 and Lemma 18, is a polynomial in q with non-negative integer coefficients. If m ≡ 82 (mod 84), then we have

Cat^{m}(E_{8}; q) = [15m + 1]_{q^{2}}[\frac{15m+4}{2}]_{q^{4}}[\frac{5m+2}{4}]_{q^{24}}[15m + 7]_{q^{2}}[\frac{5m+3}{7}]_{q^{42}} \frac{[21]_{q^{2}}}{[3]_{q^{2}} [7]_{q^{2}}}

[\frac{3m+2}{4}]_{q^{40}} \begin{matrix}[10] & q_{4} \\ [2]_{q^{4}} & [5]_{q^{4}} \end{matrix} [\frac{5m+4}{6}]_{q^{36}} \frac{[6]_{q^{6}}}{[2]_{q^{6}} [3]_{q^{6}}} [m + 1]_{q^{30}}.

If one decomposes [15m + 1]_{q^{2}} as [5m + 1]_{q^{6}} + q^{2}[5m]_{q^{6}} + q^{4}[5m]_{q^{6}}, then one sees that, by Corollary 6, this is a polynomial in q with non-negative integer coefficients. If m ≡ 83 (mod 84), then we have

Cat^{m}(E_{8}; q) = [\frac{15m+1}{14}]_{q^{28}} \frac{[14]_{q^{2}}}{[2]_{q^{2}} [7]_{q^{2}}} [15m + 4]_{q^{2}}[\frac{5m+2}{3}]_{q^{18}}[\frac{15m+7}{4}]_{q^{8}}[\frac{5m+3}{2}]_{q^{12}}

× [3m + 2]_{q^{10}}[5m + 4]_{q^{6}}[\frac{m+1}{4}]_{q^{120}} \frac{[60]_{q^{2}} [2]_{q^{2}} [3]_{q^{2}} [5]_{q^{2}}}{[10]_{q^{2}} [12]_{q^{2}} [15]_{q^{2}}} ,

which, by Corollary 6 and Lemma 21, is a polynomial in q with non-negative integer coefficients.

□

5. Auxiliary results I

This section collects several auxiliary results which allow us to reduce the problem of proving Theorem 2, or the equivalent statement (3.3), for the 26 exceptional groups listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover special choices of the parameters, Lemmas 26 and 30 afford an inductive procedure. More precisely, if we assume that we have already verified Theorem 2 for all groups of smaller rank, then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce the verification of Theorem 2 for the group that we are currently considering to a finite problem; see Remark 3. The final lemma of this section, Lemma 32, disposes of complex reflection groups with a special property satisfied by their degrees.

Let p = am + b, 0 ≤ b < m. We have

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{a+1}w_{m−b+1}c^{−a−1}, c^{a+1}w_{m−b+2}c^{−a−1}, …, c^{a+1}w_{m}c^{−a−1},
c^{a}w_{1}c^{−a}, …, c^{a}w_{m−b}c^{−a}), (5.1)

where ∗ stands for the element of W which is needed to complete the product of the components to c.

Lemma 26. It suffices to check (3.3) for p a divisor of mh. More precisely, let p be a divisor of mh, and let k be another positive integer with gcd(k, mh/p) = 1, then we have

Cat^{m}(W; q)|_{q=e^{2πip/mh}} = Cat^{m}(W; q)|_{q=e^{2πikp/mh}}  (5.2)

and

| Fix_{NC^{m}(W)}(ϕ^{p})| = | Fix_{NC^{m}(W)}(ϕ^{kp})|.  (5.3)

Proof. For (5.2), this follows immediately from

lim_{q→ζ} \frac{[α]_{q}}{[β]_{q}} = \begin{cases}\frac{α}{β} & if α ≡ β ≡ 0 (mod d), \\ 1 & otherwise, \end{cases}  (5.4)

where ζ is a primitive d-th root of unity and α, β are non-negative integers such that α ≡ β (mod d).

In order to establish (5.3), suppose that x ∈ Fix_{NC^{m}(W)}(ϕ^{p}), that is, x ∈ NC^{m}(W) and ϕ^{p}(x) = x. It obviously follows that ϕ^{kp}(x) = x, so that x ∈ Fix_{NC^{m}(W)}(ϕ^{kp}). To establish the converse, note that, if gcd(k, mh/p) = 1, then there exists k′ with k′k ≡ 1 (mod \frac{mh}{p}). It follows that, if x ∈ Fix_{NC^{m}(W)}(ϕ^{kp}), that is, if x ∈ NC^{m}(W) and ϕ^{kp}(x) = x, then x = ϕ^{k′kp}(x) = ϕ^{p}(x), whence x ∈ Fix_{NC^{m}(W)}(ϕ^{p}). □

Lemma 27. Let p be a divisor of mh. If p is divisible by m, then (3.3) is true.

Proof. According to (5.1), the action of ϕ^{p} on NC^{m}(W) is described by

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{p/m}w_{1}c^{−p/m}, …, c^{p/m}w_{m}c^{−p/m}).

Hence, if (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, then each individual w_{i} must be fixed under conjugation by c^{p/m}.

Using the notation W′ = Cent_{W} (c^{p/m}), the previous observation means that w_{i} ∈ W′, i = 1, 2, …, m. Springer [32, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) proved that W′ is a well-generated complex reflection group whose degrees coincide with those degrees of W that are divisible by mh/p. It was furthermore shown in [10, Lemma 3.3] that

NC(W) ∩ W′ = NC(W′).  (5.5)

Hence, the tuples (w_{0}; w_{1}, …, w_{m}) fixed by ϕ^{p} are in fact identical with the elements of NC^{m}(W′), which implies that

| Fix_{NC^{m}(W)}(ϕ^{p})| = |NC^{m}(W′)|.  (5.6)

Application of Theorem 1 with W replaced by W′ and of the “limit rule” (5.4) then yields that

|NC^{m}(W′)| = _{∏_{1≤i≤n} ; \frac{mh}{p} |d_{i}} \frac{mh + d_{i}}{d_{i}} = Cat^{m}(W; q)|_{q=e^{2πip/mh}}.  (5.7)

Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of the lemma. □

Lemma 28. Equation (3.3) holds for all divisors p of m.

Proof. Using (5.4) and the fact that the degrees of irreducible well-generated complex reflection groups satisfy d_{i} < h for all i < n, we see that

Cat^{m}(W; q)|_{q=e^{2πip/mh}} = \begin{cases}m + 1 & if m = p, \\ 1 & if m ≠ p. \end{cases}

On the other hand, if (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, then, because of the action (5.1), we must have w_{1} = w_{p+1} = ⋯ = w_{m−p+1} and w_{1} = cw_{m−p+1}c^{−1}. In particular, w_{1} ∈ Cent_{W} (c). By the theorem of Springer cited in the proof of Lemma 27, the subgroup Cent_{W} (c) is itself a complex reflection group whose degrees are those degrees of W that are divisible by h. The only such degree is h itself, hence Cent_{W} (c) is the cyclic group generated by c. Moreover, by (5.5), we obtain that w_{1} = ε, the identity element of W, or w_{1} = c. Therefore, for m = p the set Fix_{NC^{m}(W)}(ϕ^{p}) consists of the m + 1 elements (w_{0}; w_{1}, …, w_{m}) obtained by choosing w_{i} = c for a particular i between 0 and m, all other w_{j}’s being equal to ε, while, for m ≠ p, we have

Fix_{NC^{m}(W)}(ϕ^{p}) = {(c; ε, …, ε)},

whence the result. □

Lemma 29. Let W be an irreducible well-generated complex reflection group all of whose degrees are divisible by d. Then each element of W is fixed under conjugation by c^{h/d}.

Proof. By the theorem of Springer cited in the proof of Lemma 27, the subgroup W′ = Cent_{W} (c^{h/d}) is itself a complex reflection group whose degrees are those degrees of W that are divisible by d. Thus, by our assumption, the degrees of W′ coincide with the degrees of W, and hence W′ must be equal to W. Phrased differently, each element of W is fixed under conjugation by c^{h/d}, as claimed. □

Lemma 30. Let W be an irreducible well-generated complex reflection group of rank n, and let p = m_{1}h_{1} be a divisor of mh, where m = m_{1}m_{2} and h = h_{1}h_{2}. Without loss of generality, we assume that gcd(h_{1}, m_{2}) = 1. Suppose that Theorem 2 has already been verified for all irreducible well-generated complex reflection groups with rank < n. If h_{2} does not divide all degrees d_{i}, then Equation (3.3) is satisfied.

Proof. Let us write h_{1} = am_{2} + b, with 0 ≤ b < m_{2}. The condition gcd(h_{1}, m_{2}) = 1 translates into gcd(b, m_{2}) = 1. From (5.1), we infer that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{a+1}w_{m−m_{1}b+1}c^{−a−1}, c^{a+1}w_{m−m_{1}b+2}c^{−a−1}, …, c^{a+1}w_{m}c^{−a−1},
c^{a}w_{1}c^{−a}, …, c^{a}w_{m−m_{1}b}c^{−a}). (5.8)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{a+1}w_{i+m−m_{1}b}c^{−a−1}, i = 1, 2, …, m_{1}b,
w_{i} = c^{a}w_{i−m_{1}b}c^{−a}, i = m_{1}b + 1, m_{1}b + 2, …, m,

which, after iteration, implies in particular that

w_{i} = c^{b(a+1)+(m_{2}−b)a}w_{i}c^{−b(a+1)−(m_{2}−b)a} = c^{h_{1}}w_{i}c^{−h_{1}}, i = 1, 2, …, m.

It is at this point where we need gcd(b, m_{2}) = 1. The last equation shows that each w_{i}, i = 1, 2, …, m, and thus also w_{0}, lies in Cent_{W} (c^{h_{1}}). By the theorem of Springer cited in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group, W′ say, whose degrees are those degrees of W that are divisible by h/h_{1} = h_{2}. Since, by assumption, h_{2} does not divide all degrees, W′ has rank strictly less than n. Again by assumption, we know that Theorem 2 is true for W′, so that in particular,

| Fix_{NC^{m}(W′)}(ϕ^{p})| = Cat^{m}(W′; q)|_{q=e^{2πip/mh}}.

The arguments above together with (5.5) show that Fix_{NC^{m}(W)}(ϕ^{p}) = Fix_{NC^{m}(W′)}(ϕ^{p}). On the other hand, using (5.4) it is straightforward to see that

Cat^{m}(W; q)|_{q=e^{2πip/mh}} = Cat^{m}(W′; q)|_{q=e^{2πip/mh}}.

This proves (3.3) for our particular p, as required. □

Lemma 31. Let W be an irreducible well-generated complex reflection group of rank n, and let p = m_{1}h_{1} be a divisor of mh, where m = m_{1}m_{2} and h = h_{1}h_{2}. We assume that gcd(h_{1}, m_{2}) = 1. If m_{2} > n then

Fix_{NC^{m}(W)}(ϕ^{p}) = {(c; ε, …, ε)}.

Proof. Let us suppose that (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(W)}(ϕ^{p}) and that there exists a j ≥ 1 such that w_{j} ≠ ε. By (5.8), it then follows for such a j that also w_{k} ≠ ε for all k ≡ j − lm_{1}b (mod m), where, as before, b is defined as the unique integer with h_{1} = am_{2} + b and 0 ≤ b < m_{2}. Since, by assumption, gcd(b, m_{2}) = 1, there are exactly m_{2} such k’s which are distinct mod m. However, this implies that the sum of the absolute lengths of the w_{i}’s, 0 ≤ i ≤ m, is at least m_{2} > n, a contradiction to Remark 1.(2). □

Remark 3. (1) If we put ourselves in the situation of the assumptions of Lemma 30, then we may conclude that equation (3.3) only needs to be checked for pairs (m_{2}, h_{2}) subject to the following restrictions:

m_{2} ≥ 2, gcd(h_{1}, m_{2}) = 1, and h_{2} divides all degrees of W .  (5.9)

Indeed, Lemmas 27 and 30 together imply that equation (3.3) is always satisfied in all other cases.

(2) Still putting ourselves in the situation of Lemma 30, if m_{2} > n and m_{2}h_{2} does not divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says that in this case the left-hand side of (3.3) equals 1, while a straightforward computation using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.

(3) It should be observed that this leaves a finite number of choices for m_{2} to consider, whence a finite number of choices for (m_{1}, m_{2}, h_{1}, h_{2}). Altogether, there remains a finite number of choices for p = h_{1}m_{1} to be checked.

Lemma 32. Let W be an irreducible well-generated complex reflection group of rank n with the property that d_{i} | h for i = 1, 2, …, n. Then Theorem 2 is true for this group W.

Proof. By Lemma 26, we may restrict ourselves to divisors p of mh.

Suppose that e^{2πip/mh} is a d_{i}-th root of unity for some i. In other words, mh/p divides d_{i}. Since d_{i} is a divisor of h by assumption, the integer mh/p also divides h. But this is equivalent to saying that m divides p, and equation (3.3) holds by Lemma 27.

Now assume that mh/p does not divide any of the d_{i}’s. Then, by (5.4), the righthand side of (3.3) equals 1. On the other hand, (c; ε, …, ε) is always an element of Fix_{NC^{m}(W)}(ϕ^{p}). To see that there are no others, we make appeal to the classification of all irreducible well-generated complex reflection groups, which we recalled in Section 2. Inspection reveals that all groups satisfying the hypotheses of the lemma have rank n ≤ 2. Except for the groups contained in the infinite series G(d, 1, n) and G(e, e, n) for which Theorem 2 has been established in [19], these are the groups G_{5}, G_{6}, G_{9}, G_{10}, G_{14}, G_{17}, G_{18}, G_{21}. We now discuss these groups case by case, keeping the notation of Lemma 30. In order to simplify the argument, we note that Lemma 31 implies that equation (3.3) holds if m_{2} > 2, so that in the following arguments we always may assume that m_{2} = 2.

Case G_{5}. The degrees are 6, 12, and therefore Remark 3.(1) implies that equation (3.3) is always satisfied.

Case G_{6}. The degrees are 4, 12, and therefore, according to Remark 3.(1), we need only consider the case where h_{2} = 4 and m_{2} = 2, that is, p = 3m/2. Then (5.8) becomes

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

(5.10)

If (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p} and not equal to (c; ε, …, ε), there must exist an i with 1 ≤ i ≤ \frac{m}{2} such that ℓ_{T} (w_{i}) = ℓ_{T} (w_{\frac{m}{2} +i}) = 1, w_{\frac{m}{2} +i} = cw_{i}c^{−1}, w_{i}w_{\frac{m}{2} +i} = w_{i}cw_{i}c^{−1} = c, and all w_{j}, with j ≠ i, \frac{m}{2} + i, equal ε. However, with the help of the GAP package CHEVIE [14, 27], one verifies that there is no w_{i} in G_{6} such that

ℓ_{T} (w_{i}) = 1 and w_{i}cw_{i}c^{−1} = c

are simultaneously satisfied. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{9}. The degrees are 8, 24, and therefore, according to Remark 3.(1), we need only consider the case where h_{2} = 8 and m_{2} = 2, that is, p = 3m/2. This is the same p as for G_{6}. Again, CHEVIE finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{10}. The degrees are 12, 24, and therefore Remark 3.(1) implies that equation (3.3) is always satisfied.

Case G_{14}. The degrees are 6, 24, and therefore Remark 3.(1) implies that equation (3.3) is always satisfied.

Case G_{17}. The degrees are 20, 60, and therefore, according to Remark 3.(1), we need only consider the cases where h_{2} = 20 or h_{2} = 4. In the first case, p = 3m/2, which is the same p as for G_{6}. Again, CHEVIE finds no solution. In the second case, p = 15m/2. Then (5.8) becomes

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{8}w_{\frac{m}{2} +1}c^{−8}, c^{8}w_{\frac{m}{2} +2}c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{1}c^{−7}, …, c^{7}w\frac{m}{2} c^{−7}). (5.11)

By Lemma 29, every element of NC(W) is fixed under conjugation by c^{3}, and, thus, on elements fixed by ϕ^{p}, the above action of ϕ^{p} reduces to the one in (5.10). This action was already discussed in the first case. Hence, in both cases, the left-hand side of (3.3) is equal to 1, as required.

Case G_{18}. The degrees are 30, 60, and therefore Remark 3.(1) implies that equation (3.3) is always satisfied.

Case G_{21}. The degrees are 12, 60, and therefore, according to Remark 3.(1), we need only consider the cases where h_{2} = 12 or h_{2} = 4. In the first case, p = 5m/2, so that (5.8) becomes

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}). (5.12)

If (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p} and not equal to (c; ε, …, ε), there must exist an i with 1 ≤ i ≤ \frac{m}{2} such that ℓ_{T} (w_{i}) = 1 and w_{i}c^{2}w_{i}c^{−2} = c. However, with the help of the GAP package CHEVIE [14, 27], one verifies that there is no such solution to this equation. In the second case, p = 15m/2. Then (5.8) becomes the action in (5.11). By Lemma 29, every element of NC(W) is fixed under conjugation by c^{5}, and, thus, on elements fixed by ϕ^{p}, the action of ϕ^{p} in (5.11) reduces to the one in the first case. Hence, in both cases, the left-hand side of (3.3) is equal to 1, as required.

This completes the proof of the lemma. □

6. Case-by-case verification of Theorem 2

In the sequel we write ζ_{d} for a primitive d-th root of unity.

Case G_{4}. The degrees are 4, 6, and hence we have

Cat^{m}(G_{4}; q) = \frac{[6m + 6]_{q} [6m + 4]_{q}}{[6]_{q} [4]_{q}} .

Let ζ be a 6m-th root of unity. In what follows, we abbreviate the assertion that “ζ is a primitive d-th root of unity” as “ζ = ζ_{d}.” The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{4}; q) = m + 1, if ζ = ζ_{6}, ζ_{3},  (6.1a)
lim_{q→ζ} Cat^{m}(G_{4}; q) = \frac{3m+2}{2}, if ζ = ζ_{4}, 2 | m,  (6.1b)
lim_{q→ζ} Cat^{m}(G_{4}; q) = Cat^{m}(G_{4}), if ζ = −1 or ζ = 1,  (6.1c)
lim_{q→ζ} Cat^{m}(G_{4}; q) = 1, otherwise.  (6.1d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.1). The only cases not covered by Lemmas 27 and 28 are the ones in (6.1b) and (6.1d). On the other hand, the only case left to consider according to Remark 3 is the case where h_{2} = m_{2} = 2, that is the case (6.1b) where p = 3m/2. In particular, m must be divisible by 2. The action of ϕ^{p} is the same as the one in (5.10). With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{4} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to m/2 elements in NC^{m}(G_{4}) since the index i ranges from 1 to m/2.

Hence, in total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(G_{4})}(ϕ^{p}), which agrees with the limit in (6.1b).

Case G_{8}. The degrees are 8, 12, and hence we have

Cat^{m}(G_{8}; q) = \frac{[12m + 12]_{q} [12m + 8]_{q}}{[12]_{q} [8]_{q}} .

Let ζ be a 12m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{8}; q) = m + 1, if ζ = ζ_{12}, ζ_{6}, ζ_{3},  (6.2a)
lim_{q→ζ} Cat^{m}(G_{8}; q) = \frac{3m+2}{2}, if ζ = ζ_{8}, 2 | m,  (6.2b)
lim_{q→ζ} Cat^{m}(G_{8}; q) = Cat^{m}(G_{8}), if ζ = ζ_{4}, −1, 1,  (6.2c)
lim_{q→ζ} Cat^{m}(G_{8}; q) = 1, otherwise.  (6.2d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.2). The only cases not covered by Lemmas 27 and 28 are the ones in (6.2b) and (6.2d). On the other hand, the only case left to consider according to Remark 3 is the case where h_{2} = 4 and m_{2} = 2, that is the case (6.2b) where p = 3m/2. In particular, m must be divisible by 2. The action of ϕ^{p} is the same as the one in (5.10). With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{8} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to m/2 elements in NC^{m}(G_{8}) since the index i ranges from 1 to m/2.

Hence, in total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(G_{8})}(ϕ^{p}), which agrees with the limit in (6.2b).

Case G_{16}. The degrees are 20, 30, and hence we have

Cat^{m}(G_{16}; q) = \frac{[30m + 30]_{q} [30m + 20]_{q}}{[30]_{q} [20]_{q}} .

Let ζ be a 30m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{16}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{6}, ζ_{3},  (6.3a)
lim_{q→ζ} Cat^{m}(G_{16}; q) = \frac{3m+2}{2}, if ζ = ζ_{20}, ζ_{4}, 2 | m,  (6.3b)
lim_{q→ζ} Cat^{m}(G_{16}; q) = Cat^{m}(G_{16}), if ζ = ζ_{10}, ζ_{5}, −1, 1,  (6.3c)
lim_{q→ζ} Cat^{m}(G_{16}; q) = 1, otherwise.  (6.3d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.3). The only cases not covered by Lemmas 27 and 28 are the ones in (6.3b) and (6.3d). On the other hand, the only cases left to consider according to Remark 3 are the cases where h_{2} = 10 and m_{2} = 2, respectively h_{2} = m_{2} = 2. Both cases belong to (6.3b). In the first case, we have p = 3m/2, while in the second case we have p = 15m/2. In particular, m must be divisible by 2. In the first case, the action of ϕ^{p} is the same as the one in (5.10). With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{16} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to m/2 elements in NC^{m}(G_{16}) since the index i ranges from 1 to m/2. On the other hand, if p = 15m/2, then the action of ϕ^{p} is the same as the one in (5.11). By Lemma 29, every element of NC(W) is fixed under conjugation by c^{3}, and, thus, on elements fixed by ϕ^{p}, the action of ϕ^{p} reduces to the one in the first case.

Hence, in total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(G_{16})}(ϕ^{p}), which agrees with the limit in (6.3b).

Case G_{20}. The degrees are 12, 30, and hence we have

Cat^{m}(G_{20}; q) = \frac{[30m + 30]_{q} [30m + 12]_{q}}{[30]_{q} [12]_{q}} .

Let ζ be a 30m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{20}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (6.4a)
lim_{q→ζ} Cat^{m}(G_{20}; q) = \frac{5m+2}{2}, if ζ = ζ_{12}, ζ_{4}, 2 | m,  (6.4b)
lim_{q→ζ} Cat^{m}(G_{20}; q) = Cat^{m}(G_{20}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (6.4c)
lim_{q→ζ} Cat^{m}(G_{20}; q) = 1, otherwise.  (6.4d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.4). The only cases not covered by Lemmas 27 and 28 are the ones in (6.4b) and (6.4d). On the other hand, the only cases left to consider according to Remark 3 are the cases where h_{2} = 6 and m_{2} = 2, respectively h_{2} = m_{2} = 2. Both cases belong to (6.4b). In the first case, we have p = 5m/2, while in the second case we have p = 15m/2. In particular, m must be divisible by 2. In the first case, the action of ϕ^{p} is the same as the one in (5.12). With the help of CHEVIE, one finds that each of the 5 (complex) reflections in G_{20} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to m/2 elements in NC^{m}(G_{20}) since the index i ranges from 1 to m/2. On the other hand, if p = 15m/2, then the action of ϕ^{p} is the same as the one in (5.11). By Lemma 29, every element of NC(W) is fixed under conjugation by c^{5}, and, thus, on elements fixed by ϕ^{p}, the action of ϕ^{p} reduces to the one in the first case.

Hence, in total, we obtain 1 + 5\frac{m}{2} = \frac{5m+2}{2} elements in Fix_{NC^{m}(G_{20})}(ϕ^{p}), which agrees with the limit in (6.4b).

Case G_{23} = H_{3}. The degrees are 2, 6, 10, and hence we have

Cat^{m}(H_{3}; q) = \frac{[10m + 10]_{q} [10m + 6]_{q} [10m + 2]_{q}}{[10]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 10m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(H_{3}; q) = m + 1, if ζ = ζ_{10}, ζ_{5},  (6.5a)
lim_{q→ζ} Cat^{m}(H_{3}; q) = \frac{10m+6}{6}, if ζ = ζ_{6}, ζ_{3}, 3 | m,  (6.5b)
lim_{q→ζ} Cat^{m}(H_{3}; q) = Cat^{m}(H_{3}), if ζ = −1 or ζ = 1,  (6.5c)
lim_{q→ζ} Cat^{m}(H_{3}; q) = 1, otherwise.  (6.5d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.5). The only cases not covered by Lemmas 27 and 28 are the ones in (6.5b) and (6.5d). By Lemma 26, we are free to choose p = 5m/3 if ζ = ζ_{6}, respectively p = 10m/3 if ζ = ζ_{3}. In both cases, m must be divisible by 3.

We start with the case that p = 5m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{3} +1}c^{−2}, c^{2}w_{\frac{m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{3} c^{−1}).

(6.6)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{3} +i}c^{−2}, i = 1, 2, …, \frac{2m}{3},  (6.7a)
w_{i} = cw_{i− \frac{2m}{3}} c^{−1}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.7b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.7) reduce to

t_{1} = c^{2}t_{2}c^{−2},  (6.8a)
t_{2} = c^{2}t_{3}c^{−2},  (6.8b)
t_{3} = ct_{1}c^{−1}.  (6.8c)

One of these equations is in fact superfluous: if we substitute (6.8b) and (6.8c) in (6.8a), then we obtain t_{1} = c^{5}t_{1}c^{−5} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(H_{3}), we must have t_{1}t_{2}t_{3} = c. Combining this with (6.8), we infer that

t_{1}(c^{−2}t_{1}c^{2})(ct_{1}c^{−1}) = c.  (6.9)

With the help of Stembridge’s Maple package coxeter [35], one obtains five solutions for t_{1} in this equation:

t_{1} ∈ {[2], [3], [2, 1, 2], [1, 2, 3, 2, 1], [1, 3, 2, 1, 2, 1, 3]}.  (6.10)

Here we have used the short notation of coxeter: if {s_{1}, s_{2}, s_{3}} is a simple system of generators of H_{3}, corresponding to the Dynkin diagram displayed in Figure 1, then [j_{1}, j_{2}, …, j_{k}] stands for the element s_{j_{1}}s_{j_{2}} … s_{j_{k}}.

We claim that each of the above five solutions gives rise to m/3 elements of Fix_{NC^{m}(H_{3})}(ϕ^{p}). Indeed, given t_{1}, the elements t_{2} and t_{3} can be computed by (6.8a) and (6.8c), and there are m/3 possibilities to choose the index i for w_{i}.

• • •_{1} _{2} _{3}^{5}

Figure 1. The Dynkin diagram for H_{3}

In total, we obtain 1 + 5\frac{m}{3} = \frac{10m+6}{6} elements in Fix_{NC^{m}(H_{3})}(ϕ^{p}), which agrees with the limit in (6.5b).

In the case that p = 10m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{3} +1}c^{−4}, c^{4}w_{\frac{2m}{3} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{3}} c^{−3}). (6.11)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{3} +i}c^{−4}, i = 1, 2, …, \frac{m}{3},  (6.12a)
w_{i} = c^{3}w_{i− \frac{m}{3}} c^{−3}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.12b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.12) reduce to

t_{1} = c^{4}t_{3}c^{−4},  (6.13a)
t_{2} = c^{3}t_{1}c^{−3},  (6.13b)
t_{3} = c^{3}t_{2}c^{−3}.  (6.13c)

One of these equations is in fact superfluous: if we substitute (6.13b) and (6.13c) in (6.13a), then we obtain t_{1} = c^{10}t_{1}c^{−10} which is automatically satisfied since c^{10} = ε.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(H_{3}), we must have t_{1}t_{2}t_{3} = c. Combining this with (6.13), we infer that

t_{1}(c^{3}t_{1}c^{−3})(c^{−4}t_{1}c^{4}) = c.  (6.14)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 2, we see that this equation is equivalent with (6.9). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/3, and, consequently, the number of solutions to (6.14) is the same, namely \frac{5m+3}{3}, as required.

Finally, we turn to (6.5d). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 3, h_{2} = m_{2} = 2, respectively h_{2} = 2 and m_{2} = 3. These correspond to the choices p = 10m/3, p = 5m/2, respectively p = 5m/3, out of which only p = 5m/2 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by (5.12). A computation with Stembridge’s Maple package coxeter [35] finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{24}. The degrees are 4, 6, 14, and hence we have

Cat^{m}(G_{24}; q) = \frac{[14m + 14]_{q} [14m + 6]_{q} [14m + 4]_{q}}{[14]_{q} [6]_{q} [4]_{q}} .

Let ζ be a 14m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{24}; q) = m + 1, if ζ = ζ_{14}, ζ_{7},  (6.15a)
lim_{q→ζ} Cat^{m}(G_{24}; q) = \frac{7m+3}{3}, if ζ = ζ_{6}, ζ_{3}, 3 | m,  (6.15b)
lim_{q→ζ} Cat^{m}(G_{24}; q) = \frac{7m+2}{2}, if ζ = ζ_{4}, 2 | m,  (6.15c)
lim_{q→ζ} Cat^{m}(G_{24}; q) = Cat^{m}(G_{24}), if ζ = −1 or ζ = 1,  (6.15d)
lim_{q→ζ} Cat^{m}(G_{24}; q) = 1, otherwise.  (6.15e)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.15). The only cases not covered by Lemmas 27 and 28 are the ones in (6.15b), (6.15c), and (6.15e).

We first consider (6.15b). By Lemma 26, we are free to choose p = 7m/3 if ζ = ζ_{6}, respectively p = 14m/3 if ζ = ζ_{3}. In both cases, m must be divisible by 3.

We start with the case that p = 7m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{2m}{3} +1}c^{−3}, c^{3}w_{\frac{2m}{3} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{2m}{3}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{2m}{3} +i}c^{−3}, i = 1, 2, …, \frac{m}{3},  (6.16a)
w_{i} = c^{2}w_{i− \frac{m}{3}} c^{−2}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.16b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.16) reduce to

t_{1} = c^{3}t_{3}c^{−3},  (6.17a)
t_{2} = c^{2}t_{1}c^{−2},  (6.17b)
t_{3} = c^{2}t_{2}c^{−2}.  (6.17c)

One of these equations is in fact superfluous: if we substitute (6.17b) and (6.17c) in (6.17a), then we obtain t_{1} = c^{7}t_{1}c^{−7} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{24}), we must have t_{1}t_{2}t_{3} = c. Combining this with (6.17), we infer that

t_{1}(c^{2}t_{1}c^{−2})(c^{4}t_{1}c^{−4}) = c.  (6.18)

With the help of CHEVIE, one obtains 7 solutions for t_{1} in this equation:

t_{1} ∈ {[1], [2], [3], [15], [16], [19], [21]},  (6.19)

each of them giving rise to m/3 elements of Fix_{NC^{m}(G_{24})}(ϕ^{p}) since i ranges from 1 to m/3. Here we have used the short notation of CHEVIE: [j_{1}, j_{2}, …, j_{k}] stands for the element r_{j_{1}}r_{j_{2}} … r_{j_{k}}, where r_{i} is the i-th (complex) reflection corresponding to the i-th root in the internal ordering of the roots of G_{24} in CHEVIE.

In total, we obtain 1 + 7\frac{m}{3} = \frac{7m+3}{3} elements in Fix_{NC^{m}(G_{24})}(ϕ^{p}), which agrees with the limit in (6.15b).

In the case that p = 14m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{m}{3} +1}c^{−5}, c^{5}w_{\frac{m}{3} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w\frac{m}{3} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{m}{3} +i}c^{−5}, i = 1, 2, …, \frac{2m}{3},  (6.20a)
w_{i} = c^{4}w_{i− \frac{2m}{3}} c^{−4}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.20b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.20) reduce to

t_{1} = c^{5}t_{2}c^{−5},  (6.21a)
t_{2} = c^{5}t_{3}c^{−5},  (6.21b)
t_{3} = c^{4}t_{1}c^{−4}.  (6.21c)

One of these equations is in fact superfluous: if we substitute (6.21b) and (6.21c) in (6.21a), then we obtain t_{1} = c^{14}t_{1}c^{−14} which is automatically satisfied since c^{14} = ε.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{24}), we must have t_{1}t_{2}t_{3} = c. Combining this with (6.21), we infer that

t_{1}(c^{9}t_{1}c^{−9})(c^{−4}t_{1}c^{4}) = c.  (6.22)

Using that c^{7}t_{1}c^{−7} = t_{1}, due to Lemma 29 with d = 2, we see that this equation is equivalent with (6.18). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/3, and, consequently, the number of solutions to (6.22) is the same, namely \frac{7m+3}{3}, as required.

Our next case is (6.15c). By Lemma 26, we are free to choose p = 7m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{m}{2} +1}c^{−4}, c^{4}w_{\frac{m}{2} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w\frac{m}{2} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{m}{2} +i}c^{−4}, i = 1, 2, …, \frac{m}{2},  (6.23a)
w_{i} = c^{3}w_{i− \frac{m}{2}} c^{−3}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.23b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1.

Writing t_{1}, t_{2} for w_{i}, w_{i+ \frac{m}{2}} , respectively, the equations (6.23) reduce to

t_{1} = c^{4}t_{2}c^{−4},  (6.24a)
t_{2} = c^{3}t_{1}c^{−3}.  (6.24b)

One of these equations is in fact superfluous: if we substitute (6.24b) in (6.24a), then we obtain t_{1} = c^{7}t_{1}c^{−7} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{24}), we must have t_{1}t_{2} ≤_{T} c, where ≤_{T} is the partial order defined in (2.1). Combining this with (6.24), we infer that

t_{1}(c^{3}t_{1}c^{−3}) ≤_{T} c.  (6.25)

With the help of CHEVIE, one obtains 7 solutions for t_{1} in this relation:

t_{1} ∈ {[5], [6], [7], [9], [12], [29], [32]},  (6.26)

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{24})}(ϕ^{p}) since i ranges from 1 to m/2. Here we have used again the short notation of CHEVIE referring to the internal ordering of the roots of G_{24} in CHEVIE.

In total, we obtain 1 + 7\frac{m}{2} = \frac{7m+2}{2} elements in Fix_{NC^{m}(G_{24})}(ϕ^{p}), which agrees with the limit in (6.15c).

Finally, we turn to (6.15e). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 3, h_{2} = m_{2} = 2, and h_{2} = 2 and m_{2} = 3. These correspond to the choices p = 14m/3, p = 7m/2, respectively p = 7m/3, all of which have already been discussed as they do not belong to (6.15e). Hence, (3.3) must necessarily hold, as required.

Case G_{25}. The degrees are 6, 9, 12, and hence we have

Cat^{m}(G_{25}; q) = \frac{[12m + 12]_{q} [12m + 9]_{q} [12m + 6]_{q}}{[12]_{q} [9]_{q} [6]_{q}} .

Let ζ be a 12m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{25}; q) = m + 1, if ζ = ζ_{12}, ζ_{4},  (6.27a)
lim_{q→ζ} Cat^{m}(G_{25}; q) = \frac{4m+3}{3}, if ζ = ζ_{9}, 3 | m,  (6.27b)
lim_{q→ζ} Cat^{m}(G_{25}; q) = (m + 1)(2m + 1), if ζ = ζ_{6}, −1  (6.27c)
lim_{q→ζ} Cat^{m}(G_{25}; q) = Cat^{m}(G_{25}), if ζ = ζ_{3}, 1,  (6.27d)
lim_{q→ζ} Cat^{m}(G_{25}; q) = 1, otherwise.  (6.27e)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.27). The only cases not covered by Lemmas 27 and 28 are the ones in (6.27b) and (6.27e).

We first consider (6.27b). By Lemma 26, we are free to choose p = 4m/3. In particular, m must be divisible by 3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{2m}{3} +1}c^{−2}, c^{2}w_{\frac{2m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{2m}{3}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{2m}{3} +i}c^{−2}, i = 1, 2, …, \frac{m}{3},  (6.28a)
w_{i} = cw_{i− \frac{m}{3}} c^{−1}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.28b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.28) reduce to

t_{1} = c^{2}t_{3}c^{−2},  (6.29a)
t_{2} = ct_{1}c^{−1},  (6.29b)
t_{3} = ct_{2}c^{−1}.  (6.29c)

One of these equations is in fact superfluous: if we substitute (6.29b) and (6.29c) in (6.29a), then we obtain t_{1} = c^{4}t_{1}c^{−4} which is automatically satisfied due to Lemma 29 with d = 3.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{25}), we must have t_{1}t_{2}t_{3} = c. Combining this with (6.29), we infer that

t_{1}(ct_{1}c^{−1})(c^{2}t_{1}c^{−2}) = c.  (6.30)

With the help of CHEVIE, one obtains four solutions for t_{1} in this equation:

t_{1} ∈ {[1], [2], [3], [14]},  (6.31)

each of them giving rise to m/3 elements of Fix_{NC^{m}(G_{25})}(ϕ^{p}) since i ranges from 1 to m/3. Here we have used again the short notation of CHEVIE referring to the internal ordering of the roots of G_{25} in CHEVIE.

In total, we obtain 1 + 4\frac{m}{3} = \frac{4m+3}{3} elements in Fix_{NC^{m}(G_{25})}(ϕ^{p}), which agrees with the limit in (6.27b).

Finally, we turn to (6.27e). By Remark 3, the only choice for h_{2} and m_{2} to be considered are h_{2} = m_{2} = 3. This corresponds to the choice p = 4m/3, which has already been discussed as they do not belong to (6.27e). Hence, (3.3) must necessarily hold, as required.

Case G_{26}. The degrees are 6, 12, 18, and hence we have

Cat^{m}(G_{26}; q) = \frac{[18m + 18]_{q} [18m + 12]_{q} [18m + 6]_{q}}{[18]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 14m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{26}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (6.32a)
lim_{q→ζ} Cat^{m}(G_{26}; q) = \frac{3m+2}{2}, if ζ = ζ_{12}, ζ_{4}, 2 | m,  (6.32b)
lim_{q→ζ} Cat^{m}(G_{26}; q) = Cat^{m}(G_{26}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (6.32c)
lim_{q→ζ} Cat^{m}(G_{26}; q) = 1, otherwise.  (6.32d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.32). The only cases not covered by Lemmas 27 and 28 are the ones in (6.32b) and (6.32d).

We first consider (6.32b). By Lemma 26, we are free to choose p = 3m/2 if ζ = ζ_{12}, respectively p = 9m/2 if ζ = ζ_{4}. In both cases, m must be divisible by 2.

We start with the case that p = 3m/2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.33a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.33b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1.

Writing t_{1}, t_{2} for w_{i}, w_{i+ \frac{m}{2}} , respectively, the equations (6.33) reduce to

t_{1} = c^{2}t_{2}c^{−2},  (6.34a)
t_{2} = c^{1}t_{1}c^{−1}.  (6.34b)

One of these equations is in fact superfluous: if we substitute (6.34b) in (6.34a), then we obtain t_{1} = c^{3}t_{1}c^{−3} which is automatically satisfied due to Lemma 29 with d = 6.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{26}), we must have t_{1}t_{2} ≤_{T} c. Combining this with (6.34), we infer that

t_{1}(ct_{1}c^{−1}) ≤_{T} c.  (6.35)

With the help of CHEVIE, one obtains three solutions for t_{1} in this equation:

t_{1} ∈ {[2], [3], [12]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{26})}(ϕ^{p}) since i ranges from 1 to m/2. Here we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{26} in CHEVIE.

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(G_{26})}(ϕ^{p}), which agrees with the limit in (6.32b).

In the case that p = 9m/2, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{m}{2} +1}c^{−5}, c^{5}w_{\frac{m}{2} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w\frac{m}{2} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{m}{2} +i}c^{−5}, i = 1, 2, …, \frac{m}{2},  (6.36a)
w_{i} = c^{4}w_{i− \frac{m}{2}} c^{−4}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.36b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1.

Writing t_{1}, t_{2} for w_{i}, w_{i+ \frac{m}{2}} , respectively, the equations (6.36) reduce to

t_{1} = c^{5}t_{2}c^{−5},  (6.37a)
t_{2} = c^{4}t_{1}c^{−4}.  (6.37b)

One of these equations is in fact superfluous: if we substitute (6.37b) in (6.37a), then we obtain t_{1} = c^{9}t_{1}c^{−9} which is automatically satisfied due to Lemma 29 with d = 2. Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{26}), we must have t_{1}t_{2} ≤_{T} c. Combining this with (6.37), we infer that

t_{1}(c^{4}t_{1}c^{−4}) ≤_{T} c.  (6.38)

Using that c^{3}t_{1}c^{−3} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.35). Therefore, we are facing exactly the same enumeration problem here as for p = 3m/2, and, consequently, the number of solutions to (6.38) is the same, namely \frac{3m+2}{2}, as required.

Finally, we turn to (6.32d). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 6 and m_{2} = 2, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 3m/2, respectively p = 9m/2, all of which have already been discussed as they do not belong to (6.32d). Hence, (3.3) must necessarily hold, as required.

Case G_{27}. The degrees are 6, 12, 30, and hence we have

Cat^{m}(G_{27}; q) = \frac{[30m + 30]_{q} [30m + 12]_{q} [30m + 6]_{q}}{[30]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 14m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{27}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (6.39a)
lim_{q→ζ} Cat^{m}(G_{27}; q) = \frac{5m+2}{2}, if ζ = ζ_{12}, ζ_{4}, 2 | m,  (6.39b)
lim_{q→ζ} Cat^{m}(G_{27}; q) = Cat^{m}(G_{27}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (6.39c)
lim_{q→ζ} Cat^{m}(G_{27}; q) = 1, otherwise.  (6.39d)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.39). The only cases not covered by Lemmas 27 and 28 are the ones in (6.39b) and (6.39d).

We first consider (6.39b). By Lemma 26, we are free to choose p = 5m/2 if ζ = ζ_{12}, respectively p = 15m/2 if ζ = ζ_{4}. In both cases, m must be divisible by 2.

We start with the case that p = 5m/2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{2} +i}c^{−3}, i = 1, 2, …, \frac{m}{2},  (6.40a)
w_{i} = c^{2}w_{i− \frac{m}{2}} c^{−2}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.40b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1.

Writing t_{1}, t_{2} for w_{i}, w_{i+ \frac{m}{2}} , respectively, the equations (6.40) reduce to

t_{1} = c^{3}t_{2}c^{−3},  (6.41a)
t_{2} = c^{2}t_{1}c^{−2}.  (6.41b)

One of these equations is in fact superfluous: if we substitute (6.41b) in (6.41a), then we obtain t_{1} = c^{5}t_{1}c^{−5} which is automatically satisfied due to Lemma 29 with d = 6.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{27}), we must have t_{1}t_{2} ≤_{T} c. Combining this with (6.41), we infer that

t_{1}(c^{2}t_{1}c^{−2}) ≤_{T} c.  (6.42)

With the help of CHEVIE, one obtains five solutions for t_{1} in this equation:

t_{1} ∈ {[1], [2], [15], [16], [28]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{27})}(ϕ^{p}) since i ranges from 1 to m/2. Here we have used the short notation of CHEVIE referring to the internal ordering of the roots of G_{27} in CHEVIE.

In total, we obtain 1 + 5\frac{m}{2} = \frac{5m+2}{2} elements in Fix_{NC^{m}(G_{27})}(ϕ^{p}), which agrees with the limit in (6.39b).

In the case that p = 15m/2, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{8}w_{\frac{m}{2} +1}c^{−8}, c^{8}w_{\frac{m}{2} +2}c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{1}c^{−7}, …, c^{7}w\frac{m}{2} c^{−7}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{8}w_{\frac{m}{2} +i}c^{−8}, i = 1, 2, …, \frac{m}{2},  (6.43a)
w_{i} = c^{7}w_{i− \frac{m}{2}} c^{−7}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.43b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1.

Writing t_{1}, t_{2} for w_{i}, w_{i+ \frac{m}{2}} , respectively, the equations (6.43) reduce to

t_{1} = c^{8}t_{2}c^{−8},  (6.44a)
t_{2} = c^{7}t_{1}c^{−7}.  (6.44b)

One of these equations is in fact superfluous: if we substitute (6.44b) in (6.44a), then we obtain t_{1} = c^{15}t_{1}c^{−15} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{27}), we must have t_{1}t_{2} ≤_{T} c. Combining this with (6.44), we infer that

t_{1}(c^{7}t_{1}c^{−7}) ≤_{T} c.  (6.45)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.42). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/2, and, consequently, the number of solutions to (6.45) is the same, namely \frac{5m+2}{2}, as required.

Finally, we turn to (6.39d). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 6 and m_{2} = 3, h_{2} = 6 and m_{2} = 2, h_{2} = m_{2} = 3, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 5m/3, 5m/2, 10m/3, respectively 15m/2, out of which only p = 5m/3 and p = 10m/3 have not yet been discussed and belong to the current case. If p = 5m/3, the corresponding action of ϕ^{p} is given by (6.6), so that we have to solve for t_{1} with ℓ_{T} (t_{1}) = 1 in the equation (6.9). A computation with the help of CHEVIE finds no solution. If p = 10m/3, the corresponding action of ϕ^{p} is given by (6.11), so that we have to solve for t_{1} with ℓ_{T} (t_{1}) in the equation (6.14). Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with the one in (6.9). Hence, in both cases, the left-hand side of (3.3) is equal to 1, as required.

Case G_{28} = F_{4}. The degrees are 2, 6, 8, 12, and hence we have

Cat^{m}(F_{4}; q) = \frac{[12m + 12]_{q} [12m + 8]_{q} [12m + 6]_{q} [12m + 2]_{q}}{[12]_{q} [8]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 12m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(F_{4}; q) = m + 1, if ζ = ζ_{12},  (6.46a)
lim_{q→ζ} Cat^{m}(F_{4}; q) = \frac{3m+2}{2}, if ζ = ζ_{8}, 2 | m,  (6.46b)
lim_{q→ζ} Cat^{m}(F_{4}; q) = (m + 1)(2m + 1), if ζ = ζ_{6}, ζ_{3},  (6.46c)
lim_{q→ζ} Cat^{m}(F_{4}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{4},  (6.46d)
lim_{q→ζ} Cat^{m}(F_{4}; q) = Cat^{m}(F_{4}), if ζ = −1 or ζ = 1,  (6.46e)
lim_{q→ζ} Cat^{m}(F_{4}; q) = 1, otherwise.  (6.46f)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.46). The only cases not covered by Lemmas 27 and 28 are the ones in (6.46b) and (6.46f). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.47a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.47b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.48a)

(iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.48b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.48c)

and all other w_{j}’s are equal to ε.

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(F_{4}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.47)–(6.48), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(cw_{i}c^{−1}) = c,  (6.49)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(cw_{i}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.50)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{1}}(cw_{i_{1}}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = 1.  (6.51)

With the help of Stembridge’s Maple package coxeter [35], one obtains three solutions for w_{i} in (6.49):

w_{i} ∈ {[1, 2, 3, 4, 3, 2], [2, 3], [1, 3, 2, 1, 3, 4]},

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}} being a simple system of generators of F_{4}, corresponding to the Dynkin diagram displayed in Figure 2. Each of the above solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(F_{4})}(ϕ^{p}) since i ranges from 1 to m/2.

• • • •_{1} _{2} _{3} _{4}^{4}

Figure 2. The Dynkin diagram for F_{4}

There are no solutions for w_{i} in (6.50) and for w_{i_{1}} in (6.51).

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(F_{4})}(ϕ^{p}), which agrees with the limit in (6.46b).

Finally, we turn to (6.46f). By Remark 3, the are no choices for h_{2} and m_{2} to be considered. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{29}. The degrees are 4, 8, 12, 20, and hence we have

Cat^{m}(G_{29}; q) = \frac{[20m + 20]_{q} [20m + 12]_{q} [20m + 8]_{q} [20m + 4]_{q}}{[20]_{q} [12]_{q} [8]_{q} [4]_{q}} .

Let ζ be a 20m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{29}; q) = m + 1, if ζ = ζ_{20}, ζ_{10}, ζ_{5},  (6.52a)
lim_{q→ζ} Cat^{m}(G_{29}; q) = \frac{5m+3}{3}, if ζ = ζ_{12}, ζ_{6}, ζ_{3}, 3 | m,  (6.52b)
lim_{q→ζ} Cat^{m}(G_{29}; q) = \frac{5m+2}{2}, if ζ = ζ_{8}, 2 | m,  (6.52c)
lim_{q→ζ} Cat^{m}(G_{29}; q) = Cat^{m}(G_{29}), if ζ = ζ_{4}, −1, 1,  (6.52d)
lim_{q→ζ} Cat^{m}(G_{29}; q) = 1, otherwise.  (6.52e)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.52). The only cases not covered by Lemmas 27 and 28 are the ones in (6.52b), (6.52c), and (6.52e).

We begin with the case in (6.52b). By Lemma 26, we are free to choose p = 5m/3 if ζ = ζ_{12}, we are free to choose p = 10m/3 if ζ = ζ_{6}, we are free to choose p = 20m/3 if ζ = ζ_{3}. In particular, in all three cases, m must be divisible by 3.

We start with the case that p = 5m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{3} +1}c^{−2}, c^{2}w_{\frac{m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{3} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{3} +i}c^{−2}, i = 1, 2, …, \frac{2m}{3},  (6.53a)
w_{i} = cw_{i− \frac{2m}{3}} c^{−1}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.53b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1,  (6.54a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{29}), we must have w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c.

Together with equations (6.53)–(6.54), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) = c.  (6.55)

With the help of CHEVIE, one obtains five solutions for w_{i} in (6.55):

w_{i} ∈ {[1], [2], [8], [25], [31]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{29} in CHEVIE. Each of the above solutions for w_{i} gives rise to m/3 elements of Fix_{NC^{m}(G_{29})}(ϕ^{p}) since i ranges from 1 to m/3.

In total, we obtain 1 + 5\frac{m}{3} = \frac{5m+3}{3} elements in Fix_{NC^{m}(G_{29})}(ϕ^{p}), which agrees with the limit in (6.52b).

In the case that p = 10m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{3} +1}c^{−4}, c^{4}w_{\frac{2m}{3} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{3}} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{3} +i}c^{−4}, i = 1, 2, …, \frac{m}{3},  (6.56a)
w_{i} = c^{3}w_{i− \frac{m}{3}} c^{−3}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.56b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.56) reduce to

t_{1} = c^{4}t_{3}c^{−4},  (6.57a)
t_{2} = c^{3}t_{1}c^{−3},  (6.57b)
t_{3} = c^{3}t_{2}c^{−3}.  (6.57c)

One of these equations is in fact superfluous: if we substitute (6.57b) and (6.57c) in (6.57a), then we obtain t_{1} = c^{10}t_{1}c^{−10} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{29}), we must have t_{1}t_{2}t_{3} ≤_{T} c. Combining this with (6.57), we infer that

t_{1}(c^{3}t_{1}c^{−3})(c^{6}t_{1}c^{−6}) ≤_{T} c.  (6.58)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 4, we see that this equation is equivalent with (6.55). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/3, and, consequently, the number of solutions to (6.58) is the same, namely \frac{5m+3}{3}, as required.

In the case that p = 20m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{7}w_{\frac{m}{3} +1}c^{−7}, c^{7}w_{\frac{m}{3} +2}c^{−7}, …, c^{7}w_{m}c^{−7}, c^{6}w_{1}c^{−6}, …, c^{6}w\frac{m}{3} c^{−6}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{7}w_{\frac{m}{3} +i}c^{−7}, i = 1, 2, …, \frac{2m}{3},  (6.59a)
w_{i} = c^{6}w_{i− \frac{2m}{3}} c^{−6}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.59b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.59) reduce to

t_{1} = c^{7}t_{2}c^{−7},  (6.60a)
t_{2} = c^{7}t_{3}c^{−7},  (6.60b)
t_{3} = c^{6}t_{1}c^{−6}.  (6.60c)

One of these equations is in fact superfluous: if we substitute (6.60b) and (6.60c) in (6.60a), then we obtain t_{1} = c^{20}t_{1}c^{−20} which is automatically satisfied since c^{20} = ε.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{29}), we must have t_{1}t_{2}t_{3} ≤_{T} c. Combining this with (6.60), we infer that

t_{1}(c^{13}t_{1}c^{−13})(c^{6}t_{1}c^{−6}) ≤_{T} c.  (6.61)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 4, we see that this equation is equivalent with (6.55). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/3, and, consequently, the number of solutions to (6.61) is the same, namely \frac{5m+3}{3}, as required.

Next we discuss the case in (6.52c). By Lemma 26, we are free to choose p = 5m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{2} +i}c^{−3}, i = 1, 2, …, \frac{m}{2},  (6.62a)
w_{i} = c^{2}w_{i− \frac{m}{2}} c^{−2}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.62b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.63a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.63b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.63c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{29}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.62)–(6.63), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(c^{2}w_{i}c^{−2}) = c,  (6.64)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(c^{2}w_{i}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.65)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{2}} = c^{3}w_{i_{2}}c^{−3},

w_{i_{1}}w_{i_{2}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2}) = c, and ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = 1. (6.66)

With the help of CHEVIE, one obtains five solutions for w_{i} in (6.65):

w_{i} ∈ {[4], [9], [14], [27], [32]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{29} in CHEVIE. Each of these solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(G_{29})}(ϕ^{p}) since i ranges from 1 to m/2.

There are no solutions for w_{i} in (6.64) and for (w_{i_{1}}, w_{i_{2}}) in (6.66).

In total, we obtain 1 + 5\frac{m}{2} = \frac{5m+2}{2} elements in Fix_{NC^{m}(G_{29})}(ϕ^{p}), which agrees with the limit in (6.52c).

Finally, we turn to (6.52e). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 3, h_{2} = 2 and m_{2} = 3, h_{2} = 4 and m_{2} = 2, h_{2} = 4 and m_{2} = 3, respectively h_{2} = m_{2} = 4. These correspond to the choices p = 20m/3, p = 10m/3, p = 5m/2, p = 5m/3, respectively p = 5m/4, out of which only p = 5m/4 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{3m}{4} +1}c^{−2}, c^{2}w_{\frac{3m}{4} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{3m}{4}} c^{−1}),

so that we have to solve

t_{1}(ct_{1}c^{−1})(c^{2}t_{1}c^{−2})(c^{3}t_{1}c^{−3}) = c

for t_{1} with ℓ_{T} (t_{1}). A computation with the help of CHEVIE finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{30} = H_{4}. The degrees are 2, 12, 20, 30, and hence we have

Cat^{m}(H_{4}; q) = \frac{[30m + 30]_{q} [30m + 20]_{q} [30m + 12]_{q} [30m + 2]_{q}}{[30]_{q} [20]_{q} [12]_{q} [2]_{q}} .

Let ζ be a 30m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(H_{4}; q) = m + 1, if ζ = ζ_{30}, ζ_{15},  (6.67a)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{3m+2}{2}, if ζ = ζ_{20}, 2 | m,  (6.67b)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{5m+2}{2}, if ζ = ζ_{12}, 2 | m,  (6.67c)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{10}, ζ_{5},  (6.67d)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(m+1)(5m+2)}{2}, if ζ = ζ_{6}, ζ_{3},  (6.67e)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(3m+2)(5m+2)}{4}, if ζ = ζ_{4}, 2 | m,  (6.67f)
lim_{q→ζ} Cat^{m}(H_{4}; q) = Cat^{m}(H_{4}), if ζ = −1 or ζ = 1,  (6.67g)
lim_{q→ζ} Cat^{m}(H_{4}; q) = 1, otherwise.  (6.67h)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.67). The only cases not covered by Lemmas 27 and 28 are the ones in (6.67b), (6.67c), (6.67f), and (6.67h).

We begin with the case in (6.67b). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.68a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.68b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.69a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.69b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.69c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(H_{4}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.68)–(6.69), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(cw_{i}c^{−1}) = c,  (6.70)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(cw_{i}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.71)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{1}}(cw_{i_{1}}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = 1.  (6.72)

With the help of Stembridge’s Maple package coxeter [35], one obtains three solutions for w_{i} in (6.70):

w_{i} ∈ {[1, 2, 3, 4, 3, 2, 1, 2], [2, 1, 2, 3], [1, 3, 2, 1, 2, 1, 3, 4]},

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}} being a simple system of generators of H_{4}, corresponding to the Dynkin diagram displayed in Figure 3. Each of the above solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(H_{4})}(ϕ^{p}) since i ranges from 1 to m/2.

• • • •_{1} _{2} _{3} _{4}^{5}

Figure 3. The Dynkin diagram for H_{4}

There are no solutions for w_{i} in (6.71) and for w_{i_{1}} in (6.72).

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(H_{4})}(ϕ^{p}), which agrees with the limit in (6.67b).

Next we discuss the case in (6.67c). By Lemma 26, we are free to choose p = 5m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{2} +i}c^{−3}, i = 1, 2, …, \frac{m}{2},  (6.73a)
w_{i} = c^{2}w_{i− \frac{m}{2}} c^{−2}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.73b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.74a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.74b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.74c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(H_{4}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.73)–(6.74), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(c^{2}w_{i}c^{−2}) = c,  (6.75)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(c^{2}w_{i}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.76)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{1}}(c^{2}w_{i_{1}}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = 1.  (6.77)

With the help of Stembridge’s Maple package coxeter [35], one obtains five solutions for w_{i} in (6.75):

w_{i} ∈ {[1, 3, 2, 1, 2, 1, 3, 2], [1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3], [2, 1, 2, 3, 2, 1, 2, 4], [2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 4], [1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3]},

where we used again coxeter’s short notation, {s_{1}, s_{2}, s_{3}, s_{4}} being a simple system of generators of H_{4}, corresponding to the Dynkin diagram displayed in Figure 3. Each of these solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(H_{4})}(ϕ^{p}) since i ranges from 1 to m/2.

There are no solutions for w_{i} in (6.76) and for w_{i_{1}} in (6.77).

In total, we obtain 1 + 5\frac{m}{2} = \frac{5m+2}{2} elements in Fix_{NC^{m}(H_{4})}(ϕ^{p}), which agrees with the limit in (6.67c).

Finally we discuss the case in (6.67f). By Lemma 26, we are free to choose p = 15m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{8}w_{\frac{m}{2} +1}c^{−8}, c^{8}w_{\frac{m}{2} +2}c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{1}c^{−7}, …, c^{7}w\frac{m}{2} c^{−7}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{8}w_{\frac{m}{2} +i}c^{−8}, i = 1, 2, …, \frac{m}{2},  (6.78a)
w_{i} = c^{7}w_{i− \frac{m}{2}} c^{−7}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.78b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.79a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.79b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.79c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(H_{4}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.78)–(6.79), this implies that

w_{i} = c^{15}w_{i}c^{−15} and w_{i}(c^{7}w_{i}c^{−7}) = c,  (6.80)

respectively that

w_{i} = c^{15}w_{i}c^{−15}, w_{i}(c^{7}w_{i}c^{−7}) = c, and ℓ_{T} (w_{i}) = 1,  (6.81)

respectively that

w_{i_{1}} = c^{15}w_{i_{1}}c^{−15}, w_{i_{2}} = c^{15}w_{i_{2}}c^{−15}, w_{i_{1}}w_{i_{2}}(c^{7}w_{i_{1}}c^{−7})(c^{7}w_{i_{2}}c^{−7}) = c.  (6.82)

Here, the first equations in both (6.80) and (6.81), and the first two equations in (6.82) are automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains eight solutions for w_{i} in (6.80):

w_{i} ∈ {[1, 3, 2, 1, 2, 1, 3, 2], [1, 2, 3, 4, 3, 2, 1, 2], [2, 1, 2, 3],

[1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3], [1, 3, 2, 1, 2, 1, 3, 4], [2, 1, 2, 3, 2, 1, 2, 4], [2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 4], [1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3]}, (6.83)

where {s_{1}, s_{2}, s_{3}, s_{4}} is a simple system of generators of H_{4}, corresponding to the Dynkin diagram displayed in Figure 3, and each of them gives rise to m/2 elements of Fix_{NC^{m}(H_{4})}(ϕ^{p}) since i ranges from 1 to m/2. Furthermore, one obtains 15 solutions for w_{i} in (6.81):

w_{i} ∈ {[2], [3], [4], [2, 1, 2], [3, 2, 1, 2, 3], [1, 3, 2, 1, 2, 1, 3], [2, 1, 2, 3, 2, 1, 2], [1, 2, 3, 4, 3, 2, 1],

[2, 1, 3, 2, 1, 2, 1, 3, 2], [1, 4, 3, 2, 1, 2, 1, 3, 4], [2, 1, 2, 3, 4, 3, 2, 1, 2],

[1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4], [1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 3], [2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 2, 4], [1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(H_{4})}(ϕ^{p}) since i ranges from 1 to m/2, and one obtains 30 pairs (w_{i_{1}}, w_{i_{2}}) of solutions in (6.82):

(w_{i_{1}}, w_{i_{2}}) ∈ {([2], [2, 1, 3, 2, 1, 2, 1, 3, 2]), ([2], [2, 1, 2, 3, 4, 3, 2, 1, 2]), ([3], [3, 2, 1, 2, 3]),

([3], [1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3]), ([4], [1, 4, 3, 2, 1, 2, 1, 3, 4]), ([4], [2, 1, 2, 3, 4, 3, 2, 1, 2]), ([2, 1, 2], [3]), ([2, 1, 2], [1, 4, 3, 2, 1, 2, 1, 3, 4]),

([3, 2, 1, 2, 3], [2, 1, 3, 2, 1, 2, 1, 3, 2]), ([3, 2, 1, 2, 3], [1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 3]),

([1, 3, 2, 1, 2, 1, 3], [2]), ([1, 3, 2, 1, 2, 1, 3], [4]), ([2, 1, 2, 3, 2, 1, 2], [4]), ([2, 1, 2, 3, 2, 1, 2], [2, 1, 2]), ([1, 2, 3, 4, 3, 2, 1], [2]), ([1, 2, 3, 4, 3, 2, 1], [3, 2, 1, 2, 3]), ([2, 1, 3, 2, 1, 2, 1, 3, 2], [1, 3, 2, 1, 2, 1, 3]), ([2, 1, 3, 2, 1, 2, 1, 3, 2], [2, 1, 2, 3, 2, 1, 2]),

([1, 4, 3, 2, 1, 2, 1, 3, 4], [2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 2, 4]), ([1, 4, 3, 2, 1, 2, 1, 3, 4], [1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3]),

([2, 1, 2, 3, 4, 3, 2, 1, 2], [2, 1, 2, 3, 2, 1, 2]), ([2, 1, 2, 3, 4, 3, 2, 1, 2], [2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 2, 4]),

([1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4], [3]),

([1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4], [1, 2, 3, 4, 3, 2, 1]),
([1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 3], [1, 3, 2, 1, 2, 1, 3]),
([1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 3], [1, 2, 3, 4, 3, 2, 1]),
([2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 2, 4], [2, 1, 2]),

([2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 2, 4], [1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4]), ([1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3], [1, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4]),

([1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3], [1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 3])}, (6.84)

each of them giving rise to \binom{m/2}{2} elements of Fix_{NC^{m}(H_{4})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2}.

In total, we obtain 1+(15+8)\frac{m}{2} +30\binom{m/2}{2} = \frac{(3m+2)(5m+2)}{4} elements in Fix_{NC^{m}(H_{4})}(ϕ^{p}), which agrees with the limit in (6.67f).

Finally, we turn to (6.67h). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 15m/2, respectively p = 15m/4, out of which only p = 15m/4 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{m}{4} +1}c^{−4}, c^{4}w_{\frac{m}{4} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w\frac{m}{4} c^{−3}), so that we have to solve

t_{1}(c^{11}t_{1}c^{−11})(c^{7}t_{1}c^{−7})(c^{3}t_{1}c^{−3}) = c

for t_{1} with ℓ_{T} (t_{1}). A computation with Stembridge’s Maple package coxeter [35] finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{32}. The degrees are 12, 18, 24, 30, and hence we have

Cat^{m}(G_{32}; q) = \frac{[30m + 30]_{q} [30m + 24]_{q} [30m + 18]_{q} [30m + 12]_{q}}{[30]_{q} [24]_{q} [18]_{q} [12]_{q}} .

Let ζ be a 30m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{32}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (6.85a)
lim_{q→ζ} Cat^{m}(G_{32}; q) = \frac{5m+4}{4}, if ζ = ζ_{24}, ζ_{8}, 4 | m,  (6.85b)
lim_{q→ζ} Cat^{m}(G_{32}; q) = \frac{5m+3}{3}, if ζ = ζ_{18}, ζ_{9}, 3 | m,  (6.85c)
lim_{q→ζ} Cat^{m}(G_{32}; q) = \frac{(5m+4)(5m+2)}{8}, if ζ = ζ_{12}, ζ_{4}, 2 | m,  (6.85d)
lim_{q→ζ} Cat^{m}(G_{32}; q) = Cat^{m}(G_{32}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (6.85e)
lim_{q→ζ} Cat^{m}(G_{32}; q) = 1, otherwise.  (6.85f)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.85). The only cases not covered by Lemmas 27 and 28 are the ones in (6.85b), (6.85c), (6.85d), and (6.85f).

We begin with the case in (6.85b). By Lemma 26, we are free to choose p = 5m/4 if ζ = ζ_{24}, and we are free to choose p = 15m/4 if ζ = ζ_{8}. In particular, in all both cases, m must be divisible by 4.

We start with the case that p = 5m/4. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{3m}{4} +1}c^{−2}, c^{2}w_{\frac{3m}{4} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{3m}{4}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{3m}{4} +i}c^{−2}, i = 1, 2, …, \frac{m}{4},  (6.86a)
w_{i} = cw_{i− \frac{m}{4}} c^{−1}, i = \frac{m}{4} + 1, \frac{m}{4} + 2, …, m.  (6.86b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{4} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) = 1,  (6.87a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{32}), we must have

w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} = c.

Together with equations (6.86)–(6.87), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(cw_{i}c^{−1})(c^{2}w_{i}c^{−2})(c^{3}w_{i}c^{−3}) = c.  (6.88)

With the help of CHEVIE, one obtains five solutions for w_{i} in (6.88):

w_{i} ∈ {[1], [2], [3], [4], [27]},  (6.89)

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{32} in CHEVIE. Each of the above solutions for w_{i} gives rise to m/4 elements of Fix_{NC^{m}(G_{32})}(ϕ^{p}) since i ranges from 1 to m/4.

In total, we obtain 1 + 5\frac{m}{4} = \frac{5m+4}{4} elements in Fix_{NC^{m}(G_{32})}(ϕ^{p}), which agrees with the limit in (6.85b).

In the case that p = 15m/4, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{m}{4} +1}c^{−4}, c^{4}w_{\frac{m}{4} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w\frac{m}{4} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{m}{4} +i}c^{−4}, i = 1, 2, …, \frac{3m}{4},  (6.90a)
w_{i} = c^{3}w_{i− \frac{3m}{4}} c^{−3}, i = \frac{3m}{4} + 1, \frac{3m}{4} + 2, …, m.  (6.90b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{4} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) = 1.

Writing t_{1}, t_{2}, t_{3}, t_{4} for w_{i}, w_{i+ \frac{m}{4}} , w_{i+ \frac{2m}{4}} , w_{i+ \frac{3m}{4}} , respectively, the equations (6.90) reduce to

t_{1} = c^{4}t_{2}c^{−4},  (6.91a)
t_{2} = c^{4}t_{3}c^{−4},  (6.91b)
t_{3} = c^{4}t_{4}c^{−4},  (6.91c)
t_{4} = c^{3}t_{1}c^{−3}.  (6.91d)

One of these equations is in fact superfluous: if we substitute (6.91b)–(6.91d) in (6.91a), then we obtain t_{1} = c^{15}t_{1}c^{−15} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{32}), we must have t_{1}t_{2}t_{3}t_{4} = c. Combining this with (6.91), we infer that

t_{1}(c^{11}t_{1}c^{−11})(c^{7}t_{1}c^{−7})(c^{3}t_{1}c^{−3}) = c.  (6.92)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.88). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/4, and, consequently, the number of solutions to (6.92) is the same, namely \frac{5m+4}{4}, as required.

Next we consider the case in (6.85c). By Lemma 26, we are free to choose p = 5m/3 if ζ = ζ_{18}, and we are free to choose p = 10m/3 if ζ = ζ_{9}. In particular, in both cases, m must be divisible by 3.

We start with the case that p = 5m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{3} +1}c^{−2}, c^{2}w_{\frac{m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{3} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{3} +i}c^{−2}, i = 1, 2, …, \frac{2m}{3},  (6.93a)
w_{i} = cw_{i− \frac{2m}{3}} c^{−1}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.93b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1,  (6.94a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{32}), we must have w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c.

Together with equations (6.93)–(6.94), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) ≤_{T} c.  (6.95)

With the help of CHEVIE, one obtains three solutions for w_{i} in (6.95):

w_{i} ∈ {[1], [2], [3], [4], [27]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{32} in CHEVIE. Each of the above solutions for w_{i} gives rise to m/3 elements of Fix_{NC^{m}(G_{32})}(ϕ^{p}) since i ranges from 1 to m/3.

In total, we obtain 1 + 5\frac{m}{3} = \frac{5m+3}{3} elements in Fix_{NC^{m}(G_{32})}(ϕ^{p}), which agrees with the limit in (6.85c).

In the case that p = 10m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{3} +1}c^{−4}, c^{4}w_{\frac{2m}{3} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{3}} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{3} +i}c^{−4}, i = 1, 2, …, \frac{m}{3},  (6.96a)
w_{i} = c^{3}w_{i− \frac{m}{3}} c^{−3}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.96b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1.

Writing t_{1}, t_{2}, t_{3} for w_{i}, w_{i+ \frac{m}{3}} , w_{i+ \frac{2m}{3}} , respectively, the equations (6.96) reduce to

t_{1} = c^{4}t_{3}c^{−4},  (6.97a)
t_{2} = c^{3}t_{1}c^{−3},  (6.97b)
t_{3} = c^{3}t_{2}c^{−3}.  (6.97c)

One of these equations is in fact superfluous: if we substitute (6.97b) and (6.97c) in (6.97a), then we obtain t_{1} = c^{10}t_{1}c^{−10} which is automatically satisfied due to Lemma 29 with d = 2.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{32}), we must have t_{1}t_{2}t_{3} ≤_{T} c. Combining this with (6.97), we infer that

t_{1}(c^{3}t_{1}c^{−3})(c^{6}t_{1}c^{−6}) ≤_{T} c.  (6.98)

Using that c^{5}t_{1}c^{−5} = t_{1}, due to Lemma 29 with d = 4, we see that this equation is equivalent with (6.95). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/3, and, consequently, the number of solutions to (6.98) is the same, namely \frac{5m+3}{3}, as required.

Next we discuss the case in (6.85d). By Lemma 26, we are free to choose p = 5m/2 if ζ = ζ_{12}, and we are free to choose p = 15m/2 if ζ = ζ_{4}. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}). (6.99)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{2} +i}c^{−3}, i = 1, 2, …, \frac{m}{2},  (6.100a)
w_{i} = c^{2}w_{i− \frac{m}{2}} c^{−2}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.100b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.101a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.101b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.101c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{32}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} = c. Together with equations (6.100)–(6.101), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(c^{2}w_{i}c^{−2}) = c,  (6.102)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(c^{2}w_{i}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.103)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{2}} = c^{3}w_{i_{2}}c^{−3},

w_{i_{1}}w_{i_{2}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2}) = c, and ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = 1. (6.104)

With the help of CHEVIE, one obtains ten solutions for w_{i} in (6.102):

w_{i} ∈ {[3, 4], [1, 20], [4, 7], [1, 2], [2, 3], [4, 27], [3, 23], [1, 23], [2, 16], [12, 27]},  (6.105)

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{32} in CHEVIE, one obtains solutions for w_{i} in (6.103):

w_{i} ∈ {[4] [3] [20] [1] [7] [2] [16] [12] [27] [23]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{32})}(ϕ^{p}) since i ranges from 1 to m/2, and one obtains 25 pairs (w_{i_{1}}, w_{i_{2}}) satisfying (6.104):

(w_{i_{1}}, w_{i_{2}}) ∈ {([4], [20]), ([4], [7]), ([4], [27]), ([3], [4]), ([3], [12]), ([3], [23]), ([20], [3]), ([20], [1]), ([1], [20]), ([1], [2]), ([1], [23]), ([7], [4]), ([7], [1]), ([2], [3]), ([2], [7]),

([2], [16]), ([16], [4]), ([16], [2]), ([12], [2]), ([12], [27]), ([27], [1]), ([27], [16]),

([27], [12]), ([23], [3]), ([23], [27])}, (6.106)

each of them giving rise to \binom{m/2}{2} elements of Fix_{NC^{m}(G_{32})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2}.

In total, we obtain 1 + 20\frac{m}{2} + 25\binom{m/2}{2} = \frac{(5m+4)(5m+2)}{8} elements in Fix_{NC^{m}(G_{32})}(ϕ^{p}), which agrees with the limit in (6.85d).

If p = 15m/2, then, from (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{8}w_{\frac{m}{2} +1}c^{−8}, c^{8}w_{\frac{m}{2} +2}c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{1}c^{−7}, …, c^{7}w\frac{m}{2} c^{−7}).

Using that c^{5}wc^{−5} = w for all w ∈ NC(G_{32}, due to Lemma 29 with d = 6, we see that this action is identical with the one in (6.99). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/2, and, consequently, the number of elements in Fix_{NC^{m}(G_{32})}(ϕ^{p}) is the same, namely \frac{(5m+4)(5m+2)}{8}, as required.

Finally, we turn to (6.85f). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 2 and m_{2} = 4, h_{2} = m_{2} = 2, h_{2} = m_{2} = 3, h_{2} = 6 and m_{2} = 4, h_{2} = 6 and m_{2} = 3, respectively h_{2} = 6 and m_{2} = 2. These correspond to the choices p = 15m/4, p = 15m/2, p = 10m/3, p = 5m/4, p = 5m/3, respectively p = 5m/2, all of which have already been discussed as they do not belong to (6.85f). Hence, (3.3) must necessarily hold, as required.

Case G_{33}. The degrees are 4, 6, 10, 12, 18, and hence we have

Cat^{m}(G_{33}; q) = \frac{[18m + 18]_{q} [18m + 12]_{q} [18m + 10]_{q} [18m + 6]_{q} [18m + 4]_{q}}{[18]_{q} [12]_{q} [10]_{q} [6]_{q} [4]_{q}} .

Let ζ be a 18m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{33}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (6.107a)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{3m+2}{2}, if ζ = ζ_{12}, 2 | m,  (6.107b)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{9m+5}{5}, if ζ = ζ_{10}, ζ_{5}, 5 | m,  (6.107c)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{(m+1)(3m+2)(3m+1)}{2}, if ζ = ζ_{6}, ζ_{3},  (6.107d)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{(3m+2)(9m+2)}{4}, if ζ = ζ_{4}, 2 | m,  (6.107e)
lim_{q→ζ} Cat^{m}(G_{33}; q) = Cat^{m}(G_{33}), if ζ = −1 or ζ = 1,  (6.107f)
lim_{q→ζ} Cat^{m}(G_{33}; q) = 1, otherwise.  (6.107g)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.107). The only cases not covered by Lemmas 27 and 28 are the ones in (6.107b), (6.107c), (6.107e), and (6.107g).

We begin with the case in (6.107b). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.108a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.108b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.109a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.109b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.109c)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{33}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c. Together with equations (6.108)–(6.109), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(cw_{i}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i}) = 2,  (6.110)

respectively that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(cw_{i}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.111)

respectively that

w_{i_{1}} = c^{3}w_{i_{1}}c^{−3}, w_{i_{1}}(cw_{i_{1}}c^{−1}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = 1.  (6.112)

With the help of CHEVIE, one obtains three solutions for w_{i} in (6.110):

w_{i} ∈ {[1, 44], [2, 4], [5, 42]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{33} in CHEVIE. Each of them gives rise to m/2 elements of Fix_{NC^{m}(G_{33})}(ϕ^{p}) since i ranges from 1 to m/2.

There are no solutions to (6.111) and to (6.112).

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(G_{33})}(ϕ^{p}), which agrees with the limit in (6.107b).

Next we turn to the case in (6.107c). By Lemma 26, we are free to choose p = 9m/5 if ζ = ζ_{10}, and we are free to choose p = 18m/5 if ζ = ζ_{5}. In particular, in all both cases, m must be divisible by 5.

We start with the case that p = 9m/5. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{m}{5} +1}c^{−2}, c^{2}w_{\frac{m}{5} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{5} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{5} +i}c^{−2}, i = 1, 2, …, \frac{4m}{5},  (6.113a)
w_{i} = cw_{i− \frac{4m}{5}} c^{−1}, i = \frac{4m}{5} + 1, \frac{4m}{5} + 2, …, m.  (6.113b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1,  (6.114a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε.

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{33}), we must have

w_{i}w_{i+ \frac{m}{5}} w_{i+ \frac{2m}{5}} w_{i+ \frac{3m}{5}} w_{i+ \frac{4m}{5}} = c.

Together with equations (6.113)–(6.114), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{7}w_{i}c^{−7})(c^{5}w_{i}c^{−5})(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) = c.  (6.115)

With the help of CHEVIE, one obtains nine solutions for w_{i} in (6.115):

w_{i} ∈ {[5], [4], [1], [2], [10], [27], [42], [44], [52]},  (6.116)

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{33} in CHEVIE. Each of the above solutions for w_{i} gives rise to m/5 elements of Fix_{NC^{m}(G_{33})}(ϕ^{p}) since i ranges from 1 to m/5.

In total, we obtain 1 + 9\frac{m}{5} = \frac{9m+5}{5} elements in Fix_{NC^{m}(G_{33})}(ϕ^{p}), which agrees with the limit in (6.107c).

In the case that p = 18m/5, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{5} +1}c^{−4}, c^{4}w_{\frac{2m}{5} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{5}} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{5} +i}c^{−4}, i = 1, 2, …, \frac{3m}{5},  (6.117a)
w_{i} = c^{3}w_{i− \frac{3m}{5}} c^{−3}, i = \frac{3m}{5} + 1, \frac{3m}{5} + 2, …, m.  (6.117b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1.

Writing t_{1}, t_{2}, t_{3}, t_{4}, t_{5} for w_{i}, w_{i+ \frac{m}{5}} , w_{i+ \frac{2m}{5}} , w_{i+ \frac{3m}{5}} , w_{i+ \frac{4m}{5}} , respectively, the equations (6.117) reduce to

t_{1} = c^{4}t_{3}c^{−4},  (6.118a)
t_{2} = c^{4}t_{4}c^{−4},  (6.118b)
t_{3} = c^{4}t_{5}c^{−4},  (6.118c)
t_{4} = c^{3}t_{1}c^{−3},  (6.118d)
t_{5} = c^{3}t_{2}c^{−3}.  (6.118e)

One of these equations is in fact superfluous: if we substitute (6.118b)–(6.118e) in (6.118a), then we obtain t_{1} = c^{18}t_{1}c^{−18} which is automatically satisfied since c^{18} = ε.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{33}), we must have t_{1}t_{2}t_{3}t_{4}t_{5} = c. Combining this with (6.118), we infer that

t_{1}(c^{7}t_{1}c^{−7})(c^{14}t_{1}c^{−14})(c^{3}t_{1}c^{−3})(c^{10}t_{1}c^{−10}) = c.  (6.119)

Using that c^{9}t_{1}c^{−9} = t_{1}, due to Lemma 29 with d = 2, we see that this equation is equivalent with (6.115). Therefore, we are facing exactly the same enumeration problem here as for p = 9m/5, and, consequently, the number of solutions to (6.119) is the same, namely \frac{9m+5}{5}, as required.

Next we consider the case in (6.107e). By Lemma 26, we are free to choose p = 9m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{m}{2} +1}c^{−5}, c^{5}w_{\frac{m}{2} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w\frac{m}{2} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{m}{2} +i}c^{−5}, i = 1, 2, …, \frac{m}{2},  (6.120a)
w_{i} = c^{4}w_{i− \frac{m}{2}} c^{−4}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.120b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.121a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.121b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.121c)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{33}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c. Together with equations (6.120)–(6.121), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{4}w_{i}c^{−4}) ≤_{T} c, and ℓ_{T} (w_{i}) = 2,  (6.122)

respectively that

w_{i} = c^{9}w_{i}c^{−9}, w_{i}(c^{4}w_{i}c^{−4}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.123)

respectively that

w_{i_{1}} = c^{9}w_{i_{1}}c^{−9}, w_{i_{2}} = c^{9}w_{i_{2}}c^{−9},

w_{i_{1}}w_{i_{2}}(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = 1. (6.124)

With the help of CHEVIE, one obtains 21 solutions for w_{i} in (6.122):

w_{i} ∈ {[4, 5], [1, 20], [5, 7], [1, 2], [2, 4], [10, 259], [27, 208], [27, 44], [4, 39], [5, 42], [1, 39], [5, 52], [10, 208], [1, 44], [42, 113], [4, 113], [2, 27], [10, 42], [2, 49], [52, 61], [44, 123]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{33} in CHEVIE, one obtains 18 solutions for w_{i} in (6.123):

w_{i} ∈ {[5], [4], [20], [208], [259], [1], [2], [7], [10], [27], [39], [42], [49], [44], [52], [113], [61], [123]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{33})}(ϕ^{p}) since i ranges from 1 to m/2, and one obtains 54 pairs (w_{i_{1}}, w_{i_{2}}) satisfying (6.124):

(w_{i_{1}}, w_{i_{2}}) ∈ {([5], [20]), ([5], [7]), ([5], [42]), ([5], [52]), ([4], [5]), ([4], [10]), ([4], [39]), ([4], [113]), ([20], [4]), ([20], [1]), ([208], [27]), ([208], [52]), ([259], [10]), ([259], [27]), ([1], [20]), ([1], [2]), ([1], [39]), ([1], [44]), ([2], [4]),

([2], [7]), ([2], [27]), ([2], [49]), ([7], [5]), ([7], [1]), ([10], [208]), ([10], [259]), ([10], [2]), ([10], [42]), ([27], [5]), ([27], [208]), ([27], [44]), ([27], [61]), ([39], [4]), ([39], [42]), ([42], [1]), ([42], [27]), ([42], [113]), ([42], [123]),

([49], [5]), ([49], [2]), ([44], [4]), ([44], [259]), ([44], [52]), ([44], [123]), ([52], [1]), ([52], [10]), ([52], [49]),

([52], [61]), ([113], [42]), ([113], [44]), ([61], [2]), ([61], [52]), ([123], [10]), ([123], [44])},

each of them giving rise to \binom{m/2}{2} elements of Fix_{NC^{m}(G_{33})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ m.

In total, we obtain 1+(21+18)\frac{m}{2}+54\binom{m/2}{2} = \frac{(3m+2)(9m+2)}{4} elements in Fix_{NC^{m}(G_{33})}(ϕ^{p}), which agrees with the limit in (6.107e).

Finally, we turn to (6.107g). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 18m/5, p = 9m/5, p = 9m/4, respectively p = 9m/2, out of which only p = 9m/4 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{3m}{4} +1}c^{−3}, c^{3}w_{\frac{3m}{4} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{3m}{4}} c^{−2}),

so that we have to solve

t_{1}(c^{2}t_{1}c^{−2})(c^{4}t_{1}c^{−4})(c^{6}t_{1}c^{−6}) ≤_{T} c

for t_{1} with ℓ_{T} (t_{1}). A computation with the help of CHEVIE finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{34}. The degrees are 6, 12, 18, 24, 30, 42, and hence we have

Cat^{m}(G_{34}; q) = \frac{[42m + 42]_{q} [42m + 30]_{q} [42m + 24]_{q}}{[42]_{q} [30]_{q} [24]_{q}}

× \frac{[42m + 18]_{q} [42m + 12]_{q} [42m + 6]_{q}}{[18]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 42m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(G_{34}; q) = m + 1, if ζ = ζ_{42}, ζ_{21}, ζ_{14}, ζ_{7},  (6.125a)
lim_{q→ζ} Cat^{m}(G_{34}; q) = \frac{7m+5}{5}, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5}, 5 | m,  (6.125b)
lim_{q→ζ} Cat^{m}(G_{34}; q) = \frac{7m+4}{4}, if ζ = ζ_{24}, ζ_{8}, 4 | m,  (6.125c)
lim_{q→ζ} Cat^{m}(G_{34}; q) = \frac{7m+3}{3}, if ζ = ζ_{18}, ζ_{9}, 3 | m,  (6.125d)
lim_{q→ζ} Cat^{m}(G_{34}; q) = \frac{(7m+4)(7m+2)}{8}, if ζ = ζ_{12}, ζ_{4}, 2 | m,  (6.125e)
lim_{q→ζ} Cat^{m}(G_{34}; q) = Cat^{m}(G_{34}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (6.125f)
lim_{q→ζ} Cat^{m}(G_{34}; q) = 1, otherwise.  (6.125g)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.125). The only cases not covered by Lemmas 27 and 28 are the ones in (6.125b), (6.125c), (6.125d), (6.125e), and (6.125g).

We begin with the case in (6.125b). By Lemma 26, we are free to choose p = 7m/5 if ζ = ζ_{30}, we are free to choose p = 14m/5 if ζ = ζ_{15}, we are free to choose p = 21m/5 if ζ = ζ_{10}, and we are free to choose p = 42m/5 if ζ = ζ_{5}. In particular, in all cases, m must be divisible by 5.

We start with the case that p = 7m/5. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{3m}{5} +1}c^{−2}, c^{2}w_{\frac{3m}{5} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{3m}{5}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{3m}{5} +i}c^{−2}, i = 1, 2, …, \frac{2m}{5},  (6.126a)
w_{i} = cw_{i− \frac{2m}{5}} c^{−1}, i = \frac{2m}{5} + 1, \frac{2m}{5} + 2, …, m.  (6.126b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1,  (6.127a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have

w_{i}w_{i+ \frac{m}{5}} w_{i+ \frac{2m}{5}} w_{i+ \frac{3m}{5}} w_{i+ \frac{4m}{5}} ≤_{T} c.

Together with equations (6.126)–(6.127), this implies that

w_{i} = c^{7}w_{i}c^{−7} and w_{i}(c^{4}w_{i}c^{−4})(cw_{i}c^{−1})(c^{5}w_{i}c^{−5})(c^{2}w_{i}c^{−2}) ≤_{T} c.  (6.128)

With the help of CHEVIE, one obtains seven solutions for w_{i} in (6.128):

w_{i} ∈ {[4], [5], [6], [2], [11], [44], [63], [74]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{34} in CHEVIE. Each of the above solutions for w_{i} gives rise to m/5 elements of Fix_{NC^{m}(G_{34})}(ϕ^{p}) since i ranges from 1 to m/5.

In total, we obtain 1 + 7\frac{m}{5} = \frac{7m+5}{5} elements in Fix_{NC^{m}(G_{34})}(ϕ^{p}), which agrees with the limit in (6.125b).

In the case that p = 14m/5, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{5} +1}c^{−3}, c^{3}w_{\frac{m}{5} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{5} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{5} +i}c^{−3}, i = 1, 2, …, \frac{4m}{5},  (6.129a)
w_{i} = c^{2}w_{i− \frac{4m}{5}} c^{−2}, i = \frac{4m}{5} + 1, \frac{4m}{5} + 2, …, m.  (6.129b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1.

Writing t_{1}, t_{2}, t_{3}, t_{4}, t_{5} for w_{i}, w_{i+ \frac{m}{5}} , w_{i+ \frac{2m}{5}} , w_{i+ \frac{3m}{5}} , w_{i+ \frac{4m}{5}} , respectively, the equations (6.129) reduce to

t_{1} = c^{3}t_{2}c^{−3},  (6.130a)
t_{2} = c^{3}t_{3}c^{−3},  (6.130b)
t_{3} = c^{3}t_{4}c^{−3},  (6.130c)
t_{4} = c^{3}t_{5}c^{−3},  (6.130d)
t_{5} = c^{2}t_{1}c^{−2}.  (6.130e)

One of these equations is in fact superfluous: if we substitute (6.130b)–(6.130e) in (6.130a), then we obtain t_{1} = c^{14}t_{1}c^{−14} which is automatically satisfied due to Lemma 29 with d = 3.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have t_{1}t_{2}t_{3}t_{4}t_{5} ≤_{T} c. Combining this with (6.130), we infer that

t_{1}(c^{11}t_{1}c^{−11})(c^{8}t_{1}c^{−8})(c^{5}t_{1}c^{−5})(c^{2}t_{1}c^{−2}) ≤_{T} c.  (6.131)

Using that c^{7}t_{1}c^{−7} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.128). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/5, and, consequently, the number of solutions to (6.131) is the same, namely \frac{7m+5}{5}, as required.

In the case that p = 21m/5, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{4m}{5} +1}c^{−5}, c^{5}w_{\frac{4m}{5} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w_{\frac{4m}{5}} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{4m}{5} +i}c^{−5}, i = 1, 2, …, \frac{m}{5},  (6.132a)
w_{i} = c^{4}w_{i− \frac{m}{5}} c^{−4}, i = \frac{m}{5} + 1, \frac{m}{5} + 2, …, m.  (6.132b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1.

Writing t_{1}, t_{2}, t_{3}, t_{4}, t_{5} for w_{i}, w_{i+ \frac{m}{5}} , w_{i+ \frac{2m}{5}} , w_{i+ \frac{3m}{5}} , w_{i+ \frac{4m}{5}} , respectively, the equations (6.132) reduce to

t_{1} = c^{5}t_{5}c^{−5},  (6.133a)
t_{2} = c^{4}t_{1}c^{−4},  (6.133b)
t_{3} = c^{4}t_{2}c^{−4},  (6.133c)
t_{4} = c^{4}t_{3}c^{−4},  (6.133d)
t_{5} = c^{4}t_{4}c^{−4}.  (6.133e)

One of these equations is in fact superfluous: if we substitute (6.133b)–(6.133e) in (6.133a), then we obtain t_{1} = c^{21}t_{1}c^{−21} which is automatically satisfied due to Lemma 29 with d = 6.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have t_{1}t_{2}t_{3}t_{4}t_{5} ≤_{T} c. Combining this with (6.133), we infer that

t_{1}(c^{4}t_{1}c^{−4})(c^{8}t_{1}c^{−8})(c^{12}t_{1}c^{−12})(c^{16}t_{1}c^{−16}) ≤_{T} c.  (6.134)

Using that c^{7}t_{1}c^{−7} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.128). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/5, and, consequently, the number of solutions to (6.134) is the same, namely \frac{7m+5}{5}, as required.

In the case that p = 42m/5, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{9}w_{\frac{3m}{5} +1}c^{−9}, c^{9}w_{\frac{3m}{5} +2}c^{−9}, …, c^{9}w_{m}c^{−9}, c^{8}w_{1}c^{−8}, …, c^{8}w_{\frac{3m}{5}} c^{−8}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{9}w_{\frac{3m}{5} +i}c^{−9}, i = 1, 2, …, \frac{2m}{5},  (6.135a)
w_{i} = c^{8}w_{i− \frac{2m}{5}} c^{−8}, i = \frac{2m}{5} + 1, \frac{2m}{5} + 2, …, m.  (6.135b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: either all the w_{i}’s are equal to ε, or there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1.

Writing t_{1}, t_{2}, t_{3}, t_{4}, t_{5} for w_{i}, w_{i+ \frac{m}{5}} , w_{i+ \frac{2m}{5}} , w_{i+ \frac{3m}{5}} , w_{i+ \frac{4m}{5}} , respectively, the equations (6.135) reduce to

t_{1} = c^{9}t_{4}c^{−9},  (6.136a)
t_{2} = c^{9}t_{5}c^{−9},  (6.136b)
t_{3} = c^{8}t_{1}c^{−8},  (6.136c)
t_{4} = c^{8}t_{2}c^{−8},  (6.136d)
t_{5} = c^{8}t_{3}c^{−8}.  (6.136e)

One of these equations is in fact superfluous: if we substitute (6.136b)–(6.136e) in (6.136a), then we obtain t_{1} = c^{42}t_{1}c^{−42} which is automatically satisfied since c^{42} = ε.

Since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have t_{1}t_{2}t_{3}t_{4}t_{5} ≤_{T} c. Combining this with (6.136), we infer that

t_{1}(c^{25}t_{1}c^{−25})(c^{8}t_{1}c^{−8})(c^{33}t_{1}c^{−33})(c^{16}t_{1}c^{−16}) ≤_{T} c.  (6.137)

Using that c^{7}t_{1}c^{−7} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.128). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/5, and, consequently, the number of solutions to (6.137) is the same, namely \frac{7m+5}{5}, as required.

Next we consider the case in (6.125c). By Lemma 26, we are free to choose p = 7m/4 if ζ = ζ_{24}, and we are free to choose p = 21m/4 if ζ = ζ_{8}. In both cases, m must be divisible by 4.

We start with the case that p = 7m/4. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{4} +1}c^{−2}, c^{2}w_{\frac{m}{4} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{4} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{4} +i}c^{−2}, i = 1, 2, …, \frac{3m}{4},  (6.138a)
w_{i} = cw_{i− \frac{3m}{4}} c^{−1}, i = \frac{3m}{4} + 1, \frac{3m}{4} + 2, …, m.  (6.138b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) = 1,  (6.139a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have

w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} ≤_{T} c.

Together with equations (6.138)–(6.139), this implies that

w_{i} = c^{7}w_{i}c^{−7} and w_{i}(c^{5}w_{i}c^{−5})(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) ≤_{T} c.  (6.140)

With the help of CHEVIE, one obtains seven solutions for w_{i} in (6.140):

w_{i} ∈ {[1], [2], [28], [34], [61], [46], [168]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{34} in CHEVIE. Each of them gives rise to m/4 elements of Fix_{NC^{m}(G_{34})}(ϕ^{p}) since i ranges from 1 to m/4.

In total, we obtain 1 + 7\frac{m}{4} = \frac{7m+4}{4} elements in Fix_{NC^{m}(G_{34})}(ϕ^{p}), which agrees with the limit in (6.125c).

If p = 21m/4, then, from (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{6}w_{\frac{3m}{4} +1}c^{−6}, c^{6}w_{\frac{3m}{4} +2}c^{−6}, …, c^{6}w_{m}c^{−6}, c^{5}w_{1}c^{−5}, …, c^{5}w_{\frac{3m}{4}} c^{−5}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{6}w_{\frac{3m}{4} +i}c^{−6}, i = 1, 2, …, \frac{m}{4},  (6.141a)
w_{i} = c^{5}w_{i− \frac{3m}{4}} c^{−5}, i = \frac{m}{4} + 1, \frac{m}{4} + 2, …, m.  (6.141b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) = 1,  (6.142a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have

w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} ≤_{T} c.

Together with equations (6.141)–(6.142), this implies that

w_{i} = c^{21}w_{i}c^{−21} and w_{i}(c^{5}w_{i}c^{−5})(c^{10}w_{i}c^{−10})(c^{15}w_{i}c^{−15}) ≤_{T} c.  (6.143)

Using that c^{7}t_{1}c^{−7} = t_{1}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.140). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/4, and, consequently, the number of solutions to (6.137) is the same, namely \frac{7m+4}{4}, as required.

Our next case is the case in (6.125d). By Lemma 26, we are free to choose p = 7m/3 if ζ = ζ_{18}, and we are free to choose p = 14m/3 if ζ = ζ_{9}. In both cases, m must be divisible by 3.

We start with the case that p = 7m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{2m}{3} +1}c^{−3}, c^{3}w_{\frac{2m}{3} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{2m}{3}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{2m}{3} +i}c^{−3}, i = 1, 2, …, \frac{m}{3},  (6.144a)
w_{i} = c^{2}w_{i− \frac{m}{3}} c^{−2}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.144b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 2,  (6.145a)

(iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1,  (6.145b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{1}+ \frac{2m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{2m}{3}} ) = 1,

(6.145c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c,

respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{3}} w_{i_{2}+ \frac{m}{3}} w_{i_{1}+ \frac{2m}{3}} w_{i_{2}+ \frac{2m}{3}} = c. Together with equations (6.144)– (6.145), this implies that

w_{i} = c^{7}w_{i}c^{−7} and w_{i}(c^{2}w_{i}c^{−2})(c^{4}w_{i}c^{−4}) = c,  (6.146)

respectively that

w_{i} = c^{7}w_{i}c^{−7}, w_{i}(c^{2}w_{i}c^{−2})(c^{4}w_{i}c^{−4}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.147)

respectively that

w_{i_{1}} = c^{7}w_{i_{1}}c^{−7}, w_{i_{2}} = c^{7}w_{i_{2}}c^{−7},

and w_{i_{1}}w_{i_{2}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2})(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4}) = c. (6.148)

With the help of CHEVIE, one obtains 21 solutions for w_{i} in (6.147):

w_{i} ∈ {[4] [5] [6] [11] [44] [63] [74]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{34} in CHEVIE. Each of them gives rise to m/3 elements of Fix_{NC^{m}(G_{34})}(ϕ^{p}) since i ranges from 1 to m/3.

There are no solutions w_{i} in (6.146) and for (w_{i_{1}}, w_{i_{2}}) in (6.148).

In total, we obtain 1 + 7\frac{m}{3} = \frac{7m+3}{3} elements in Fix_{NC^{m}(G_{34})}(ϕ^{p}), which agrees with the limit in (6.125d).

In the case that p = 14m/3, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{m}{3} +1}c^{−5}, c^{5}w_{\frac{m}{3} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w\frac{m}{3} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{m}{3} +i}c^{−5}, i = 1, 2, …, \frac{2m}{3},  (6.149a)
w_{i} = c^{4}w_{i− \frac{2m}{3}} c^{−4}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.149b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 2,  (6.150a)

(iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1,  (6.150b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{1}+ \frac{2m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{2m}{3}} ) = 1,

(6.150c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c,

respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{3}} w_{i_{2}+ \frac{m}{3}} w_{i_{1}+ \frac{2m}{3}} w_{i_{2}+ \frac{2m}{3}} = c. Together with equations (6.149)– (6.150), this implies that

w_{i} = c^{14}w_{i}c^{−14} and w_{i}(c^{9}w_{i}c^{−9})(c^{4}w_{i}c^{−4}) = c,  (6.151)

respectively that

w_{i} = c^{14}w_{i}c^{−14}, w_{i}(c^{9}w_{i}c^{−9})(c^{4}w_{i}c^{−4}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.152)

respectively that

w_{i_{1}} = c^{14}w_{i_{1}}c^{−14}, w_{i_{2}} = c^{14}w_{i_{2}}c^{−14},

and w_{i_{1}}w_{i_{2}}(c^{9}w_{i_{1}}c^{−9})(c^{9}w_{i_{2}}c^{−9})(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4}) = c. (6.153)

Using that c^{7}wc^{−7} = w for all w ∈ NC(G_{34}, due to Lemma 29 with d = 6, we see that this equation is equivalent with (6.146). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/3, and, consequently, the number of solutions to (6.151) is the same, namely \frac{7m+3}{3}, as required.

Next we consider the case in (6.125e). By Lemma 26, we are free to choose p = 7m/2 if ζ = ζ_{12}, and we are free to choose p = 21m/2 if ζ = ζ_{4}. In both cases, m must be divisible by 2.

We begin with the case that p = 7m/2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{m}{2} +1}c^{−4}, c^{4}w_{\frac{m}{2} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w\frac{m}{2} c^{−3}). (6.154)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{m}{2} +i}c^{−4}, i = 1, 2, …, \frac{m}{2},  (6.155a)
w_{i} = c^{3}w_{i− \frac{m}{2}} c^{−3}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.155b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) ≤ 3,  (6.156a)

(iii) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{1} = ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) ≥ 1, ℓ_{2} = ℓ_{T} (w_{i_{2}}) == ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) ≥ 1, ℓ_{1} + ℓ_{2} ≤ 3,

(6.156b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1}, i_{2}, i_{3} with 1 ≤ i_{1} < i_{2} < i_{3} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) = 1, (6.156c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(G_{34}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c, respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} = c

. Together with equations (6.155)–(6.156), this implies that

w_{i} = c^{7}w_{i}c^{−7} and w_{i}(c^{4}w_{i}c^{−4}) ≤_{T} c,  (6.157)

respectively that

w_{i_{1}} = c^{7}w_{i_{1}}c^{−7}, w_{i_{2}} = c^{7}w_{i_{2}}c^{−7}, and w_{i_{1}}w_{i_{2}}(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4}) ≤_{T} c, (6.158)

respectively that

w_{i_{1}} = c^{7}w_{i_{1}}c^{−7}, w_{i_{2}} = c^{7}w_{i_{2}}c^{−7}, w_{i_{3}} = c^{7}w_{i_{3}}c^{−7},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4})(c^{4}w_{i_{3}}c^{−4}) = c. (6.159)

With the help of CHEVIE, one obtains 14 solutions for w_{i} in (6.157) with ℓ_{T} (w_{i}) = 1:

w_{i} ∈ {[14], [35], [1], [2], [10], [24], [28], [34], [39], [56], [61], [46], [168], [105]},

where we have again used the short notation of CHEVIE referring to the internal ordering of the roots of G_{34} in CHEVIE, one obtains 21 solutions for w_{i} in (6.157) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 14], [1, 35], [1, 24], [1, 34], [14, 61], [2, 39], [10, 34], [2, 56], [2, 35], [14, 46], [35, 168], [34, 56], [2, 61], [28, 56], [10, 28], [10, 61], [34, 105], [28, 105], [24, 61], [39, 168], [24, 46]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(G_{34})}(ϕ^{p}) since i ranges from 1 to m/2, and one obtains 49 pairs (w_{i_{1}}, w_{i_{2}}) satisfying (6.158):

(w_{i_{1}}, w_{i_{2}}) ∈ {([14], [1]), ([14], [61]), ([14], [46]), ([35], [1]), ([35], [46]), ([35], [168]), ([1], [14]), ([1], [35]), ([1], [24]), ([1], [34]), ([2], [35]), ([2], [39]), ([2], [56]), ([2], [61]), ([10], [28]), ([10], [34]), ([10], [61]), ([24], [28]), ([24], [61]),

([24], [46]), ([28], [1]), ([28], [10]), ([28], [56]), ([28], [105]), ([34], [39]), ([34], [56]), ([34], [46]), ([34], [105]), ([39], [1]), ([39], [2]), ([39], [168]), ([56], [2]), ([56], [34]), ([56], [168]), ([61], [10]), ([61], [24]), ([61], [168]),

([61], [105]), ([46], [14]), ([46], [2]), ([46], [10]), ([46], [24]), ([168], [14]), ([168], [35]), ([168], [28]),

([168], [39]), ([105], [2]), ([105], [28]), ([105], [34])},

each of them giving rise to \binom{m/2}{2} elements of Fix_{NC^{m}(G_{34})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ m.

There are no solutions for w_{i} with ℓ_{T} (w_{i}) = 3 in (6.157), and hence no solutions for (w_{i_{1}}, w_{i_{2}}) with ℓ_{T} (w_{i_{1}}) + ℓ_{T} (w_{i_{2}}) = 3 in (6.158), and no solutions for (w_{i_{1}}, w_{i_{2}}, w_{i_{3}}) in (6.159).

In total, we obtain 1+(14+21)\frac{m}{2}+49\binom{m/2}{2} = \frac{(7m+2)(7m+4)}{8} elements in Fix_{NC^{m}(G_{34})}(ϕ^{p}), which agrees with the limit in (6.125e).

If p = 21m/2, from (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{11}w_{\frac{m}{2} +1}c^{−11}, c^{11}w_{\frac{m}{2} +2}c^{−11}, …, c^{11}w_{m}c^{−11}, c^{10}w_{1}c^{−10}, …, c^{10}w\frac{m}{2} c^{−10}).

Using that c^{7}wc^{−7} = w for all w ∈ NC(G_{34}, due to Lemma 29 with d = 6, we see that this action is identical with the one in (6.154). Therefore, we are facing exactly the same enumeration problem here as for p = 7m/2, and, consequently, the number of elements in Fix_{NC^{m}(G_{34})}(ϕ^{p}) is the same, namely \frac{(7m+2)(7m+4)}{8}, as required.

Finally, we turn to (6.125g). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, h_{2} = m_{2} = 2, h_{2} = 3 and m_{2} = 5, h_{2} = m_{2} = 3, h_{2} = 6 and m_{2} = 6, h_{2} = 6 and m_{2} = 5, h_{2} = 6 and m_{2} = 4, h_{2} = 6 and m_{2} = 3, respectively h_{2} = 6 and m_{2} = 2, . These correspond to the choices p = 42m/5, p = 21m/5, p = 21m/4, p = 21m/2, p = 14m/3, p = 14m/5, p = 7m/6, p = 7m/5, p = 7m/4, p = 7m/3, respectively p = 7m/2, out of which only p = 7m/6 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{5m}{6} +1}c^{−2}, c^{2}w_{\frac{5m}{6} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{5m}{6}} c^{−1}),

so that we have to solve

t_{1}(ct_{1}c^{−1})(c^{2}t_{1}c^{−2})(c^{3}t_{1}c^{−3})(c^{4}t_{1}c^{−4})(c^{5}t_{1}c^{−5}) = c.

A computation with the help of CHEVIE finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{35} = E_{6}. The degrees are 2, 5, 6, 8, 9, 12, and hence we have

Cat^{m}(E_{6}; q) = \frac{[12m + 12]_{q} [12m + 9]_{q} [12m + 8]_{q} [12m + 6]_{q} [12m + 5]_{q} [12m + 2]_{q}}{[12]_{q} [9]_{q} [8]_{q} [6]_{q} [5]_{q} [2]_{q}} .

Let ζ be a 12m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(E_{6}; q) = m + 1, if ζ = ζ_{12},  (6.160a)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{4m+3}{3}, if ζ = ζ_{9}, 3 | m,  (6.160b)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{3m+2}{2}, if ζ = ζ_{8}, 2 | m,  (6.160c)
lim_{q→ζ} Cat^{m}(E_{6}; q) = (m + 1)(2m + 1), if ζ = ζ_{6},  (6.160d)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{12m+5}{5}, if ζ = ζ_{5}, 5 | m,  (6.160e)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{4},  (6.160f)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(4m+3)(2m+1)}{3}, if ζ = ζ_{3},  (6.160g)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(3m+2)(2m+1)(6m+1)}{2}, if ζ = −1,  (6.160h)
lim_{q→ζ} Cat^{m}(E_{6}; q) = Cat^{m}(E_{6}), if ζ = 1,  (6.160i)
lim_{q→ζ} Cat^{m}(E_{6}; q) = 1, otherwise.  (6.160j)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.160). The only cases not covered by Lemmas 27 and 28 are the ones in (6.160b), (6.160c), (6.160e), and (6.160j).

We begin with the case in (6.160b). By Lemma 26, we are free to choose p = 4m/3. In particular, m must be divisible by 3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{2m}{3} +1}c^{−2}, c^{2}w_{\frac{2m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{2m}{3}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{2m}{3} +i}c^{−2}, i = 1, 2, …, \frac{m}{3},  (6.161a)
w_{i} = cw_{i− \frac{m}{3}} c^{−1}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.161b)

There are four distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 2,  (6.162a)

and all other w_{j}’s are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{3} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) = 1,  (6.162b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{3} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{3}} ) = ℓ_{T} (w_{i_{1}+ \frac{2m}{3}} ) = ℓ_{T} (w_{i_{2}+ \frac{2m}{3}} ) = 1, (6.162c)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{6}), we must have w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c,

respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{3}} w_{i_{2}+ \frac{m}{3}} w_{i_{1}+ \frac{2m}{3}} w_{i_{2}+ \frac{2m}{3}} = c. Together with equations (6.161)– (6.162), this implies that

w_{i} = c^{4}w_{i}c^{−4} and w_{i}(cw_{i}c^{−1})(c^{2}w_{i}c^{−2}) = c,  (6.163)

respectively that

w_{i} = c^{4}w_{i}c^{−4}, w_{i}(cw_{i}c^{−1})(c^{2}w_{i}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i}) = 1,  (6.164)

respectively that

w_{i_{1}} = c^{4}w_{i_{1}}c^{−4}, w_{i_{1}}(cw_{i_{1}}c^{−1})(c^{2}w_{i_{1}}c^{−2}) ≤_{T} c, and ℓ_{T} (w_{i_{1}}) = 1.  (6.165)

With the help of Stembridge’s Maple package coxeter [35], one obtains four solutions for w_{i} in (6.163):

w_{i} ∈ {[1, 2, 3, 4, 5, 6, 5, 4, 2, 3], [3, 4], [2, 4, 2, 5], [1, 3, 4, 5, 4, 3, 1, 6]},

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}} being a simple system of generators of E_{6}, corresponding to the Dynkin diagram displayed in Figure 4. Each of the above solutions for w_{i} gives rise to m/3 elements of Fix_{NC^{m}(E_{6})}(ϕ^{p}) since i ranges from 1 to m/3.

• • • • • •_{1} _{3} _{4} _{5} _{6}^{2}

Figure 4. The Dynkin diagram for E_{6}

There are no solutions for w_{i} in (6.164) and for w_{i_{1}} in (6.165).

In total, we obtain 1 + 4\frac{m}{3} = \frac{4m+3}{3} elements in Fix_{NC^{m}(E_{6})}(ϕ^{p}), which agrees with the limit in (6.160b).

Next we discuss the case in (6.160c). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.166a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.166b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 3,  (6.167a)

and all other w_{j}’s are equal to ε, (iii) there is a j with 1 ≤ j ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{j}) = ℓ_{T} (w_{j+ \frac{m}{2}} ) ≤ 2.  (6.167b)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{6}), we must have w_{i}w_{i+ \frac{m}{2}} = c, respectively w_{j}w_{j+ \frac{m}{2}} ≤_{T} c. Together with equations (6.166)–(6.167), this implies that

w_{i} = c^{3}w_{i}c^{−3} and w_{i}(cw_{i}c^{−1}) = c,  (6.168)

respectively that

w_{j} = c^{3}w_{j}c^{−3}, w_{j}(cw_{j}c^{−1}) ≤_{T} c, and 1 ≤ ℓ_{T} (w_{j}) ≤ 2.  (6.169)

With the help of Stembridge’s Maple package coxeter [35], one obtains three solutions for w_{i} in (6.168):

w_{i} ∈ {[1, 3, 4, 3, 5], [1, 2, 3, 4, 3, 5, 6, 5, 4, 3, 1], [2, 3, 4, 5, 4, 2, 6]},

where we used again coxeter’s short notation, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}} being a simple system of generators of E_{6}, corresponding to the Dynkin diagram displayed in Figure 4. Each of these solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(E_{6})}(ϕ^{p}) since i ranges from 1 to m/2.

There are no solutions for w_{j} in (6.169).

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(E_{6})}(ϕ^{p}), which agrees with the limit in (6.160c).

Finally we discuss the case in (6.160e). By Lemma 26, we are free to choose p = 12m/5. In particular, m must be divisible by 5. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{3m}{5} +1}c^{−3}, c^{3}w_{\frac{3m}{5} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{3m}{5}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{3m}{5} +i}c^{−3}, i = 1, 2, …, \frac{2m}{5},  (6.170a)
w_{i} = c^{2}w_{i− \frac{2m}{5}} c^{−2}, i = \frac{2m}{5} + 1, \frac{2m}{5} + 2, …, m.  (6.170b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1,  (6.171)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{6}), we must have

w_{i}w_{i+ \frac{m}{5}} w_{i+ \frac{2m}{5}} w_{i+ \frac{3m}{5}} w_{i+ \frac{4m}{5}} ≤_{T} c.

Together with equations (6.170)–(6.171), this implies that

w_{i} = c^{12}w_{i}c^{−12} and w_{i}(c^{7}w_{i}c^{−7})(c^{2}w_{i}c^{−2})(c^{9}w_{i}c^{−9})(c^{4}w_{i}c^{−4}) ≤_{T} c.  (6.172)

Here, the first equation is automatically satisfied since c^{12} = ε.

With the help of Stembridge’s Maple package coxeter [35], one obtains 12 solutions for w_{i} in (6.172):

w_{i} ∈ {[1], [3], [4], [5], [6], [2, 4, 2], [3, 4, 3], [2, 4, 5, 4, 2], [1, 3, 4, 5, 4, 3, 1], [1, 3, 4, 5, 6, 5, 4, 3, 1], [2, 3, 4, 5, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1]},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}} is a simple system of generators of E_{6}, corresponding to the Dynkin diagram displayed in Figure 4, and each of them gives rise to m/5 elements of Fix_{NC^{m}(E_{6})}(ϕ^{p}) since i ranges from 1 to m/5.

In total, we obtain 1 + 12\frac{m}{5} = \frac{12m+5}{5} elements in Fix_{NC^{m}(E_{6})}(ϕ^{p}), which agrees with the limit in (6.160e).

Finally, we turn to (6.160j). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 5 and h_{2} = 2 and m_{2} = 5. These correspond to the choices p = 12m/5, respectively p = 6m/5, out which only p = 6m/5 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{4m}{5} +1}c^{−2}, c^{2}w_{\frac{4m}{5} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{4m}{5}} c^{−1}),

so that we have to solve

t_{1}(ct_{1}c^{−1})(c^{2}t_{1}c^{−2})(c^{3}t_{1}c^{−3})(c^{4}t_{1}c^{−4}) ≤_{T} c

for t_{1} with ℓ_{T} (t_{1}) = 1. A computation with the help of Stembridge’s Maple package coxeter [35] finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

Case G_{36} = E_{7}. The degrees are 2, 6, 8, 10, 12, 14, 18, and hence we have

Cat^{m}(E_{7}; q) = \frac{[18m + 18]_{q} [18m + 14]_{q} [18m + 12]_{q}}{[18]_{q} [14]_{q} [12]_{q}}

× \frac{[18m + 10]_{q} [18m + 8]_{q} [18m + 6]_{q} [18m + 2]_{q}}{[10]_{q} [8]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 18m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(E_{7}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (6.173a)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{9m+7}{7}, if ζ = ζ_{14}, ζ_{7}, 7 | m,  (6.173b)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{3m+2}{2}, if ζ = ζ_{12}, 2 | m,  (6.173c)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{9m+5}{5}, if ζ = ζ_{10}, ζ_{5}, 5 | m,  (6.173d)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{9m+4}{4}, if ζ = ζ_{8}, 4 | m,  (6.173e)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{(m + 1)(3m + 2)(3m + 1)}{2}, if ζ = ζ_{6}, ζ_{3},  (6.173f)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{(3m + 2)(9m + 4)}{8}, if ζ = ζ_{4}, 2 | m,  (6.173g)
lim_{q→ζ} Cat^{m}(E_{7}; q) = Cat^{m}(E_{7}), if ζ = −1 or ζ = 1,  (6.173h)
lim_{q→ζ} Cat^{m}(E_{7}; q) = 1, otherwise.  (6.173i)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.173). The only cases not covered by Lemmas 27 and 28 are the ones in (6.173b), (6.173c), (6.173d), (6.173e), (6.173g), and (6.173i).

We begin with the case in (6.173b). By Lemma 26, we are free to choose p = 9m/7 if ζ = ζ_{14}, respectively p = 18m/7 if ζ = ζ_{7}. In both cases, m must be divisible by 7.

We start with the case that p = 9m/7. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{5m}{7} +1}c^{−2}, c^{2}w_{\frac{5m}{7} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{5m}{7}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{5m}{7} +i}c^{−2}, i = 1, 2, …, \frac{2m}{7},  (6.174a)
w_{i} = cw_{i− \frac{2m}{7}} c^{−1}, i = \frac{2m}{7} + 1, \frac{2m}{7} + 2, …, m.  (6.174b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{7} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{7}} ) = ℓ_{T} (w_{i+ \frac{2m}{7}} ) = ℓ_{T} (w_{i+ \frac{3m}{7}} )

= ℓ_{T} (w_{i+ \frac{4m}{7}} ) = ℓ_{T} (w_{i+ \frac{5m}{7}} ) = ℓ_{T} (w_{i+ \frac{6m}{7}} ) = 1, (6.175)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have

w_{i}w_{i+ \frac{m}{7}} w_{i+ \frac{2m}{7}} w_{i+ \frac{3m}{7}} w_{i+ \frac{4m}{7}} w_{i+ \frac{5m}{7}} w_{i+ \frac{6m}{7}} = c.

Together with equations (6.174)–(6.175), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{5}w_{i}c^{−5})(cw_{i}c^{−1})(c^{6}w_{i}c^{−6})(c^{2}w_{i}c^{−2})(c^{7}w_{i}c^{−7})(c^{3}w_{i}c^{−3}) = c.

(6.176)

Here, the first equation is automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains 9 solutions for w_{i} in (6.176):

w_{i} ∈ {[4], [5], [6], [7], [3, 4, 3], [2, 4, 5, 4, 2], [1, 3, 4, 5, 6, 5, 4, 3, 1], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1]}, (6.177)

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} being a simple system of generators of E_{7}, corresponding to the Dynkin diagram displayed in Figure 5. Each of the above solutions for w_{i} gives rise to m/7 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/7.

• • • • • • •_{1} _{3} _{4} _{5} _{6} _{7}^{2}

Figure 5. The Dynkin diagram for E_{7}

In total, we obtain 1 + 9\frac{m}{7} = \frac{9m+7}{7} elements in Fix_{NC^{m}(E_{7})}(ϕ^{p}), which agrees with the limit in (6.173b).

In the case that p = 18m/7, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{3m}{7} +1}c^{−3}, c^{3}w_{\frac{3m}{7} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{3m}{7}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{3m}{7} +i}c^{−3}, i = 1, 2, …, \frac{4m}{7},  (6.178a)
w_{i} = c^{2}w_{i− \frac{4m}{7}} c^{−2}, i = \frac{4m}{7} + 1, \frac{4m}{7} + 2, …, m.  (6.178b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{7} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{7}} ) = ℓ_{T} (w_{i+ \frac{2m}{7}} ) = ℓ_{T} (w_{i+ \frac{3m}{7}} )

= ℓ_{T} (w_{i+ \frac{4m}{7}} ) = ℓ_{T} (w_{i+ \frac{5m}{7}} ) = ℓ_{T} (w_{i+ \frac{6m}{7}} ) = 1, (6.179)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have

w_{i}w_{i+ \frac{m}{7}} w_{i+ \frac{2m}{7}} w_{i+ \frac{3m}{7}} w_{i+ \frac{4m}{7}} w_{i+ \frac{5m}{7}} w_{i+ \frac{6m}{7}} = c.

Together with equations (6.178)–(6.179), this implies that

w_{i} = c^{18}w_{i}c^{−18}

and w_{i}(c^{5}w_{i}c^{−5})(c^{10}w_{i}c^{−10})(c^{15}w_{i}c^{−15})(c^{2}w_{i}c^{−2})(c^{7}w_{i}c^{−7})(c^{12}w_{i}c^{−12}) = c. (6.180)

Here, the first equation is automatically satisfied since c^{18} = ε. Due to Lemma 29 with d = 2, we have c^{9}w_{i}c^{−9} = w_{i}, hence also c^{10}w_{i}c^{−10} = cw_{i}c^{−1}, etc., so that (6.180) reduces to (6.176). Therefore, we are facing exactly the same enumeration problem here as for p = 9m/7, and, consequently, the number of solutions to (6.180) is the same, namely \frac{9m+7}{7}, as required.

Next we discuss the case in (6.173c). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.181a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.181b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) ≤ 3.  (6.182)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c. Together with equations (6.181)–(6.182), this implies that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(cw_{i}c^{−1}) ≤_{T} c, and 1 ≤ ℓ_{T} (w_{i}) ≤ 3.  (6.183)

With the help of Stembridge’s Maple package coxeter [35], one obtains three solutions for w_{i} in (6.183) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 3, 4, 5, 4, 3], [2, 3, 4, 5, 6, 5, 4, 2], [1, 2, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1]},

where we used again coxeter’s short notation, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} being a simple system of generators of E_{7}, corresponding to the Dynkin diagram displayed in Figure 5, and none if ℓ_{T} (w_{i}) = 1 or ℓ_{T} (w_{i}) = 3. Each of the solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/2.

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(E_{7})}(ϕ^{p}), which agrees with the limit in (6.173c).

Next we consider the case in (6.173d). By Lemma 26, we are free to choose p = 9m/5 if ζ = ζ_{10}, respectively p = 18m/5 if ζ = ζ_{5}. In both cases, m must be divisible by 5.

We start with the case that p = 9m/5. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{5} +1}c^{−2}, c^{2}w_{\frac{m}{5} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{5} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{5} +i}c^{−2}, i = 1, 2, …, \frac{4m}{5},  (6.184a)
w_{i} = cw_{i− \frac{4m}{5}} c^{−1}, i = \frac{4m}{5} + 1, \frac{4m}{5} + 2, …, m.  (6.184b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1,  (6.185a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have

w_{i}w_{i+ \frac{m}{5}} w_{i+ \frac{2m}{5}} w_{i+ \frac{3m}{5}} w_{i+ \frac{4m}{5}} ≤_{T} c.

Together with equations (6.184)–(6.185), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{7}w_{i}c^{−7})(c^{5}w_{i}c^{−5})(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) ≤_{T} c.  (6.186)

With the help of Stembridge’s Maple package coxeter [35], one obtains 9 solutions for w_{i} in (6.186):

w_{i} ∈ {[4], [5], [6], [7], [3, 4, 3], [2, 4, 5, 4, 2], [1, 3, 4, 5, 6, 5, 4, 3, 1], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1]},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} is a simple system of generators of E_{7}, corresponding to the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/5 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/5.^{4}

In total, we obtain 1 + 9\frac{m}{5} = \frac{9m+5}{5} elements in Fix_{NC^{m}(E_{7})}(ϕ^{p}), which agrees with the limit in (6.173d).

4Miraculously, these are exactly the same solutions as in the case of (6.173b). We have no explanation for this phenomenon.

In the case that p = 18m/5, we infer from (5.1) that

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{5} +1}c^{−4}, c^{4}w_{\frac{2m}{5} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{5}} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{5} +i}c^{−4}, i = 1, 2, …, \frac{3m}{5},  (6.187a)
w_{i} = c^{3}w_{i− \frac{3m}{5}} c^{−3}, i = \frac{3m}{5} + 1, \frac{3m}{5} + 2, …, m.  (6.187b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{5} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{5}} ) = ℓ_{T} (w_{i+ \frac{2m}{5}} ) = ℓ_{T} (w_{i+ \frac{3m}{5}} ) = ℓ_{T} (w_{i+ \frac{4m}{5}} ) = 1,  (6.188a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have

w_{i}w_{i+ \frac{m}{5}} w_{i+ \frac{2m}{5}} w_{i+ \frac{3m}{5}} w_{i+ \frac{4m}{5}} ≤_{T} c.

Together with equations (6.187)–(6.188), this implies that

w_{i} = c^{18}w_{i}c^{−18} and w_{i}(c^{7}w_{i}c^{−7})(c^{14}w_{i}c^{−14})(c^{3}w_{i}c^{−3})(c^{10}w_{i}c^{−10}) ≤_{T} c. (6.189)

Here, the first equation is automatically satisfied since c^{18} = ε. Due to Lemma 29 with d = 2, we have c^{9}w_{i}c^{−9} = w_{i}, hence also c^{14}w_{i}c^{−14} = c^{5}w_{i}c^{−5}, etc., so that (6.189) reduces to (6.186). Therefore, we are facing exactly the same enumeration problem here as for p = 9m/5, and, consequently, the number of solutions to (6.189) is the same, namely \frac{9m+5}{5}, as required.

Our next case is the case in (6.173e). By Lemma 26, we are free to choose p = 9m/4. In particular, m must be divisible by 4. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{3m}{4} +1}c^{−3}, c^{3}w_{\frac{3m}{4} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{3m}{4}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{3m}{4} +i}c^{−3}, i = 1, 2, …, \frac{m}{4},  (6.190a)
w_{i} = c^{2}w_{i− \frac{m}{4}} c^{−2}, i = \frac{m}{4} + 1, \frac{m}{4} + 2, …, m.  (6.190b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{4} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) = 1,  (6.191)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} ≤_{T} c. Together with equations (6.190)–(6.191), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{2}w_{i}c^{−2})(c^{4}w_{i}c^{−4})(c^{6}w_{i}c^{−6}) ≤_{T} c.  (6.192)

Here, the first equation in (6.192) is automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains 9 solutions for w_{i} in (6.192) with ℓ_{T} (w_{i}) = 1:

w_{i} ∈ {[1], [3], [2, 4, 2], [3, 4, 5, 4, 3], [1, 3, 4, 5, 4, 3, 1], [2, 4, 5, 6, 5, 4, 2], [2, 3, 4, 5, 6, 5, 4, 2, 3], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4]},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} is a simple system of generators of E_{7}, corresponding to the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/4 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/4. Hence, we obtain 1 + 9\frac{m}{4} = \frac{9m+4}{4} elements in Fix_{NC^{m}(E_{7})}(ϕ^{p}), which agrees with the limit in (6.173e).

Finally we discuss the case in (6.173g). By Lemma 26, we are free to choose p = 9m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{m}{2} +1}c^{−5}, c^{5}w_{\frac{m}{2} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w\frac{m}{2} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{m}{2} +i}c^{−5}, i = 1, 2, …, \frac{m}{2},  (6.193a)
w_{i} = c^{4}w_{i− \frac{m}{2}} c^{−4}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.193b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 3,  (6.194a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 2,  (6.194b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) = 1,  (6.194c)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε,

(v) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.194d)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (vi) there are i_{1} and i_{2} with 1 ≤ i_{1}, i_{2} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = 2 and ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = 1,  (6.194e)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (vii) there are i_{1}, i_{2}, i_{3} with 1 ≤ i_{1} < i_{2} < i_{3} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) = 1, (6.194f)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε.

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{7}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c, respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} ≤_{T} c.

Together with equations (6.193)–(6.194), this implies that

w_{i} = c^{9}w_{i}c^{−9} and w_{i}(c^{4}w_{i}c^{−4}) ≤_{T} c,  (6.195)

respectively that

w_{i_{1}} = c^{9}w_{i_{1}}c^{−9}, w_{i_{2}} = c^{9}w_{i_{2}}c^{−9}, and w_{i_{1}}w_{i_{2}}(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4}) ≤_{T} c, (6.196)

respectively that

w_{i_{1}} = c^{9}w_{i_{1}}c^{−9}, w_{i_{2}} = c^{9}w_{i_{2}}c^{−9}, w_{i_{3}} = c^{9}w_{i_{3}}c^{−9},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}(c^{4}w_{i_{1}}c^{−4})(c^{4}w_{i_{2}}c^{−4})(c^{4}w_{i_{3}}c^{−4}) ≤_{T} c. (6.197)

Here, the first equation in (6.195), the first two in (6.196), and the first three in (6.197), are all automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains 9 solutions for w_{i} in (6.195) with ℓ_{T} (w_{i}) = 1:

w_{i} ∈ {[1], [3], [2, 4, 2], [3, 4, 5, 4, 3], [1, 3, 4, 5, 4, 3, 1], [2, 4, 5, 6, 5, 4, 2], [2, 3, 4, 5, 6, 5, 4, 2, 3],

[1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4]},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}} is a simple system of generators of E_{7}, corresponding to the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/2 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/2. Furthermore, one obtains 12 solutions for w_{i} in (6.195) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 2, 4, 2], [1, 3, 4, 5, 4, 3], [1, 2, 4, 5, 6, 5, 4, 2], [1, 3, 1, 4, 5, 4, 3, 1], [2, 3, 4, 5, 6, 5, 4, 2],

[1, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1], [2, 3, 4, 3, 5, 6, 5, 4, 2, 3], [1, 2, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3], [1, 3, 1, 4, 3, 5, 4, 3, 1, 6, 7, 6, 5, 4, 3, 1],

[1, 3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4]},

each of them giving rise to m/2 elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since i ranges from 1 to m/2, and one obtains 27 pairs (w_{i_{1}}, w_{i_{2}}) of solutions in (6.196) with ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = 1:

w_{i} ∈ {([1], [2, 4, 2]), ([1], [3, 4, 5, 4, 3]), ([1], [2, 4, 5, 6, 5, 4, 2]), ([3], [1, 3, 4, 5, 4, 3, 1]), ([3], [2, 4, 5, 6, 5, 4, 2]), ([3], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1]), ([2, 4, 2], [1]),

([2, 4, 2], [2, 3, 4, 5, 6, 5, 4, 2, 3]), ([2, 4, 2], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1]),
([3, 4, 5, 4, 3], [1, 3, 4, 5, 4, 3, 1]), ([3, 4, 5, 4, 3], [2, 3, 4, 5, 6, 5, 4, 2, 3]),
([3, 4, 5, 4, 3], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1]), ([1, 3, 4, 5, 4, 3, 1], [1]),
([1, 3, 4, 5, 4, 3, 1], [3]), ([1, 3, 4, 5, 4, 3, 1], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4]),
([2, 4, 5, 6, 5, 4, 2], [1]), ([2, 4, 5, 6, 5, 4, 2], [2, 3, 4, 5, 6, 5, 4, 2, 3]),
([2, 4, 5, 6, 5, 4, 2], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4]), ([2, 3, 4, 5, 6, 5, 4, 2, 3], [3]),
([2, 3, 4, 5, 6, 5, 4, 2, 3], [2, 4, 2]), ([2, 3, 4, 5, 6, 5, 4, 2, 3], [3, 4, 5, 4, 3]),
([1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [3]), ([1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [3, 4, 5, 4, 3]),
([1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4]),
([1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4], [2, 4, 2]),
([1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4], [1, 3, 4, 5, 4, 3, 1]),

([1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4], [2, 4, 5, 6, 5, 4, 2])},

each of them giving rise to \binom{m/2}{2} elements of Fix_{NC^{m}(E_{7})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2}.

There are no solutions for w_{i} in (6.195) with ℓ_{T} (w_{i}) = 3, and hence there are no solutions for w_{i_{1}}, w_{i_{2}} in (6.196) if we are in case (vi), or for w_{i_{1}}, w_{i_{2}}, w_{i_{3}} in (6.197).

In total, we obtain 1+(9+12)\frac{m}{2} +27\binom{m/2}{2} = \frac{(3m+2)(9m+4)}{8} elements in Fix_{NC^{m}(E_{7})}(ϕ^{p}), which agrees with the limit in (6.173g).

Finally, we turn to (6.173i). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 7, h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 7, h_{2} = 2 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 18m/7, p = 18m/5, p = 9m/7, p = 9m/5, p = 9m/4, respectively p = 9m/2, all of which have already been discussed as they do not belong to (6.173i). Hence, (3.3) must necessarily hold, as required.

Case G_{37} = E_{8}. The degrees are 2, 8, 12, 14, 18, 20, 24, 30, and hence we have

Cat^{m}(E_{8}; q) = \frac{[30m + 30]_{q} [30m + 24]_{q} [30m + 20]_{q} [30m + 18]_{q}}{[30]_{q} [24]_{q} [20]_{q} [18]_{q}}

× \frac{[30m + 14]_{q} [30m + 12]_{q} [30m + 8]_{q} [30m + 2]_{q}}{[14]_{q} [12]_{q} [8]_{q} [2]_{q}} .

Let ζ be a 30m-th root of unity. The following cases on the right-hand side of (3.3) occur:

lim_{q→ζ} Cat^{m}(E_{8}; q) = m + 1, if ζ = ζ_{30}, ζ_{15},  (6.198a)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{5m+4}{4}, if ζ = ζ_{24}, 4 | m,  (6.198b)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{3m+2}{2}, if ζ = ζ_{20}, 2 | m,  (6.198c)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{5m+3}{3}, if ζ = ζ_{18}, ζ_{9}, 3 | m,  (6.198d)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{15m+7}{7}, if ζ = ζ_{14}, ζ_{7}, 7 | m,  (6.198e)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(5m+4)(5m+2)}{8}, if ζ = ζ_{12}, 2 | m,  (6.198f)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{10}, ζ_{5},  (6.198g)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(5m + 4)(15m + 4)}{16}, if ζ = ζ_{8}, 4 | m,  (6.198h)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(m + 1)(5m + 4)(5m + 3)(5m + 2)}{24}, if ζ = ζ_{6}, ζ_{3},  (6.198i)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(5m + 4)(3m + 2)(5m + 2)(15m + 4)}{64}, if ζ = ζ_{4}, 2 | m, (6.198j)
lim_{q→ζ} Cat^{m}(E_{8}; q) = Cat^{m}(E_{8}), if ζ = −1 or ζ = 1,  (6.198k)
lim_{q→ζ} Cat^{m}(E_{8}; q) = 1, otherwise.  (6.198l)

We must now prove that the left-hand side of (3.3) in each case agrees with the values exhibited in (6.198). The only cases not covered by Lemmas 27 and 28 are the ones in (6.198b), (6.198c), (6.198d), (6.198e), (6.198f), (6.198h), (6.198j), and (6.198l).

We begin with the case in (6.198b). By Lemma 26, we are free to choose p = 5m/4. In particular, m must be divisible by 4. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{3m}{4} +1}c^{−2}, c^{2}w_{\frac{3m}{4} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw_{\frac{3m}{4}} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{3m}{4} +i}c^{−2}, i = 1, 2, …, \frac{m}{4},  (6.199a)
w_{i} = cw_{i− \frac{m}{4}} c^{−1}, i = \frac{m}{4} + 1, \frac{m}{4} + 2, …, m.  (6.199b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{4} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) ≤ 2.  (6.200)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} ≤_{T} c.

Together with equations (6.199)–(6.200), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(cw_{i}c^{−1})(c^{2}w_{i}c^{−2})(c^{3}w_{i}c^{−3}) ≤_{T} c.  (6.201)

With the help of Stembridge’s Maple package coxeter [35], one obtains 5 solutions for w_{i} in (6.201) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 4, 3, 5], [2, 4, 5, 4, 2, 6], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8]},

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} being a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6. Each of the above solutions for w_{i} gives rise to m/4 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/4.

• • • • • • • •_{1} _{3} _{4} _{5} _{6} _{7} _{8}^{2}

Figure 6. The Dynkin diagram for E_{8}

There are no solutions for w_{i} in (6.201) with ℓ_{T} (w_{i}) = 1.

In total, we obtain 1 + 5\frac{m}{4} = \frac{5m+4}{4} elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198b).

Next we discuss the case in (6.198c). By Lemma 26, we are free to choose p = 3m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{2} +1}c^{−2}, c^{2}w_{\frac{m}{2} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{2} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{2} +i}c^{−2}, i = 1, 2, …, \frac{m}{2},  (6.202a)
w_{i} = cw_{i− \frac{m}{2}} c^{−1}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.202b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) ≤ 4.  (6.203)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c. Together with equations (6.202)–(6.203), this implies that

w_{i} = c^{3}w_{i}c^{−3}, w_{i}(cw_{i}c^{−1}) ≤_{T} c, and 1 ≤ ℓ_{T} (w_{i}) ≤ 4.  (6.204)

With the help of Stembridge’s Maple package coxeter [35], one obtains three solutions for w_{i} in (6.204) with ℓ_{T} (w_{i}) = 4:

w_{i} ∈ {[1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 6, 5, 4, 2, 7], [1, 2, 4, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1, 8]},

where we used again coxeter’s short notation, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} being a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6.

Each of these solutions for w_{i} gives rise to m/2 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/2. There are no solutions for w_{i} in (6.204) with 1 ≤ ℓ_{T} (w_{i}) ≤ 3.

In total, we obtain 1 + 3\frac{m}{2} = \frac{3m+2}{2} elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198c).

Next we consider the case in (6.198d). If ζ = ζ_{18}, then, by Lemma 26, we are free to choose p = 5m/3, whereas, for ζ = ζ_{9}, we can choose p = 10m/3. In particular, in both cases m must be divisible by 3.

First, let p = 5m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m})) = (∗; c^{2}w_{\frac{m}{3} +1}c^{−2}, c^{2}w_{\frac{m}{3} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{3} c^{−1}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{2}w_{\frac{m}{3} +i}c^{−2}, i = 1, 2, …, \frac{2m}{3},  (6.205a)
w_{i} = cw_{i− \frac{2m}{3}} c^{−1}, i = \frac{2m}{3} + 1, \frac{2m}{3} + 2, …, m.  (6.205b)

There several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{3} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) ≤ 2.  (6.206)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c.

Together with equations (6.205)–(6.206), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(c^{3}w_{i}c^{−3})(cw_{i}c^{−1}) ≤_{T} c.  (6.207)

With the help of Stembridge’s Maple package coxeter [35], one obtains five solutions for w_{i} in (6.207) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 4, 3, 5], [2, 4, 5, 4, 2, 6], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8]},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} is a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/3 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/3. There are no solutions for w_{i} in (6.207) with ℓ_{T} (w_{i}) = 1.

In total, we obtain 1 + 5\frac{m}{3} = \frac{5m+3}{3} elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198d).

Now let p = 10m/3. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{2m}{3} +1}c^{−4}, c^{4}w_{\frac{2m}{3} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w_{\frac{2m}{3}} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{2m}{3} +i}c^{−4}, i = 1, 2, …, \frac{m}{3},  (6.208a)
w_{i} = c^{3}w_{i− \frac{m}{3}} c^{−3}, i = \frac{m}{3} + 1, \frac{m}{3} + 2, …, m.  (6.208b)

There several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{3} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{3}} ) = ℓ_{T} (w_{i+ \frac{2m}{3}} ) ≤ 2.  (6.209)

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{3}} w_{i+ \frac{2m}{3}} ≤_{T} c.

Together with equations (6.208)–(6.209), this implies that

w_{i} = c^{10}w_{i}c^{−10} and w_{i}(c^{3}w_{i}c^{−3})(c^{6}w_{i}c^{−6}) ≤_{T} c.  (6.210)

Due to Lemma 29 with d = 2, we have c^{15}w_{i}c^{−15} = w_{i}, hence also c^{5}w_{i}c^{−5} = w_{i}, so that (6.210) reduces to (6.207). Therefore, we are facing exactly the same enumeration problem here as for p = 5m/3, and, consequently, the number of solutions to (6.210) is the same, namely \frac{5m+3}{3}, as required.

Our next case is the case in (6.198e). If ζ = ζ_{14}, then, by Lemma 26, we are free to choose p = 15m/7, whereas, for ζ = ζ_{7}, we can choose p = 30m/7. In particular, in both cases m must be divisible by 7.

First, let p = 15m/7. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{6m}{7} +1}c^{−3}, c^{3}w_{\frac{6m}{7} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w_{\frac{6m}{7}} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{6m}{7} +i}c^{−3}, i = 1, 2, …, \frac{m}{7},  (6.211a)
w_{i} = c^{2}w_{i− \frac{m}{7}} c^{−2}, i = \frac{m}{7} + 1, \frac{m}{7} + 2, …, m.  (6.211b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{7} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{7}} ) = ℓ_{T} (w_{i+ \frac{2m}{7}} ) = ℓ_{T} (w_{i+ \frac{3m}{7}} )

= ℓ_{T} (w_{i+ \frac{4m}{7}} ) = ℓ_{T} (w_{i+ \frac{5m}{7}} ) = ℓ_{T} (w_{i+ \frac{6m}{7}} ) = 1, (6.212)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{7}} w_{i+ \frac{2m}{7}} w_{i+ \frac{3m}{7}} w_{i+ \frac{4m}{7}} w_{i+ \frac{5m}{7}} w_{i+ \frac{6m}{7}} ≤_{T} c.

Together with the equations (6.211)–(6.212), this implies that

w_{i} = c^{15}w_{i}c^{−15}

and w_{i}(c^{2}w_{i}c^{−2})(c^{4}w_{i}c^{−4})(c^{6}w_{i}c^{−6})(c^{8}w_{i}c^{−8})(c^{10}w_{i}c^{−10})(c^{12}w_{i}c^{−12}) ≤_{T} c. (6.213)

Here, the first equation is automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains 15 solutions for w_{i} in (6.213):

w_{i} ∈ {[4], [5], [6], [7], [8], [3, 4, 3], [2, 4, 5, 4, 2], [3, 4, 5, 4, 3], [2, 4, 5, 6, 5, 4, 2],

[1, 3, 4, 5, 6, 5, 4, 3, 1], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4]},

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} being a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/7 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/7.

In total, we obtain 1 + 15\frac{m}{7} = \frac{15m+7}{7} elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198e).

Now let p = 30m/7. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{5}w_{\frac{5m}{7} +1}c^{−5}, c^{5}w_{\frac{5m}{7} +2}c^{−5}, …, c^{5}w_{m}c^{−5}, c^{4}w_{1}c^{−4}, …, c^{4}w_{\frac{5m}{7}} c^{−4}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{5}w_{\frac{5m}{7} +i}c^{−5}, i = 1, 2, …, \frac{2m}{7},  (6.214a)
w_{i} = c^{4}w_{i− \frac{2m}{7}} c^{−4}, i = \frac{2m}{7} + 1, \frac{2m}{7} + 2, …, m.  (6.214b)

There are two distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{7} such that

ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{7}} ) = ℓ_{T} (w_{i+ \frac{2m}{7}} ) = ℓ_{T} (w_{i+ \frac{3m}{7}} )

= ℓ_{T} (w_{i+ \frac{4m}{7}} ) = ℓ_{T} (w_{i+ \frac{5m}{7}} ) = ℓ_{T} (w_{i+ \frac{6m}{7}} ) = 1, (6.215)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε.

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{7}} w_{i+ \frac{2m}{7}} w_{i+ \frac{3m}{7}} w_{i+ \frac{4m}{7}} w_{i+ \frac{5m}{7}} w_{i+ \frac{6m}{7}} ≤_{T} c.

Together with the equations (6.214)–(6.215), this implies that

w_{i} = c^{30}w_{i}c^{−30}

and w_{i}(c^{17}w_{i}c^{−17})(c^{4}w_{i}c^{−4})(c^{21}w_{i}c^{−21})(c^{8}w_{i}c^{−8})(c^{25}w_{i}c^{−25})(c^{12}w_{i}c^{−12}) ≤_{T} c. (6.216)

Here, the first equation is automatically satisfied since c^{30} = ε. Moreover, due to Lemma 29 with d = 2, we have c^{17}w_{i}c^{−17} = c^{2}w_{i}c^{−2}, etc., so that (6.216) reduces to (6.213). Therefore, we are facing exactly the same enumeration problem here as for p = 15m/7, and, consequently, the number of solutions to (6.216) is the same, namely \frac{15m+7}{7}, as required.

We now turn to the case in (6.198f). By Lemma 26, we are free to choose p = 5m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{3}w_{\frac{m}{2} +1}c^{−3}, c^{3}w_{\frac{m}{2} +2}c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{1}c^{−2}, …, c^{2}w\frac{m}{2} c^{−2}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{3}w_{\frac{m}{2} +i}c^{−3}, i = 1, 2, …, \frac{m}{2},  (6.217a)
w_{i} = c^{2}w_{i− \frac{m}{2}} c^{−2}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.217b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m: (i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) ≤ 4,  (6.218a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iii) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{1} := ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) ≥ 1, ℓ_{2} := ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) ≥ 1, and ℓ_{1} + ℓ_{2} ≤ 4,

(6.218b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1}, i_{2}, i_{3} with 1 ≤ i_{1} < i_{2} < i_{3} ≤ \frac{m}{2} such that

ℓ_{1} := ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) ≥ 1, ℓ_{2} := ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) ≥ 1, ℓ_{3} := ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) ≥ 1, and ℓ_{1} + ℓ_{2} + ℓ_{3} ≤ 4, (6.218c)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε,

(v) there are i_{1}, i_{2}, i_{3}, i_{4} with 1 ≤ i_{1} < i_{2} < i_{3} < i_{4} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{4}})

= ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{4}+ \frac{m}{2}} ) = 1, (6.218d)

and all other w_{j}’s are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respec-

tively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c, respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} ≤_{T} c,

respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} w_{i_{4}+ \frac{m}{2}} = c.

Together with the equations (6.217)–(6.218), this implies that

w_{i} = c^{5}w_{i}c^{−5} and w_{i}(c^{2}w_{i}c^{−2}) ≤_{T} c,  (6.219)

respectively that

w_{i_{1}} = c^{5}w_{i_{1}}c^{−5}, w_{i_{2}} = c^{5}w_{i_{2}}c^{−5}, and w_{i_{1}}w_{i_{2}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2}) ≤_{T} c, (6.220)

respectively that

w_{i_{1}} = c^{5}w_{i_{1}}c^{−5}, w_{i_{2}} = c^{5}w_{i_{2}}c^{−5}, w_{i_{3}} = c^{5}w_{i_{3}}c^{−5},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2})(c^{2}w_{i_{3}}c^{−2}) ≤_{T} c, (6.221) respectively that

w_{i_{1}} = c^{5}w_{i_{1}}c^{−5}, w_{i_{2}} = c^{5}w_{i_{2}}c^{−5}, w_{i_{3}} = c^{5}w_{i_{3}}c^{−5}, w_{i_{4}} = c^{5}w_{i_{4}}c^{−5},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}}(c^{2}w_{i_{1}}c^{−2})(c^{2}w_{i_{2}}c^{−2})(c^{2}w_{i_{3}}c^{−2})(c^{2}w_{i_{4}}c^{−2}) = c. (6.222)

With the help of Stembridge’s Maple package coxeter [35], one obtains 10 solutions for w_{i} in (6.219) with ℓ_{T} (w_{i}) = 2:

w_{i} ∈ {[1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 4, 3, 5], [2, 4, 5, 4, 2, 6], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [2, 4, 2, 3, 4, 5, 6, 5, 4, 3], [1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2],

[2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1]}, (6.223)

and one obtains 10 solutions for w_{i} in (6.219) with ℓ_{T} (w_{i}) = 4:

w_{i} ∈ {[1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 8], [1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 7, 6, 5, 4, 2, 3, 4],

[1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[1, 2, 3, 4, 2, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4], [4, 2, 3, 4, 5, 6],
[2, 3, 1, 4, 2, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 5, 4, 2, 3, 1, 4, 5, 6, 7],

[3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 8]}, (6.224)

where we have again used the short notation of coxeter, {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} being a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/2 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/2. There are no solutions for w_{i} in (6.219) with ℓ_{T} (w_{i}) = 1 or ℓ_{T} (w_{i}) = 3.

Consequently, there are no solutions in Cases (iv) and (v), and the only possible solutions occurring in Case (iii) are pairs (w_{i_{1}}, w_{i_{2}}) of elements of (6.223) whose product is in (6.224). Another computation using Stembridge’s Maple package coxeter finds the following 25 pairs meeting that description:

w_{i} ∈ {([1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4]),

([2, 4, 5, 4, 2, 6], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4]),

([3, 4, 3, 5], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5]),
([1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5]),

([3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4]),

([2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4]),

([1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4]),

([1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 4, 3, 5]), ([2, 4, 2, 3, 4, 5, 6, 5, 4, 3], [3, 4, 3, 5]),

([2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [3, 4, 3, 5]),

([1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [2, 4, 5, 4, 2, 6]), ([3, 4, 3, 5], [2, 4, 5, 4, 2, 6]),
([1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [2, 4, 5, 4, 2, 6]),
([3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7]),
([2, 4, 5, 4, 2, 6], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7]),
([2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7]),

([1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8]),
([1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8]),

([2, 4, 2, 3, 4, 5, 6, 5, 4, 3], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8]), ([2, 4, 5, 4, 2, 6], [2, 4, 2, 3, 4, 5, 6, 5, 4, 3]),

([2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [2, 4, 2, 3, 4, 5, 6, 5, 4, 3]),

([1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2]),
([1, 3, 4, 5, 6, 5, 4, 3, 1, 7], [1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2]),
([3, 4, 3, 5], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1]),

([2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1])},

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} is a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to \binom{m/2}{2} elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2}.

In total, we obtain

1 + 20\frac{m}{2} + 25\binom{m/2}{2} = \frac{(5m + 4)(5m + 2)}{8}

elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198f).

Next we consider the case in (6.198h). By Lemma 26, we are free to choose p = 15m/4. In particular, m must be divisible by 4. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{4}w_{\frac{m}{4} +1}c^{−4}, c^{4}w_{\frac{m}{4} +2}c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{1}c^{−3}, …, c^{3}w\frac{m}{4} c^{−3}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{4}w_{\frac{m}{4} +i}c^{−4}, i = 1, 2, …, \frac{3m}{4},  (6.225a)
w_{i} = c^{3}w_{i− \frac{3m}{4}} c^{−3}, i = \frac{3m}{4} + 1, \frac{3m}{4} + 2, …, m.  (6.225b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m, which we summarise as follows:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{4} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{4}} ) = ℓ_{T} (w_{i+ \frac{2m}{4}} ) = ℓ_{T} (w_{i+ \frac{3m}{4}} ) ≤ 2,  (6.226a)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iii) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{4} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{4}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{4}} ) = ℓ_{T} (w_{i_{1}+ \frac{2m}{4}} ) = ℓ_{T} (w_{i_{2}+ \frac{2m}{4}} ) = ℓ_{T} (w_{i_{1}+ \frac{3m}{4}} ) = ℓ_{T} (w_{i_{2}+ \frac{3m}{4}} ) = 1, (6.226b)

and all other w_{j} are equal to ε. Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have

w_{i}w_{i+ \frac{m}{4}} w_{i+ \frac{2m}{4}} w_{i+ \frac{3m}{4}} ≤_{T} c,

or

w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{4}} w_{i_{2}+ \frac{m}{4}} w_{i_{1}+ \frac{2m}{4}} w_{i_{2}+ \frac{2m}{4}} w_{i_{1}+ \frac{3m}{4}} w_{i_{2}+ \frac{3m}{4}} = c.

CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 125

Together with equations (6.225)–(6.226), this implies that

w_{i} = c^{15}w_{i}c^{−15} and w_{i}(c^{11}w_{i}c^{−11})(c^{7}w_{i}c^{−7})(c^{3}w_{i}c^{−3}) ≤_{T} c,  (6.227)

or that

w_{i_{1}} = c^{15}w_{i_{1}}c^{−15}, w_{i_{1}} = c^{15}w_{i_{2}}c^{−15},

and w_{i_{1}}w_{i_{2}}(c^{11}w_{i_{1}}c^{−11})(c^{11}w_{i_{2}}c^{−11})(c^{7}w_{i_{1}}c^{−7})(c^{7}w_{i_{2}}c^{−7})(c^{3}w_{i_{1}}c^{−3})(c^{3}w_{i_{2}}c^{−3}) = c. (6.228)

Here, the first equation in (6.227) and the first two equations in (6.228) are automatically satisfied due to Lemma 29 with d = 2.

With the help of Stembridge’s Maple package coxeter [35], one obtains 30 solutions for w_{i} in (6.227) with ℓ_{T} (w_{i}) = 1:

w_{i} ∈ {[4], [5], [6], [7], [8], [3, 4, 3], [4, 5, 4], [5, 6, 5], [6, 7, 6], [7, 8, 7], [2, 4, 5, 4, 2], [3, 4, 5, 4, 3], [2, 4, 5, 6, 5, 4, 2], [1, 3, 4, 5, 6, 5, 4, 3, 1], [4, 2, 3, 4, 5, 4, 2, 3, 4], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1], [5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5],

[2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1], [4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4],

[1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[3, 1, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[2, 4, 3, 1, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7]},

one obtains 45 solutions for w_{i} in (6.227) with ℓ_{T} (w_{i}) = 2 and w_{i} of type A_{1}^{2} (as a parabolic Coxeter element; see the end of Section 2):

w_{i} ∈ {[1, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 4, 6, 5, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 6, 5, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 7, 6],

[1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[1, 3, 1, 4, 2, 7, 6, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 6, 8, 7],

[1, 3, 4, 3, 5, 4, 3, 6, 5, 4, 3, 1], [1, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 1, 4, 3, 5, 4, 3, 1, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 8],

[1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [2, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[2, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[2, 3, 4, 2, 3, 1, 4, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3],

[2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 5, 4, 2, 6, 5, 4, 2],

[2, 4, 3, 1, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [2, 4, 5, 4, 2, 7, 8, 7],

[3, 1, 4, 2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [3, 1, 4, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 7, 8], [3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4],

[3, 4, 3, 6, 7, 6], [3, 4, 5, 4, 3, 7, 8, 7], [4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],
[4, 2, 3, 4, 5, 4, 2, 3, 4, 7], [4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4], [4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],
[4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5], [4, 5, 4, 8], [4, 7, 8, 7],

[5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [5, 4, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5]}, (6.229)

and one obtains 20 solutions for w_{i} in (6.227) with ℓ_{T} (w_{i}) = 2 and w_{i} of type A_{2}:

w_{i} ∈ {[1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 8],

[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1], [1, 2, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3],

[1, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7],
[1, 4, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[2, 3, 4, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8],

[2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [2, 4, 5, 4, 2, 6], [3, 4, 2, 3, 4, 5, 4, 2], [3, 4, 3, 5], [3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2], [4, 5], [5, 6], [6, 7], [7, 8]}, (6.230)

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} is a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/4 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/4.

The number of solutions in Case (iii) can be computed from our knowledge of the solutions in Case (ii) according to type, using some elementary counting arguments. Namely, the number of solutions of (6.228) is equal to

45 · 2 + 20 · 3 = 150,

since an element of type A_{1}^{2} can be decomposed in two ways into a product of two elements of absolute length 1, while for an element of type A_{2} this can be done in 3 ways.

In total, we obtain 1 + (30 + 45 + 20)\frac{m}{4} + 150\binom{m/4}{2} = \frac{(5m+4)(15m+4)}{16} elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198h).

Finally we discuss the case in (6.198j). By Lemma 26, we are free to choose p = 15m/2. In particular, m must be divisible by 2. From (5.1), we infer

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{8}w_{\frac{m}{2} +1}c^{−8}, c^{8}w_{\frac{m}{2} +2}c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{1}c^{−7}, …, c^{7}w\frac{m}{2} c^{−7}).

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ϕ^{p}, we obtain the system of equations

w_{i} = c^{8}w_{\frac{m}{2} +i}c^{−8}, i = 1, 2, …, \frac{m}{2},  (6.231a)
w_{i} = c^{7}w_{i− \frac{m}{2}} c^{−7}, i = \frac{m}{2} + 1, \frac{m}{2} + 2, …, m.  (6.231b)

There are several distinct possibilities for choosing the w_{i}’s, 1 ≤ i ≤ m:

(i) all the w_{i}’s are equal to ε (and w_{0} = c), (ii) there is an i with 1 ≤ i ≤ \frac{m}{2} such that

1 ≤ ℓ_{T} (w_{i}) = ℓ_{T} (w_{i+ \frac{m}{2}} ) ≤ 4,  (6.232a)

(iii) there are i_{1} and i_{2} with 1 ≤ i_{1} < i_{2} ≤ \frac{m}{2} such that

ℓ_{1} := ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) ≥ 1, ℓ_{2} := ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) ≥ 1, and ℓ_{1} + ℓ_{2} ≤ 4,

(6.232b)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε, (iv) there are i_{1}, i_{2}, i_{3} with 1 ≤ i_{1} < i_{2} < i_{3} ≤ \frac{m}{2} such that

ℓ_{1} := ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) ≥ 1, ℓ_{2} := ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) ≥ 1, ℓ_{3} := ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) ≥ 1, and ℓ_{1} + ℓ_{2} + ℓ_{3} ≤ 4, (6.232c)

and the other w_{j}’s, 1 ≤ j ≤ m, are equal to ε,

(v) there are i_{1}, i_{2}, i_{3}, i_{4} with 1 ≤ i_{1} < i_{2} < i_{3} < i_{4} ≤ \frac{m}{2} such that

ℓ_{T} (w_{i_{1}}) = ℓ_{T} (w_{i_{2}}) = ℓ_{T} (w_{i_{3}}) = ℓ_{T} (w_{i_{4}})

= ℓ_{T} (w_{i_{1}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{2}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{3}+ \frac{m}{2}} ) = ℓ_{T} (w_{i_{4}+ \frac{m}{2}} ) = 1, (6.232d)

and all other w_{j}’s are equal to ε.

Moreover, since (w_{0}; w_{1}, …, w_{m}) ∈ NC^{m}(E_{8}), we must have w_{i}w_{i+ \frac{m}{2}} ≤_{T} c, respectively w_{i_{1}}w_{i_{2}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} ≤_{T} c, respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} ≤_{T} c,

respectively

w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}}w_{i_{1}+ \frac{m}{2}} w_{i_{2}+ \frac{m}{2}} w_{i_{3}+ \frac{m}{2}} w_{i_{4}+ \frac{m}{2}} = c.

Together with equations (6.231)–(6.232), this implies that

w_{i} = c^{15}w_{i}c^{−15} and w_{i}(c^{7}w_{i}c^{−7}) ≤_{T} c,  (6.233)

respectively that

w_{i_{1}} = c^{15}w_{i_{1}}c^{−15}, w_{i_{2}} = c^{15}w_{i_{2}}c^{−15}, and w_{i_{1}}w_{i_{2}}(c^{7}w_{i_{1}}c^{−7})(c^{7}w_{i_{2}}c^{−7}) ≤_{T} c, (6.234)

respectively that

w_{i_{1}} = c^{15}w_{i_{1}}c^{−15}, w_{i_{2}} = c^{15}w_{i_{2}}c^{−15}, w_{i_{3}} = c^{15}w_{i_{3}}c^{−15},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}(c^{7}w_{i_{1}}c^{−7})(c^{7}w_{i_{2}}c^{−7})(c^{7}w_{i_{3}}c^{−7}) ≤_{T} c, (6.235)

respectively that

w_{i_{1}} = c^{15}w_{i_{1}}c^{−15}, w_{i_{2}} = c^{15}w_{i_{2}}c^{−15}, w_{i_{3}} = c^{15}w_{i_{3}}c^{−15}, w_{i_{4}} = c^{15}w_{i_{4}}c^{−15},

and w_{i_{1}}w_{i_{2}}w_{i_{3}}w_{i_{4}}(c^{7}w_{i_{1}}c^{−7})(c^{7}w_{i_{2}}c^{−7})(c^{7}w_{i_{3}}c^{−7})(c^{7}w_{i_{4}}c^{−7}) = c. (6.236)

Here, the first equation in (6.233), the first two in (6.234), the first three in (6.235), and the first four in (6.236), are all automatically satisfied due to Lemma 29 with d = 2.

128 C. KRATTENTHALER AND T. W. MÜLLER

With the help of Stembridge’s Maple package coxeter [35], one obtains 45 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 1:

w_{i} ∈ {[1], [3], [4], [5], [6], [7], [8], [2, 4, 2], [3, 4, 3], [4, 5, 4], [5, 6, 5], [6, 7, 6], [7, 8, 7],

[2, 4, 5, 4, 2], [3, 4, 5, 4, 3], [1, 3, 4, 5, 4, 3, 1], [2, 4, 5, 6, 5, 4, 2], [3, 4, 5, 6, 5, 4, 3], [1, 3, 4, 5, 6, 5, 4, 3, 1], [4, 2, 3, 4, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 5, 4, 2, 3], [2, 4, 5, 6, 7, 6, 5, 4, 2],

[1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1], [1, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5],

[4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1], [4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],
[4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],
[3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],
[3, 1, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[2, 4, 3, 1, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7]},

one obtains 150 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 2 and w_{i} of type A_{1}^{2}:

w_{i} ∈ {[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 4, 6, 5, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],

[1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 4, 5, 4, 2], [1, 2, 4, 5, 6, 5, 4, 2],

[1, 2, 6, 5, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 7, 6], [1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[1, 3, 1, 4, 2, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 8],

[1, 3, 1, 4, 2, 7, 6, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 6, 8, 7],

[1, 3, 1, 4, 3, 1, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 3, 1, 4, 3, 5, 4, 3, 1, 6, 5, 4, 3, 1], [1, 3, 1, 4, 3, 5, 4, 3, 1, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],
[1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 3, 1, 4, 5, 6, 5, 4, 3, 1], [1, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 3, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 3, 4, 3, 5, 4, 3, 1],

[1, 3, 4, 3, 5, 4, 3, 6, 5, 4, 3, 1], [1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 3, 4, 5, 4, 3, 1, 7],

[1, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 1], [1, 3, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1], [1, 3, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 3, 4, 5, 6, 5, 4, 3, 1, 8],

[1, 3, 4, 5, 6, 5, 7, 6, 5, 4, 3, 1], [1, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 1, 4, 3, 5, 4, 3, 1, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[1, 4], [1, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5], [1, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 5, 4, 2, 3, 1, 4, 5], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 8], [1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 6], [1, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7], [1, 8],

[2, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [2, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[2, 3, 4, 2, 3, 1, 4, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 3, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 3, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 6, 7, 6, 5, 4, 2, 3], [2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3], [2, 3, 4, 3, 5, 6, 7, 6, 5, 4, 2, 3], [2, 3, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 5, 4, 6, 5, 4, 2, 3], [2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 6, 5, 4, 2, 3, 8], [2, 3, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3], [2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 4, 2, 3, 4],

[2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 5, 4, 2, 6, 5, 4, 2], [2, 4, 2, 5, 6, 5, 4, 2], [2, 4, 2, 5, 6, 7, 6, 5, 4, 2], [2, 4, 2, 6], [2, 4, 2, 8],

[2, 4, 3, 1, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [2, 4, 5, 4, 2, 7, 8, 7],

[2, 4, 5, 4, 2, 7], [2, 4, 5, 4, 2, 8], [2, 4, 5, 4, 6, 5, 4, 2], [2, 4, 5, 6, 5, 7, 6, 5, 4, 2], [3, 1, 4, 2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 4, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [3, 1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [3, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 7, 8], [3, 4, 2, 3, 4, 5, 4, 2, 6, 5, 4, 2, 3, 4], [3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [3, 4, 3, 5, 4, 3], [3, 4, 3, 6, 7, 6],

[3, 4, 3, 6], [3, 4, 3, 7], [3, 4, 5, 4, 3, 7, 8, 7], [3, 4, 5, 4, 6, 5, 4, 3], [3, 4, 5, 6, 5, 4, 3, 8], [3, 5], [3, 7],

[4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4], [4, 2, 3, 4, 5, 4, 2, 3, 4, 7], [4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 8],

[4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 4], [4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4], [4, 2, 3, 4],

[4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4], [4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5], [4, 5, 4, 8], [4, 7, 8, 7],

[4, 7], [4, 8], [5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [5, 4, 2, 3, 4, 5], [5, 4, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5], [5, 8], [6, 5, 4, 2, 3, 4, 5, 6]},

one obtains 100 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 2 and w_{i} of type A_{2}:

w_{i} ∈ {[1, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [1, 2, 3, 1, 4, 5, 6, 5, 4, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 2, 3, 1, 4, 5, 6, 5, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [1, 2, 3, 4, 3, 1, 5, 6, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 3, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 8], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 2, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 2, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 2, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1],

130 C. KRATTENTHALER AND T. W. MÜLLER

[1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1],

[1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 3, 4, 3, 1, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2],

[1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [1, 3, 4, 5, 4, 3, 1, 6, 7, 6], [1, 3, 4, 5, 4, 3, 1, 6], [1, 3, 4, 5, 4, 3],
[1, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7, 8, 7], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7],
[1, 3, 4, 5, 6, 5, 4, 3], [1, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1, 8], [1, 4, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 8], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8, 7], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5],

[1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8],

[2, 3, 1, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1],

[2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [2, 3, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5],

[2, 3, 4, 5, 6, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 5, 4, 2, 3, 7, 8, 7], [2, 3, 4, 5, 6, 5, 4, 2, 3, 7],

[2, 3, 4, 5, 6, 5, 4, 2], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2],

[2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],
[2, 4, 2, 3, 4, 5, 4, 3], [2, 4, 2, 3, 4, 5, 6, 5, 4, 3], [2, 4, 2, 5, 6, 5], [2, 4, 2, 5], [2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3],

[2, 4, 5, 4, 2, 6, 7, 6], [2, 4, 5, 4, 2, 6], [2, 4, 5, 6, 5, 4, 2, 7], [3, 1, 4, 2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8], [3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6],
[3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 4, 2, 3, 4, 5, 4, 2], [3, 4, 2, 3, 4, 5, 6, 5, 4, 2],
[3, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2], [3, 4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [3, 4, 3, 5, 6, 5],
[3, 4, 3, 5], [3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2], [3, 4, 5, 4, 3, 6], [3, 4, 5, 4], [3, 4],

[4, 2, 3, 1, 4, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 7, 6], [4, 2, 3, 4, 5, 4, 2, 3, 4, 6],

[4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5], [4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7, 8, 7], [4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7],

[4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 8], [4, 5], [5, 6], [6, 7], [7, 8]},

one obtains 75 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 3 and w_{i} of type A_{1}^{3}:

w_{i} ∈ {[1, 2, 3, 1, 4, 2, 5, 4, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 1, 4, 3, 5, 4, 3, 6, 5, 4, 3, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 3, 1, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3, 1],
[1, 2, 3, 4, 2, 5, 4, 2, 3, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 4, 8, 7, 6, 5, 4, 2, 3, 1],
[1, 2, 3, 4, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1],
[1, 2, 3, 4, 3, 5, 4, 3, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 6, 5, 4, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 4, 2, 3, 1, 4, 3, 5, 4, 3, 1, 6, 5, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 4, 2, 3, 1, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 4, 2, 3, 1, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],

[1, 2, 4, 2, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 4, 2, 5, 6, 5, 4, 2], [1, 2, 4, 5, 4, 2, 8],

[1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],
[1, 3, 1, 4, 2, 3, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 8],
[1, 3, 1, 4, 2, 3, 7, 6, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 6, 8, 7],
[1, 3, 1, 4, 2, 5, 4, 2, 3, 1, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 8],

[1, 3, 1, 4, 3, 1, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [1, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 3, 1, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1], [1, 3, 4, 3, 5, 4, 3, 1, 7], [1, 3, 4, 3, 5, 4, 3, 6, 5, 4, 3, 1, 8],

[1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 8], [1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 4, 8], [1, 5, 4, 2, 3, 1, 4, 5, 7],
[1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7],
[2, 3, 1, 4, 2, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],
[2, 3, 1, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[2, 3, 4, 2, 3, 1, 4, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 3, 6, 5, 4, 2, 3, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 6, 7, 6, 5, 4, 2, 3], [2, 3, 4, 3, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 5, 4, 6, 5, 4, 2, 3, 8], [2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3], [2, 4, 2, 3, 1, 4, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],

[2, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8], [2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 4, 2, 3, 4, 8], [2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4], [3, 1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 4, 5, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [3, 1, 4, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[3, 1, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8],

[3, 1, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8],

[3, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 7, 8], [3, 4, 3, 5, 4, 3, 7, 8, 7],

[3, 4, 5, 4, 6, 5, 4, 3, 8], [4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],
[4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 4, 2, 3, 4], [4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [4, 2, 3, 4, 6],

[4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [5, 4, 2, 3, 4, 5, 7, 8, 7], [5, 4, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 8], [5, 6, 5, 4, 2, 3, 4, 5, 6]},

one obtains 165 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 3 and w_{i} of type A_{1} ∗ A_{2}:

w_{i} ∈ {[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 3, 1, 4, 2, 5, 4, 2, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],

[1, 2, 3, 1, 4, 2, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1],
[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3], [1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 1, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 2, 3, 1, 4, 5, 4, 6, 5, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 1, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3],

[1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1, 8], [1, 2, 3, 4, 3, 1, 5, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 2, 3, 4, 3, 5, 4, 3, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4],
[1, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1],
[1, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4],
[1, 2, 3, 4, 5, 4, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 2, 3, 1, 8],
[1, 2, 3, 4, 5, 6, 5, 4, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 2, 3, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3],
[1, 2, 4, 2, 3, 1, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 7, 8],
[1, 2, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 2, 4, 2, 3, 1, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 7, 6], [1, 2, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4],

132 C. KRATTENTHALER AND T. W. MÜLLER

[1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 1, 4, 8], [1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 4],

[1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 4], [1, 2, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1, 8],

[1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 4, 2, 6], [1, 2, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1],

[1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],

[1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 3, 1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[1, 3, 1, 4, 2, 5, 4, 2, 3, 1, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 6, 8, 7], [1, 3, 1, 4, 3, 1, 5, 4, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[1, 3, 1, 4, 3, 5, 4, 3, 1, 6, 5, 4, 3, 1, 7, 8, 7], [1, 3, 1, 4, 3, 5, 4, 3, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 3, 1, 4, 3, 5, 4, 3, 6, 5, 4, 3, 1], [1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 3, 1, 4, 5, 6, 5, 4, 3, 1, 7],

[1, 3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3], [1, 3, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 3, 4, 2, 3, 4, 5, 4, 2, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 3, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 3, 4, 2, 3, 5, 4, 2, 3, 1, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],

[1, 3, 4, 2, 3, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],

[1, 3, 4, 3, 1, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4], [1, 3, 4, 3, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2],

[1, 3, 4, 3, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [1, 3, 4, 3, 5, 4, 3], [1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8],

[1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 1],

[1, 3, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1, 8], [1, 3, 4, 5, 6, 5, 4, 3, 8], [1, 4, 2, 3, 1, 4, 3, 5, 4, 3, 1, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 4, 2, 3, 1, 4, 3, 5, 4, 3, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 4, 2, 3, 4, 5, 4, 3, 1, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 4], [1, 4, 3, 1, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4],

[1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8, 7], [1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7],

[1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5], [1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8],

[1, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 4, 3, 1], [1, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 7, 8, 7, 6, 5, 4, 3, 1], [1, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8, 7],

[1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [1, 5, 4, 2, 3, 4, 5], [1, 6, 5, 4, 2, 3, 4, 5, 6],

[2, 3, 1, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[2, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[2, 3, 1, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [2, 3, 1, 4, 2, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 1, 4, 2, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 1, 4, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1], [2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 2, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 3, 7, 6, 5, 4, 2, 3, 8], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 3, 7, 6, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3, 4], [2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3, 7, 8, 7], [2, 3, 4, 2, 5, 4, 2, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 8],

[2, 3, 4, 2, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 6, 5, 4, 2, 3, 7], [2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 4],

[2, 3, 4, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3], [2, 3, 4, 3, 5, 6, 7, 6, 5, 4, 2, 3, 8], [2, 3, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 8],

[2, 3, 4, 5, 4, 6, 5, 4, 2], [2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 8], [2, 3, 4, 5, 6, 5, 7, 6, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 3, 1, 4, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],

[2, 4, 2, 3, 1, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [2, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [2, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 4, 2, 3, 4, 7, 8, 7], [2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 4, 2, 3, 4, 7],

[2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 7, 6, 5, 4, 2, 3, 4], [2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 4, 8], [2, 4, 2, 3, 4, 5, 6, 5, 4, 3, 8],

[2, 4, 2, 5, 4, 6, 5, 4, 2], [2, 4, 2, 5, 6, 5, 4, 2, 7], [2, 4, 2, 5, 8], [2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 8],

[2, 4, 5, 4, 2, 7, 8], [2, 4, 5, 6, 5, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [3, 1, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [3, 1, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5],

[3, 1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8], [3, 1, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6],

[3, 1, 6, 5, 4, 2, 3, 1, 4, 5, 6], [3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7],

[3, 4, 2, 3, 4, 5, 4, 2, 3, 1, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [3, 4, 2, 3, 4, 5, 4, 2, 6, 5, 4, 2, 3, 4, 7, 8, 7], [3, 4, 2, 3, 4, 5, 4, 2, 7], [3, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2], [3, 4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[3, 4, 3, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2], [3, 4, 3, 6, 7], [3, 4, 5, 4, 3, 1, 6, 5, 4, 2, 3, 1, 4, 5, 6], [3, 4, 7], [4, 2, 3, 1, 4, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [4, 5, 8], [4, 7, 8],

[1, 3, 1, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3]},

one obtains 90 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 3 and w_{i} of type A_{3}:

w_{i} ∈ {[1, 2, 3, 1, 4, 3, 1, 5, 6, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],

[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4],

[1, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 2, 3, 1, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 1, 4, 5, 6, 5, 4, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8], [1, 2, 3, 1, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5], [1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3],

[1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8],

[1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 8], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4],

[1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 2, 4, 2, 3, 1, 4, 3, 5, 4, 3, 1, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[1, 2, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 2, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 1], [1, 2, 4, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1], [1, 2, 4, 2, 3, 4, 5, 4, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[1, 2, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1, 8], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1, 7, 8, 7], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1, 7], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 3, 1], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1, 8],

[1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3], [1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2, 3, 8], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 2], [1, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3],

[1, 3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 7],

[1, 3, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 4, 2], [1, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2],

[1, 3, 4, 5, 4, 3, 1, 6, 7], [1, 3, 4, 5, 4, 3, 6], [1, 3, 4, 5, 6, 5, 4, 3, 1, 7, 8], [1, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8], [1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 8], [1, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6], [1, 5, 4, 2, 3, 1, 4, 5, 6], [1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8], [1, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7],

[2, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1, 8], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2],

[2, 3, 4, 2, 5, 6, 5, 4, 2], [2, 3, 4, 2, 5, 6, 7, 6, 5, 4, 2], [2, 3, 4, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5], [2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7, 8, 7], [2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7], [2, 3, 4, 5, 6, 5, 4, 2, 3, 7, 8], [2, 3, 4, 5, 6, 5, 4, 2, 7], [2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 8], [2, 4, 2, 3, 4, 5, 4, 3, 6],

[2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3], [2, 4, 2, 5, 6], [2, 4, 5, 4, 2, 6, 7], [3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[3, 1, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[3, 1, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7],

[3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8],

134 C. KRATTENTHALER AND T. W. MÜLLER

[3, 4, 2, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1], [3, 4, 2, 3, 4, 5, 4, 2, 6, 7, 6], [3, 4, 2, 3, 4, 5, 4, 2, 6], [3, 4, 2, 3, 4, 5, 4, 6, 5, 4, 2],

[3, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7], [3, 4, 3, 5, 6], [3, 4, 5], [4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 7],

[4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7, 8], [4, 2, 3, 4, 5, 6, 5], [4, 2, 3, 4, 5], [5, 4, 2, 3, 4, 5, 6]},

one obtains 15 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 4 and w_{i} of type A_{1}^{2} ∗ A_{2}:

w_{i} ∈ {[1, 2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 3, 1, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[1, 2, 3, 4, 5, 4, 6, 5, 4, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 3, 1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7], [1, 3, 1, 4, 2, 3, 5, 4, 2, 3, 1, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1, 7, 6, 8, 7],

[1, 4, 3, 1, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4],

[1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8, 7],
[1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8],
[2, 4, 2, 3, 1, 4, 5, 4, 3, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8],
[2, 4, 2, 3, 1, 4, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7],
[3, 1, 4, 2, 3, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],
[3, 1, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 8],
[3, 4, 2, 3, 4, 5, 4, 2, 3, 1, 4, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8],

[4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 4, 2, 3, 4, 7, 8, 7]}, (6.237)

one obtains 45 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 4 and w_{i} of type A_{1} ∗ A_{3}:

w_{i} ∈ {[1, 2, 3, 1, 4, 2, 5, 4, 2, 6, 5, 4, 2, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 1, 4, 2, 5, 4, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 1, 4, 3, 1, 5, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7],
[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 2, 3, 1, 4, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 8], [1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3, 8, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 2, 3, 5, 4, 6, 5, 7, 6, 5, 4, 2, 3, 4, 5, 8, 7, 6, 5, 4, 2, 3, 1], [1, 2, 3, 4, 2, 5, 4, 2, 3, 4, 6, 5, 4, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1],

[1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8], [1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[1, 2, 3, 4, 3, 5, 4, 6, 5, 4, 3, 7, 6, 5, 4, 2, 3, 4], [1, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 7, 6, 5, 4, 2, 3],

[1, 2, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3], [1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 7, 6, 5, 4, 2, 3, 4, 8],

[1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 8], [1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [1, 3, 1, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 2, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 4],

[1, 3, 4, 2, 3, 5, 4, 2, 3, 1, 4, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 7, 8], [1, 3, 4, 3, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 7],

[1, 3, 4, 3, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2], [1, 4, 2, 3, 1, 4, 3, 5, 4, 3, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 4, 2, 3, 4, 5, 4, 2, 3, 4, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4],

[1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 5, 7, 8], [1, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5, 4, 3, 1, 7, 8, 7],

[2, 3, 1, 4, 2, 5, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1], [2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 6, 5, 4, 3, 1, 8], [2, 3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 1], [2, 3, 4, 2, 3, 1, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 2, 3, 5, 4, 2, 3, 6, 5, 7, 6, 8, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 3, 5, 4, 2, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3],

[2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 7, 6, 5, 4, 2, 3], [2, 3, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3, 4, 7, 8, 7],
[2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 8], [2, 4, 2, 3, 1, 4, 5, 4, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8],

[2, 4, 2, 3, 1, 5, 6, 5, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 4, 2, 3, 4, 7, 8], [2, 4, 2, 3, 4, 5, 4, 6, 5, 4, 3, 8], [3, 1, 4, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4], [3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],

[3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7], [3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8, 7],

[3, 1, 4, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6], [3, 4, 2, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 7, 6, 8, 7, 6, 5, 4, 3, 1]}, (6.238)

one obtains 5 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 4 and w_{i} of type A_{2}^{2}:

w_{i} ∈ {[1, 2, 3, 4, 2, 5, 4, 2, 3, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4],

[1, 2, 4, 2, 3, 4, 5, 4, 3, 6, 5, 7, 6, 5, 4, 2, 3, 4],

[1, 3, 1, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 7, 6, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5],
[2, 3, 1, 4, 2, 5, 4, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 1],

[2, 3, 4, 2, 3, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 8]}, (6.239)

one obtains 18 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 4 and w_{i} of type A_{4}:

w_{i} ∈ {[1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 2, 3, 1, 4], [1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6],

[1, 2, 3, 4, 2, 5, 4, 2, 6, 5, 7, 8, 7, 6, 5, 4, 2, 3], [1, 2, 3, 4, 2, 5, 4, 2, 6, 7, 8, 7, 6, 5, 4, 2, 3, 4], [1, 2, 3, 4, 2, 5, 4, 6, 5, 4, 2, 7, 6, 5, 4, 2, 3, 8], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 1, 4, 5, 8], [1, 2, 4, 2, 3, 4, 5, 4, 6, 5, 7, 6, 5, 4, 3, 1],

[1, 2, 4, 2, 3, 4, 5, 4, 6, 7, 6, 5, 4, 3, 1, 8], [1, 2, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 3, 1, 7, 8],
[1, 3, 1, 4, 5, 4, 2, 3, 1, 4, 5, 6, 5, 7, 6, 8, 7, 6, 5, 4, 2, 3], [1, 4, 2, 3, 1, 4, 3, 5, 4, 6, 7, 8, 7, 6, 5, 4, 3, 1],
[1, 5, 4, 2, 3, 4, 5, 6], [2, 3, 4, 2, 5, 4, 6, 5, 4, 2], [2, 3, 4, 2, 5, 6, 5, 4, 2, 7], [2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 7, 8],

[2, 4, 2, 3, 1, 7, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [3, 1, 6, 5, 4, 2, 3, 1, 4, 5, 6, 7], [3, 4, 2, 3, 4, 5, 4, 2, 6, 7]}, (6.240)

and one obtains 5 solutions for w_{i} in (6.233) with ℓ_{T} (w_{i}) = 4 and w_{i} of type D_{4}:

w_{i} ∈ {[1, 2, 3, 1, 4, 5, 6, 7, 8, 7, 6, 5, 4, 2, 3, 1, 4, 5], [1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 2, 3, 4, 8], [1, 5, 4, 2, 3, 1, 4, 5, 6, 7], [3, 1, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6, 7, 8], [4, 2, 3, 4, 5, 6]}, (6.241)

where {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}} is a simple system of generators of E_{8}, corresponding to the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/2 elements of Fix_{NC^{m}(E_{8})}(ϕ^{p}) since i ranges from 1 to m/2. There are no solutions for w_{i} in (6.233) with w_{i} of type A_{1}^{4}.

Letting the computer find all solutions in cases (iii)–(v) would take years. However, the number of these solutions can be computed from our knowledge of the solutions in Case (ii) according to type, if this information is combined with the decomposition numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary (multiset) permutation counting. The decomposition numbers for A_{2}, A_{3}, A_{4}, and D_{4} of which we make use can be found in the appendix of [18].

To begin with, the number of solutions of (6.234) with ℓ_{1} = ℓ_{2} = 1 is equal to

n_{1,1} := 150 · 2 + 100 · N_{A_{2}}(A_{1}, A_{1}) = 600,

since an element of type A_{1}^{2} can be decomposed in two ways into a product of two elements of absolute length 1, while for an element of type A_{2} this can be done in N_{A_{2}}(A_{1}, A_{1}) = 3 ways. Similarly, the number of solutions of (6.234) with ℓ_{1} = 2 and ℓ_{2} = 1 is equal to

n_{2,1} := 75 · 3 + 165 · (1 + N_{A_{2}}(A_{1}, A_{1})) + 90 · N_{A_{3}}(A_{2}, A_{1}) = 1425,

the number of solutions of (6.234) with ℓ_{1} = 3 and ℓ_{2} = 1 is equal to

n_{3,1} := 15 · (2 + N_{A_{2}}(A_{1}, A_{1})) + 45 · (1 + N_{A_{3}}(A_{2}, A_{1})) + 5 · (2N_{A_{2}}(A_{1}, A_{1}))

+ 18 · (N_{A_{4}}(A_{3}, A_{1}) + N_{A_{4}}(A_{1} ∗ A_{2}, A_{1})) + 5 · (N_{D_{4}}(A_{3}, A_{1}) + N_{D_{4}}(A_{1}^{3}, A_{1})) = 660, the number of solutions of (6.234) with ℓ_{1} = ℓ_{2} = 2 is equal to

n_{2,2} := 15 · (2 + 2N_{A_{2}}(A_{1}, A_{1})) + 45 · (2N_{A_{3}}(A_{2}, A_{1})) + 5 · (2 + N_{A_{2}}(A_{1}, A_{1})^{2})

+ 18 · (N_{A_{4}}(A_{2}, A_{2}) + N_{A_{4}}(A_{1}^{2}, A_{1}^{2}) + 2N_{A_{4}}(A_{2}, A_{1}^{2}))

+ 5 · (N_{D_{4}}(A_{2}, A_{2}) + 2N_{D_{4}}(A_{2}, A_{1}^{2})) = 1195,

the number of solutions of (6.235) with ℓ_{1} = ℓ_{2} = ℓ_{3} = 1 is equal to

n_{1,1,1} := 75 · 3! + 165 · (3N_{A_{2}}(A_{1}, A_{1})) + 90N_{A_{3}}(A_{1}, A_{1}, A_{1}) = 3375,

the number of solutions of (6.235) with ℓ_{1} = 2 and ℓ_{2} = ℓ_{3} = 1 is equal to

n_{2,1,1} := 15·(2+N_{A_{2}}(A_{1}, A_{1})+2·2·N_{A_{2}}(A_{1}, A_{1}))+45·(2N_{A_{3}}(A_{2}, A_{1})+N_{A_{3}}(A_{1}, A_{1}, A_{1}))

+ 5 · (2N_{A_{2}}(A_{1}, A_{1}) + 2N_{A_{2}}(A_{1}, A_{1})^{2}) + 18 · (N_{A_{4}}(A_{2}, A_{1}, A_{1}) + N_{A_{4}}(A_{1}^{2}, A_{1}, A_{1}))

+ 5 · (N_{D_{4}}(A_{2}, A_{1}, A_{1}) + N_{D_{4}}(A_{1}^{2}, A_{1}, A_{1})) = 2850,

and the number of solutions of (6.236) is equal to

n_{1,1,1,1} := 15 · (12N_{A_{2}}(A_{1}, A_{1})) + 45 · (4N_{A_{3}}(A_{1}, A_{1}, A_{1})) + 5 · (6N_{A_{2}}(A_{1}, A_{1})^{2}) + 18 · N_{A_{4}}(A_{1}, A_{1}, A_{1}, A_{1}) + 5 · N_{D_{4}}(A_{1}, A_{1}, A_{1}, A_{1}) = 6750.

In total, we obtain

1+(45+150+100+75+165+90+15+45+5+18+5)\frac{m}{2}+(n_{1,1}+2n_{2,1}+2n_{3,1}+n_{2,2})\binom{m/2}{2}

+ (n_{1,1,1} + 3n_{2,1,1})\binom{m/2}{3} + n_{1,1,1,1}\binom{m/2}{4} = \frac{(5m + 4)(3m + 2)(5m + 2)(15m + 4)}{64}

elements in Fix_{NC^{m}(E_{8})}(ϕ^{p}), which agrees with the limit in (6.198j).

Finally, we turn to (6.198l). By Remark 3, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 7, h_{2} = 2 and m_{2} = 8, h_{2} = 2 and m_{2} = 7, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 30m/7, p = 15m/8, p = 15m/7, p = 15m/4, respectively 15m/2, out of which only p = 15m/8 has not yet been discussed and belongs to the current case. The corresponding action of ϕ^{p} is given by

ϕ^{p}((w_{0}; w_{1}, …, w_{m}))

= (∗; c^{2}w_{\frac{m}{8} +1}c^{−2}, c^{2}w_{\frac{m}{8} +2}c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{1}c^{−1}, …, cw\frac{m}{8} c^{−1}),

so that we have to solve

t_{1}(c^{13}t_{1}c^{−13})(c^{11}t_{1}c^{−11})(c^{9}t_{1}c^{−9})(c^{7}t_{1}c^{−7})(c^{5}t_{1}c^{−5})(c^{3}t_{1}c^{−3})(ct_{1}c^{−1}) = c  (6.242)

for t_{1} with ℓ_{T} (t_{1}) = 1. A computation with the help of Stembridge’s Maple package coxeter [35] finds no solution. Hence, the left-hand side of (3.3) is equal to 1, as required.

7. Cyclic sieving II

In this section we present the second cyclic sieving conjecture due to Bessis and Reiner [10, Conj. 6.5].

Let ψ : NC^{m}(W) → NC^{m}(W) be the map defined by

(w_{0}; w_{1}, …, w_{m}) ↦ (cw_{m}c^{−1}; w_{0}, w_{1}, …, w_{m−1}).  (7.1)

For m = 1, we have w_{0} = cw_{1}^{−1}, so that this action reduces to the inverse of the Kreweras complement K_{id}^{c} as defined by Armstrong [3, Def. 2.5.3].

It is easy to see that ψ^{(m+1)h} acts as the identity, where h is the Coxeter number of W (see (8.1) below). By slight abuse of notation as before, let C_{2} be the cyclic group of order (m + 1)h generated by ψ.

Given these definitions, we are now in the position to state the second cyclic sieving conjecture of Bessis and Reiner. By the results of [19] and of this paper, it becomes the following theorem.

Theorem 33. For an irreducible well-generated complex reflection group W and any m ≥ 1, the triple (NC^{m}(W), Cat^{m}(W; q), C_{2}), where Cat^{m}(W; q) is the q-analogue of the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.

By definition of the cyclic sieving phenomenon, we have to prove that

| Fix_{NC^{m}(W)}(ψ^{p})| = Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}},  (7.2)

for all p in the range 0 ≤ p < (m + 1)h.

8. Auxiliary results II

This section collects several auxiliary results which allow us to reduce the problem of proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional groups listed in Section 2 to a finite problem. While Lemmas 35 and 36 cover special choices of the parameters, Lemmas 34 and 37 afford an inductive procedure. More precisely, if we assume that we have already verified Theorem 33 for all groups of smaller rank, then Lemmas 34 and 37, together with Lemmas 35 and 38, reduce the verification of Theorem 33 for the group that we are currently considering to a finite problem; see Remark 4. The final lemma of this section, Lemma 39, disposes of complex reflection groups with a special property satisfied by their degrees.

Let p = a(m + 1) + b, 0 ≤ b < m + 1. We have

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{a+1}w_{m−b+1}c^{−a−1}; c^{a+1}w_{m−b+2}c^{−a−1}, …, c^{a+1}w_{m}c^{−a−1},
c^{a}w_{0}c^{−a}, …, c^{a}w_{m−b}c^{−a}). (8.1)

Lemma 34. It suffices to check (7.2) for p a divisor of (m+1)h. More precisely, let p be a divisor of (m + 1)h, and let k be another positive integer with gcd(k, (m + 1)h/p) = 1, then we have

Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}} = Cat^{m}(W; q)|_{q=e^{2πikp/(m+1)h}}  (8.2)

and

| Fix_{NC^{m}(W)}(ψ^{p})| = | Fix_{NC^{m}(W)}(ψ^{kp})|.  (8.3)

Proof. For (8.3), this follows in the same way as (5.3) in Lemma 26.

For (8.2), we must argue differently than in Lemma 26. Let us write ζ = e^{2πip/(m+1)h}. For a given group W, we write S_{1}(W) for the set of all indices i such that ζ^{d_{i}−h} = 1, and we write S_{2}(W) for the set of all indices i such that ζ^{d_{i}} = 1. By the rule of de l’Hospital, we have

Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}} = \begin{cases}0 & if |S_{1}(W)| > |S_{2}(W)|, \\ \frac{∏_{i∈S_{1}(W)}(mh+d_{i})}{∏_{i∈S_{2}(W)} d_{i}} \frac{∏_{i∉S_{1}(W)}(1−ζ^{d_{i}−h})}{∏_{i∉S_{2}(W)}(1−ζ^{d_{i}})}, & if |S_{1}(W)| = |S_{2}(W)|. \end{cases}

(8.4)

Since, by Theorem 25, Cat^{m}(W; q) is a polynomial in q, the case |S_{1}(W)| < |S_{2}(W)| cannot occur.

We claim that, for the case where |S_{1}(W)| = |S_{2}(W)|, the factors in the quotient of products

\frac{∏_{i∉S_{1}(W)}(1 − ζ^{d_{i}−h})}{∏_{i∉S_{2}(W)}(1 − ζ^{d_{i}})}

cancel pairwise. If we assume the correctness of the claim, it is obvious that we get the same result if we replace ζ by ζ^{k}, where gcd(k, (m + 1)h/p) = 1, hence establishing (8.2).

In order to see that our claim is indeed valid, we proceed in a case-by-case fashion, making appeal to the classification of irreducible well-generated complex reflection groups, which we recalled in Section 2. First of all, since d_{n} = h, the set S_{1}(W) is always non-empty as it contains the element n. Hence, if we want to have |S_{1}(W)| = |S_{2}(W)|, the set S_{2}(W) must be non-empty as well. In other words, the integer (m + 1)h/p must divide at least one of the degrees d_{1}, d_{2}, …, d_{n}. In particular, this implies that, for each fixed reflection group W of exceptional type, only a finite number of values of (m + 1)h/p has to be checked. Writing M for (m + 1)h/p, what needs to be checked is whether the multisets (that is, multiplicities of elements must be taken into account)

{(d_{i} − h) mod M : i ∉ S_{1}(W)} and {d_{i} mod M : i ∉ S_{2}(W)}

are the same. Since, for a fixed irreducible well-generated complex reflection group, there is only a finite number of possibilities for M, this amounts to a routine verification.

□

Lemma 35. Let p be a divisor of (m + 1)h. If p is divisible by m + 1, then (7.2) is true.

Proof. According to (8.1), the action of ψ^{p} on NC^{m}(W) is described by

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{p/(m+1)}w_{0}c^{−p/(m+1)}; c^{p/(m+1)}w_{1}c^{−p/(m+1)}, …, c^{p/(m+1)}w_{m}c^{−p/(m+1)}).

Hence, if (w_{0}; w_{1}, …, w_{m}) is fixed by ψ^{p}, then each individual w_{i} must be fixed under conjugation by c^{p/(m+1)}.

Using the notation W′ = Cent_{W} (c^{p/(m+1)}), the previous observation means that w_{i} ∈ W′, i = 1, 2, …, m. By the theorem of Springer cited in the proof of Lemma 27 and by (5.5), the tuples (w_{0}; w_{1}, …, w_{m}) fixed by ψ^{p} are in fact identical with the elements of NC^{m}(W′), which implies that

| Fix_{NC^{m}(W)}(ψ^{p})| = |NC^{m}(W′)|.  (8.5)

Application of Theorem 1 with W replaced by W′ and of the “limit rule” (5.4) then yields that

|NC^{m}(W′)| = _{∏_{1≤i≤n} ; \frac{(m+1)h}{p} |d_{i}} \frac{mh + d_{i}}{d_{i}} = Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}}.  (8.6)

Combining (8.5) and (8.6), we obtain (7.2). This finishes the proof of the lemma. □

Lemma 36. Equation (7.2) holds for all divisors p of m + 1.

Proof. We have

Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}} = \begin{cases}0 & if p < m + 1, \\ m + 1 & if p = m + 1. \end{cases}

Here, the first case follows from (8.4) and the fact that we have S_{1}(W) ⊇ {n} and S_{2}(W) = ∅ if p | (m + 1) and p < m + 1.

On the other hand, if (w_{0}; w_{1}, …, w_{m}) is fixed by ψ^{p}, then, because of the action (8.1), we must have w_{0} = w_{p} = ⋯ = w_{m−p+1} and w_{0} = cw_{m−p+1}c^{−1}. In particular, w_{0} ∈ Cent_{W} (c). By the theorem of Springer cited in the proof of Lemma 27, the subgroup Cent_{W} (c) is itself a complex reflection group whose degrees are those degrees of W that are divisible by h. The only such degree is h itself, hence Cent_{W} (c) is the cyclic group generated by c. Moreover, by (5.5), we obtain that w_{0} = ε or w_{0} = c. If p = m + 1, the set Fix_{NC^{m}(W)}(ψ^{p}) consists of the m + 1 elements (w_{0}; w_{1}, …, w_{m}) obtained by choosing w_{i} = c for a particular i between 0 and m, all other w_{j}’s being equal to ε. If p < m + 1, then there is no element in Fix_{NC^{m}(W)}(ψ^{p}). □

Lemma 37. Let W be an irreducible well-generated complex reflection group of rank n, and let p = m_{1}h_{1} be a divisor of (m + 1)h, where m + 1 = m_{1}m_{2} and h = h_{1}h_{2}. We assume that gcd(h_{1}, m_{2}) = 1. Suppose that Theorem 33 has already been verified for all irreducible well-generated complex reflection groups with rank < n. If h_{2} does not divide all degrees d_{i}, then equation (7.2) is satisfied.

Proof. Let us write h_{1} = am_{2} + b, with 0 ≤ b < m_{2}. The condition gcd(h_{1}, m_{2}) = 1 translates into gcd(b, m_{2}) = 1. From (8.1), we infer that

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{a+1}w_{m−m_{1}b+1}c^{−a−1}; c^{a+1}w_{m−m_{1}b+2}c^{−a−1}, …, c^{a+1}w_{m}c^{−a−1},
c^{a}w_{0}c^{−a}, …, c^{a}w_{m−m_{1}b}c^{−a}). (8.7)

Supposing that (w_{0}; w_{1}, …, w_{m}) is fixed by ψ^{p}, we obtain the system of equations

w_{i} = c^{a+1}w_{i+m−m_{1}b+1}c^{−a−1}, i = 0, 1, …, m_{1}b − 1,
w_{i} = c^{a}w_{i−m_{1}b}c^{−a}, i = m_{1}b, m_{1}b + 1, …, m,

which, after iteration, implies in particular that

w_{i} = c^{b(a+1)+(m_{2}−b)a}w_{i}c^{−b(a+1)−(m_{2}−b)a} = c^{h_{1}}w_{i}c^{−h_{1}}, i = 0, 1, …, m.

It is at this point where we need gcd(b, m_{2}) = 1. The last equation shows that each w_{i}, i = 0, 1, …, m, lies in Cent_{W} (c^{h_{1}}). By the theorem of Springer cited in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group, W′ say, whose degrees are those degrees of W that are divisible by h/h_{1} = h_{2}. Since, by assumption, h_{2} does not divide all degrees, W′ has rank strictly less than n. Again by assumption, we know that Theorem 33 is true for W′, so that in particular,

| Fix_{NC^{m}(W′)}(ψ^{p})| = Cat^{m}(W′; q)|_{q=e^{2πip/(m+1)h}}.

The arguments above together with (5.5) show that Fix_{NC^{m}(W)}(ψ^{p}) = Fix_{NC^{m}(W′)}(ψ^{p}). On the other hand, it is straightforward to see that

Cat^{m}(W; q)|_{q=e^{2πip/(m+1)h}} = Cat^{m}(W′; q)|_{q=e^{2πip/(m+1)h}}.

This proves (7.2) for our particular p, as required. □

Lemma 38. Let W be an irreducible well-generated complex reflection group of rank n, and let p = m_{1}h_{1} be a divisor of (m + 1)h, where m + 1 = m_{1}m_{2} and h = h_{1}h_{2}. We assume that gcd(h_{1}, m_{2}) = 1. If m_{2} > n then

Fix_{NC^{m}(W)}(ψ^{p}) = ∅.

Proof. Let us suppose that (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(W)}(ψ^{p}) and that there exists a j ≥ 1 such that w_{j} ≠ ε. By (8.7), it then follows for such a j that also w_{k} ≠ ε for all k ≡ j − lm_{1}b (mod m + 1), where, as before, b is defined as the unique integer with h_{1} = am_{2} + b and 0 ≤ b < m_{2}. Since, by assumption, gcd(b, m_{2}) = 1, there are exactly m_{2} such k’s which are distinct mod m + 1. However, this implies that the sum of the absolute lengths of the w_{i}’s, 0 ≤ i ≤ m, is at least m_{2} > n, a contradiction. This leaves as only possibility (w_{0}; w_{1}, …, w_{m}) = (c; ε, …, ε). However, this is clearly not an element of Fix_{NC^{m}(W)}(ψ^{p}) unless p is divisible by m + 1. This is impossible since

\frac{p}{m + 1} = \frac{m_{1}h_{1}}{m_{1}m_{2}} = \frac{h_{1}}{m_{2}}

is not an integer by our hypotheses. □

Remark 4. (1) If we put ourselves in the situation of the assumptions of Lemma 37, then we may conclude that equation (7.2) only needs to be checked for pairs (m_{2}, h_{2}) subject to the following restrictions:

m_{2} ≥ 2, gcd(h_{1}, m_{2}) = 1, and h_{2} divides all degrees of W .  (8.8)

Indeed, Lemmas 35 and 37 together imply that equation (7.2) is always satisfied except if m_{2} ≥ 2, h_{2} divides all degrees of W, and gcd(h_{1}, m_{2}) = 1.

(2) Still putting ourselves in the situation of Lemma 37, if m_{2} > n and m_{2}h_{2} does not divide any of the degrees of W, then equation (7.2) is satisfied. Indeed, Lemma 38 says that in this case the left-hand side of (7.2) equals 0, while it is obvious that in this case the right-hand side of (7.2) equals 0 as well.

(3) It should be observed that this leaves a finite number of choices for m_{2} to consider, whence a finite number of choices for (m_{1}, m_{2}, h_{1}, h_{2}). Altogether, there remains a finite number of choices for p = h_{1}m_{1} to be checked.

Lemma 39. Let W be an irreducible well-generated complex reflection group of rank n with the property that d_{i} | h for i = 1, 2, …, n. Then Theorem 33 is true for this group W.

Proof. By Lemma 34, we may restrict ourselves to divisors p of (m + 1)h.

Suppose that e^{2πip/(m+1)h} is a d_{i}-th root of unity for some i. In other words, (m+1)h/p divides d_{i}. Since d_{i} is a divisor of h by assumption, the integer (m + 1)h/p also divides h. But this is equivalent to saying that m + 1 divides p, and equation (7.2) holds by Lemma 35.

Now assume that (m + 1)h/p does not divide any of the d_{i}’s. In this case, it follows from (8.4) and the fact that we have S_{1}(W) ⊇ {n} and S_{2}(W) = ∅ that the righthand side of (7.2) equals 0. Inspection of the classification of all irreducible wellgenerated complex reflection groups, which we recalled in Section 2, reveals that all groups satisfying the hypotheses of the lemma have rank n ≤ 2. Except for the groups contained in the infinite series G(d, 1, n) and G(e, e, n) for which Theorem 2 has been established in [19], these are the groups G_{5}, G_{6}, G_{9}, G_{10}, G_{14}, G_{17}, G_{18}, G_{21}. We now discuss these groups case by case, keeping the notation of Lemma 37. In order to simplify the argument, we note that Lemma 38 implies that equation (7.2) holds if m_{2} > 2, so that in the following arguments we always may assume that m_{2} = 2.

Case G_{5}. The degrees are 6, 12, and therefore Remark 4.(1) implies that equation (7.2) is always satisfied.

Case G_{6}. The degrees are 4, 12, and therefore, according to Remark 4.(1), we need only consider the case where h_{2} = 4 and m_{2} = 2, that is, p = 3(m + 1)/2. Then (8.7) becomes

ψ^{p}((w_{0}; w_{1}, …, w_{m})) = (c^{2}w_{\frac{m+1}{2}} c^{−2}; c^{2}w_{\frac{m+3}{2}} c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{0}c^{−1}, …, cw_{\frac{m−1}{2}} c^{−1}).

(8.9)

If (w_{0}; w_{1}, …, w_{m}) is fixed by ψ^{p}, there must exist an i with 0 ≤ i ≤ \frac{m−1}{2} such that ℓ_{T} (w_{i}) = 1, w_{i}cw_{i}c^{−1} = c, and all w_{j}, j ≠ i, \frac{m+1}{2} + i, equal ε. However, with the help of CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand side of (7.2) is equal to 0, as required.

Case G_{9}. The degrees are 8, 24, and therefore, according to Remark 4.(1), we need only consider the case where h_{2} = 8 and m_{2} = 2, that is, p = 3(m + 1)/2. This is the same p as for G_{6}. Again, CHEVIE finds no solution. Hence, the left-hand side of (7.2) is equal to 0, as required.

Case G_{10}. The degrees are 12, 24, and therefore, according to Remark 4.(1), we need only consider the case where h_{2} = 12 and m_{2} = 2, that is, p = 3(m + 1)/2. This is the same p as for G_{6}. Again, CHEVIE finds no solution. Hence, the left-hand side of (7.2) is equal to 0, as required.

Case G_{14}. The degrees are 6, 24, and therefore Remark 4.(1) implies that equation (7.2) is always satisfied.

Case G_{17}. The degrees are 20, 60, and therefore, according to Remark 4.(1), we need only consider the cases where h_{2} = 20 and m_{2} = 2, respectively that h_{2} = 4 and m_{2} = 2. In the first case, p = 3(m + 1)/2, which is the same p as for G_{6}. Again, CHEVIE finds no solution. In the second case, p = 15(m + 1)/2. Then (8.7) becomes

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{8}w_{\frac{m+1}{2}} c^{−8}; c^{8}w_{\frac{m+3}{2}} c^{−8}, …, c^{8}w_{m}c^{−8}, c^{7}w_{0}c^{−7}, …, c^{7}w_{\frac{m−1}{2}} c^{−7}). (8.10)

By Lemma 29, every element of NC(W) is fixed under conjugation by c^{3}, and, thus, on elements fixed by ψ^{p}, the above action of ψ^{p} reduces to the one in (8.9). This action was already discussed in the first case. Hence, in both cases, the left-hand side of (7.2) is equal to 0, as required.

Case G_{18}. The degrees are 30, 60, and therefore Remark 4.(1) implies that equation (7.2) is always satisfied.

Case G_{21}. The degrees are 12, 60, and therefore, according to Remark 4.(1), we need only consider the cases where h_{2} = 5 and m_{2} = 2, respectively that h_{2} = 15 and m_{2} = 2. In the first case, p = 5(m + 1)/2, so that (8.7) becomes

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{3}w_{\frac{m+1}{2}} c^{−3}; c^{3}w_{\frac{m+3}{2}} c^{−3}, …, c^{3}w_{m}c^{−3}, c^{2}w_{0}c^{−2}, …, c^{2}w_{\frac{m−1}{2}} c^{−2}). (8.11)

If (w_{0}; w_{1}, …, w_{m}) is fixed by ψ^{p}, there must exist an i with 0 ≤ i ≤ \frac{m−1}{2} such that ℓ_{T} (w_{i}) = 1 and w_{i}c^{2}w_{i}c^{−2} = c. However, with the help of CHEVIE, one verifies that there is no such solution to this equation. In the second case, p = 15(m + 1)/2. Then (8.7) becomes the action in (8.10). By Lemma 29, every element of NC(W) is fixed under conjugation by c^{5}, and, thus, on elements fixed by ψ^{p}, the action of ψ^{p} in (8.10) reduces to the one in the first case. Hence, in both cases, the left-hand side of (7.2) is equal to 0, as required.

This completes the proof of the lemma. □

9. Case-by-case verification of Theorem 33

We now perform a case-by-case verification of Theorem 33. It should be observed that the action of ψ (given in (7.1)) is exactly the same as the action of ϕ (given in (3.1)) with m replaced by m + 1 on the components w_{1}, w_{2}, …, w_{m+1}, that is, if we disregard the 0-th component of the elements of the generalised non-crossing partitions involved. The only difference which arises is that, while the (m + 1)-tuples (w_{0}; w_{1}, …, w_{m}) in (7.1) must satisfy w_{0}w_{1} ⋯ w_{m} = c, for w_{1}, w_{2}, …, w_{m+1} in (3.1) we only must have w_{1}w_{2} ⋯ w_{m+1} ≤_{T} c. The condition for (w_{0}; w_{1}, …, w_{m}) of being in Fix_{NC^{m}(W)}(ψ^{p}) is therefore exactly the same as the condition on w_{1}, w_{2}, …, w_{m+1} for the element (ε; w_{1}, …, w_{m}, w_{m+1}) being in Fix_{NC^{m+1}(W)}(ϕ^{p}). Consequently, we may use the counting results from Section 6, except that we have to restrict our attention to those elements (w_{0}; w_{1}, …, w_{m}, w_{m+1}) ∈ NC^{m+1}(W) for which w_{1}w_{2} ⋯ w_{m+1} = c, or, equivalently, w_{0} = ε.

As before, we write ζ_{d} for a primitive d-th root of unity.

Case G_{4}. The degrees are 4, 6, and hence we have

Cat^{m}(G_{4}; q) = \frac{[6m + 6]_{q} [6m + 4]_{q}}{[6]_{q} [4]_{q}} .

Let ζ be a 6(m + 1)-th root of unity. As before, in what follows we abbreviate the assertion that “ζ is a primitive d-th root of unity” as “ζ = ζ_{d}.” The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{4}; q) = m + 1, if ζ = ζ_{6}, ζ_{3},  (9.1a)
lim_{q→ζ} Cat^{m}(G_{4}; q) = \frac{3m+3}{2}, if ζ = ζ_{4}, 2 | (m + 1),  (9.1b)
lim_{q→ζ} Cat^{m}(G_{4}; q) = Cat^{m}(G_{4}), if ζ = −1 or ζ = 1,  (9.1c)
lim_{q→ζ} Cat^{m}(G_{4}; q) = 0, otherwise.  (9.1d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.1). The only cases not covered by Lemmas 35 and 36 are the ones in (9.1b) and (9.1d). On the other hand, the only case left to consider according to Remark 4 is the case where h_{2} = m_{2} = 2, that is the case (9.1b) where p = 3(m + 1)/2. In particular, m + 1 must be divisible by 2. The action of ψ^{p} is the same as the one in (8.9). Hence, the counting problem is the same as there, except that the underlying group now is G_{4}. With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{4} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to (m + 1)/2 elements in Fix_{NC^{m}(G_{4})}(ψ^{p}) since the index i ranges from 0 to (m − 1)/2.

Hence, in total, we obtain 3\frac{m+1}{2} = \frac{3m+3}{2} elements in Fix_{NC^{m}(G_{4})}(ψ^{p}), which agrees with the limit in (9.1b).

Case G_{8}. The degrees are 8, 12, and hence we have

Cat^{m}(G_{8}; q) = \frac{[12m + 12]_{q} [12m + 8]_{q}}{[12]_{q} [8]_{q}} .

Let ζ be a 12(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{8}; q) = m + 1, if ζ = ζ_{12}, ζ_{6}, ζ_{3},  (9.2a)
lim_{q→ζ} Cat^{m}(G_{8}; q) = \frac{3m+3}{2}, if ζ = ζ_{8}, 2 | (m + 1),  (9.2b)
lim_{q→ζ} Cat^{m}(G_{8}; q) = Cat^{m}(G_{8}), if ζ = ζ_{4}, −1, 1,  (9.2c)
lim_{q→ζ} Cat^{m}(G_{8}; q) = 0, otherwise.  (9.2d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.2). The only cases not covered by Lemmas 35 and 36 are the ones in (9.2b) and (9.2d). On the other hand, the only case left to consider according to Remark 4 is the case where h_{2} = 4 and m_{2} = 2, that is the case (9.2b) where p = 3(m + 1)/2. In particular, m + 1 must be divisible by 2. The action of ψ^{p} is the same as the one in (8.9). Hence, the counting problem is the same as there, except that the underlying group now is G_{8}. With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{8} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to (m + 1)/2 elements in Fix_{NC^{m}(G_{8})}(ψ^{p}) since the index i ranges from 0 to (m − 1)/2.

Hence, in total, we obtain 3\frac{m+1}{2} = \frac{3m+3}{2} elements in Fix_{NC^{m}(G_{8})}(ψ^{p}), which agrees with the limit in (9.2b).

Case G_{16}. The degrees are 20, 30, and hence we have

Cat^{m}(G_{16}; q) = \frac{[30m + 30]_{q} [30m + 20]_{q}}{[30]_{q} [20]_{q}} .

Let ζ be a 30(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{16}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{6}, ζ_{3},  (9.3a)
lim_{q→ζ} Cat^{m}(G_{16}; q) = \frac{3m+3}{2}, if ζ = ζ_{20}, ζ_{4}, 2 | (m + 1),  (9.3b)
lim_{q→ζ} Cat^{m}(G_{16}; q) = Cat^{m}(G_{16}), if ζ = ζ_{10}, ζ_{5}, −1, 1,  (9.3c)
lim_{q→ζ} Cat^{m}(G_{16}; q) = 0, otherwise.  (9.3d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.3). The only cases not covered by Lemmas 35 and 36 are the ones in (9.3b) and (9.3d). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 10 and m_{2} = 2, respectively h_{2} = m_{2} = 2. Both cases belong to (9.3b). In the first case, we have p = 3(m + 1)/2, while in the second case we have p = 15(m + 1)/2. In particular, m + 1 must be divisible by 2. In the first case, the action of ψ^{p} is the same as the one in (8.9). Hence, the counting problem is the same as there, except that the underlying group now is G_{16}. With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G_{16} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to (m + 1)/2 elements in Fix_{NC^{m}(G_{16})}(ψ^{p}) since the index i ranges from 0 to (m − 1)/2. On the other hand, if p = 15(m + 1)/2, then the action of ψ^{p} is the same as the one in (8.10). By Lemma 29, every element of NC(G_{16}) is fixed under conjugation by c^{3}, and, thus, on elements fixed by ψ^{p}, the action of ψ^{p} reduces to the one in the first case.

Hence, in total, we obtain 3\frac{m+1}{2} = \frac{3m+3}{2} elements in Fix_{NC^{m}(G_{16})}(ψ^{p}), which agrees with the limit in (9.3b).

Case G_{20}. The degrees are 12, 30, and hence we have

Cat^{m}(G_{20}; q) = \frac{[30m + 30]_{q} [30m + 12]_{q}}{[30]_{q} [12]_{q}} .

Let ζ be a 30(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{20}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (9.4a)
lim_{q→ζ} Cat^{m}(G_{20}; q) = \frac{5m+5}{2}, if ζ = ζ_{12}, ζ_{4}, 2 | (m + 1),  (9.4b)
lim_{q→ζ} Cat^{m}(G_{20}; q) = Cat^{m}(G_{20}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (9.4c)
lim_{q→ζ} Cat^{m}(G_{20}; q) = 0, otherwise.  (9.4d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.4). The only cases not covered by Lemmas 35 and 36 are the ones in (9.4b) and (9.4d). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 6 and m_{2} = 2, respectively h_{2} = m_{2} = 2. Both cases belong to (9.4b). In the first case, we have p = 5(m + 1)/2, while in the second case we have p = 15(m + 1)/2. In particular, m + 1 must be divisible by 2. In the first case, the action of ψ^{p} is the same as the one in (8.11). Hence, the counting problem is the same as there, except that the underlying group now is G_{20}. With the help of CHEVIE, one finds that each of the 5 (complex) reflections in G_{20} which are less than the (chosen) Coxeter element is a valid choice for w_{i}, and each of these choices gives rise to (m + 1)/2 elements in Fix_{NC^{m}(G_{20})}(ψ^{p}) since the index i ranges from 0 to (m − 1)/2. On the other hand, if p = 15(m + 1)/2, then the action of ψ^{p} is the same as the one in (8.10). By Lemma 29, every element of NC(G_{20}) is fixed under conjugation by c^{5}, and, thus, on elements fixed by ψ^{p}, the action of ψ^{p} reduces to the one in the first case.

Hence, in total, we obtain 5\frac{m+1}{2} = \frac{5m+5}{2} elements in Fix_{NC^{m}(G_{20})}(ψ^{p}), which agrees with the limit in (9.4b).

Case G_{23} = H_{3}. The degrees are 2, 6, 10, and hence we have

Cat^{m}(H_{3}; q) = \frac{[10m + 10]_{q} [10m + 6]_{q} [10m + 2]_{q}}{[10]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 10(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(H_{3}; q) = m + 1, if ζ = ζ_{10}, ζ_{5},  (9.5a)
lim_{q→ζ} Cat^{m}(H_{3}; q) = \frac{5m+5}{3}, if ζ = ζ_{6}, ζ_{3}, 3 | (m + 1),  (9.5b)
lim_{q→ζ} Cat^{m}(H_{3}; q) = Cat^{m}(H_{3}), if ζ = −1 or ζ = 1,  (9.5c)
lim_{q→ζ} Cat^{m}(H_{3}; q) = 0, otherwise.  (9.5d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.5). The only cases not covered by Lemmas 35 and 36 are the ones in (9.5b) and (9.5d). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 3, h_{2} = 2 and m_{2} = 3, and h_{2} = m_{2} = 2. These correspond to the choices p = 10(m + 1)/3, p = 5(m + 1)/3, respectively p = 5(m + 1)/2. The first two cases belong to (9.5b), while p = 5(m + 1)/2 belongs to (9.5d).

In the case that p = 5(m + 1)/3, the action of ψ^{p} is given by

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{2}w_{\frac{m+1}{3}} c^{−2}; c^{2}w_{\frac{m+4}{3}} c^{−2}, …, c^{2}w_{m}c^{−2}, cw_{0}c^{−1}, …, cw_{\frac{m−2}{3}} c^{−1}).

Hence, for an i with 0 ≤ i ≤ \frac{m−2}{3}, we must find an element w_{i} = t_{1}, where t_{1} satisfies (6.9), and all other w_{j}, j ∉ \{i, i+ \frac{m+1}{3}, i+ \frac{2(m+1)}{3}\}, are set equal to ε. We have found five solutions to the counting problem (6.9) in (6.10). Each of them gives rise to (m + 1)/3 elements in Fix_{NC^{m}(H_{3})}(ψ^{p}) since the index i ranges from 0 to (m − 2)/3. On the other hand, if p = 10(m + 1)/3, then the action of ψ^{p} is given by

ψ^{p}((w_{0}; w_{1}, …, w_{m}))

= (c^{4}w_{\frac{2m+2}{3}} c^{−4}; c^{4}w_{\frac{2m+5}{3}} c^{−4}, …, c^{4}w_{m}c^{−4}, c^{3}w_{0}c^{−3}, …, c^{3}w_{\frac{2m−1}{3}} c^{−3}).

By Lemma 29, every element of NC(H_{3}) is fixed under conjugation by c^{5}, and, thus, on elements fixed by ψ^{p}, the action of ψ^{p} reduces to the one in the first case.

Hence, in total, we obtain 5\frac{m+1}{3} = \frac{5m+5}{3} elements in Fix_{NC^{m}(H_{3})}(ψ^{p}), which agrees with the limit in (9.5b).

If p = 5(m + 1)/2, then the action of ψ^{p} is the same as the one in (8.11). The computation at the end of Case H_{3} in Section 6 did not find any solutions, which is in agreement with (9.5d).

Case G_{24}. The degrees are 4, 6, 14, and hence we have

Cat^{m}(G_{24}; q) = \frac{[14m + 14]_{q} [14m + 6]_{q} [14m + 4]_{q}}{[14]_{q} [6]_{q} [4]_{q}} .

Let ζ be a 14(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{24}; q) = m + 1, if ζ = ζ_{14}, ζ_{7},  (9.6a)
lim_{q→ζ} Cat^{m}(G_{24}; q) = \frac{7m+7}{3}, if ζ = ζ_{6}, ζ_{3}, 3 | (m + 1),  (9.6b)
lim_{q→ζ} Cat^{m}(G_{24}; q) = Cat^{m}(G_{24}), if ζ = −1 or ζ = 1,  (9.6c)
lim_{q→ζ} Cat^{m}(G_{24}; q) = 0, otherwise.  (9.6d)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.6). The only cases not covered by Lemmas 35 and 36 are the ones in (9.6b) and (9.6d). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 3, h_{2} = 2 and m_{2} = 3, and h_{2} = m_{2} = 2. These correspond to the choices p = 14(m + 1)/3, p = 7(m + 1)/3, respectively p = 7(m + 1)/2. The first two cases belong to (9.6b), while p = 7(m + 1)/2 belongs to (9.6d).

In the case that p = 14(m+1)/3 or p = 7(m+1)/3, we have found seven solutions to the counting problem (6.18) in (6.19), and each of them gives rise to (m+1)/3 elements in Fix_{NC^{m}(G_{24})}(ψ^{p}) (in the style as discussed in Case H_{3}). Hence, in total, we obtain 7\frac{m+1}{3} = \frac{7m+7}{3} elements in Fix_{NC^{m}(G_{24})}(ψ^{p}), which agrees with the limit in (9.6b).

If p = 7(m + 1)/2, the relevant counting problem is (6.25). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{24})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be even, which contradicts the fact that ℓ_{T} (c) = n = 3. This is in agreement with the limit in (9.6d).

Case G_{25}. The degrees are 6, 9, 12, and hence we have

Cat^{m}(G_{25}; q) = \frac{[12m + 12]_{q} [12m + 9]_{q} [12m + 6]_{q}}{[12]_{q} [9]_{q} [6]_{q}} .

Let ζ be a 12(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{25}; q) = m + 1, if ζ = ζ_{12}, ζ_{4},  (9.7a)
lim_{q→ζ} Cat^{m}(G_{25}; q) = \frac{4m+4}{3}, if ζ = ζ_{9}, 3 | (m + 1),  (9.7b)
lim_{q→ζ} Cat^{m}(G_{25}; q) = (m + 1)(2m + 1), if ζ = ζ_{6}, −1  (9.7c)
lim_{q→ζ} Cat^{m}(G_{25}; q) = Cat^{m}(G_{25}), if ζ = ζ_{3}, 1,  (9.7d)
lim_{q→ζ} Cat^{m}(G_{25}; q) = 0, otherwise.  (9.7e)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.7). The only cases not covered by Lemmas 35 and 36 are the ones in (9.7b) and (9.7e). On the other hand, the only case left to consider according to Remark 4 is the case where h_{2} = m_{2} = 3. This corresponds to the choice p = 4(m+1)/3, which belongs to (9.7b). We have found four solutions to the counting problem (6.30) in (6.31), and each of them gives rise to (m + 1)/3 elements in Fix_{NC^{m}(G_{25})}(ψ^{p}) (in the style as discussed in Case H_{3}). Hence, in total, we obtain 4\frac{m+1}{3} = \frac{4m+4}{3} elements in Fix_{NC^{m}(G_{25})}(ψ^{p}), which agrees with the limit in (9.7b).

Case G_{26}. The degrees are 6, 12, 18, and hence we have

Cat^{m}(G_{26}; q) = \frac{[18m + 18]_{q} [18m + 12]_{q} [18m + 6]_{q}}{[18]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 14(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{26}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (9.8a)
lim_{q→ζ} Cat^{m}(G_{26}; q) = Cat^{m}(G_{26}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (9.8b)
lim_{q→ζ} Cat^{m}(G_{26}; q) = 0, otherwise.  (9.8c)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.8). The only case not covered by Lemmas 35 and 36 is the one in (9.8c). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 6 and m_{2} = 2, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 3(m+1)/2, respectively p = 9(m+1)/2, both of which belong to (9.8c). The relevant counting problem is (6.35). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{26})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be even, which is absurd. This is in agreement with the limit in (9.8c).

Case G_{27}. The degrees are 6, 12, 30, and hence we have

Cat^{m}(G_{27}; q) = \frac{[30m + 30]_{q} [30m + 12]_{q} [30m + 6]_{q}}{[30]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 14(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{27}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (9.9a)
lim_{q→ζ} Cat^{m}(G_{27}; q) = Cat^{m}(G_{27}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (9.9b)
lim_{q→ζ} Cat^{m}(G_{27}; q) = 0, otherwise.  (9.9c)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.9). The only case not covered by Lemmas 35 and 36 is the one in (9.9c). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 6 and m_{2} = 3, h_{2} = m_{2} = 3, h_{2} = 6 and m_{2} = 2, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 5(m + 1)/3, 10(m + 1)/3, 5(m + 1)/2, respectively 15(m + 1)/2, all of which belong to (9.9c).

If p = 5(m + 1)/3 or p = 10(m + 1)/3, the computation with the help of CHEVIE at the end of Case G_{27} in Section 6 did not find any solutions for the corresponding counting problem. This is in agreement with the limit in (9.9c).

In the case that 5(m + 1)/2 or 15(m + 1)/2, the relevant counting problem is (6.42). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{27})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be even, which is absurd. This is again in agreement with the limit in (9.9c).

Case G_{28} = F_{4}. The degrees are 2, 6, 8, 12, and hence we have

Cat^{m}(F_{4}; q) = \frac{[12m + 12]_{q} [12m + 8]_{q} [12m + 6]_{q} [12m + 2]_{q}}{[12]_{q} [8]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 12(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(F_{4}; q) = m + 1, if ζ = ζ_{12},  (9.10a)
lim_{q→ζ} Cat^{m}(F_{4}; q) = \frac{3m+3}{2}, if ζ = ζ_{8}, 2 | (m + 1),  (9.10b)
lim_{q→ζ} Cat^{m}(F_{4}; q) = (m + 1)(2m + 1), if ζ = ζ_{6}, ζ_{3},  (9.10c)
lim_{q→ζ} Cat^{m}(F_{4}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{4},  (9.10d)
lim_{q→ζ} Cat^{m}(F_{4}; q) = Cat^{m}(F_{4}), if ζ = −1 or ζ = 1,  (9.10e)
lim_{q→ζ} Cat^{m}(F_{4}; q) = 0, otherwise.  (9.10f)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.10). The only cases not covered by Lemmas 35 and 36 are the ones in (9.10b) and (9.10f). On the other hand, according to Remark 4, the are no choices for h_{2} and m_{2} left to be considered.

Case G_{29}. The degrees are 4, 8, 12, 20, and hence we have

Cat^{m}(G_{29}; q) = \frac{[20m + 20]_{q} [20m + 12]_{q} [20m + 8]_{q} [20m + 4]_{q}}{[20]_{q} [12]_{q} [8]_{q} [4]_{q}} .

Let ζ be a 20(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{29}; q) = m + 1, if ζ = ζ_{20}, ζ_{10}, ζ_{5},  (9.11a)
lim_{q→ζ} Cat^{m}(G_{29}; q) = Cat^{m}(G_{29}), if ζ = ζ_{4}, −1, 1,  (9.11b)
lim_{q→ζ} Cat^{m}(G_{29}; q) = 0, otherwise.  (9.11c)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.11). The only case not covered by Lemmas 35 and 36 is the one in (9.11c). On the other hand, the only cases left to consider according to Remark 4, the only choices for h_{2} and m_{2} to be considered are h_{2} = 1 and m_{2} = 3, h_{2} = 2 and m_{2} = 3, h_{2} = 4 and m_{2} = 3, h_{2} = 4 and m_{2} = 2, respectively h_{2} = m_{2} = 4. These correspond to the choices p = 20(m + 1)/3, p = 10(m + 1)/3, p = 5(m + 1)/3, p = 5(m + 1)/2, respectively p = 5(m + 1)/4, all of which belong to (9.11c).

In the case that p = 20(m + 1)/3, p = 10(m + 1)/3, or p = 5(m + 1)/3, the relevant counting problem is (6.55). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{27})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 3, which is absurd. This is in agreement with the limit in (9.11c).

In the case that p = 5(m + 1)/2, the relevant counting problem is (6.64), for which we did not find any solutions. This is again in agreement with the limit in (9.11c).

In the case that p = 5(m + 1)/4, the computation at the end of Case G_{29} in Section 6 did not find any solutions, which is as well in agreement with the limit in (9.11c).

Case G_{30} = H_{4}. The degrees are 2, 12, 20, 30, and hence we have

Cat^{m}(H_{4}; q) = \frac{[30m + 30]_{q} [30m + 20]_{q} [30m + 12]_{q} [30m + 2]_{q}}{[30]_{q} [20]_{q} [12]_{q} [2]_{q}} .

Let ζ be a 30(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(H_{4}; q) = m + 1, if ζ = ζ_{30}, ζ_{15},  (9.12a)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{3m+3}{2}, if ζ = ζ_{20}, 2 | (m + 1),  (9.12b)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{5m+5}{2}, if ζ = ζ_{12}, 2 | (m + 1),  (9.12c)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{10}, ζ_{5},  (9.12d)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(m+1)(5m+2)}{2}, if ζ = ζ_{6}, ζ_{3},  (9.12e)
lim_{q→ζ} Cat^{m}(H_{4}; q) = \frac{(m+1)(15m+1)}{4}, if ζ = ζ_{4}, 2 | (m + 1),  (9.12f)
lim_{q→ζ} Cat^{m}(H_{4}; q) = Cat^{m}(H_{4}), if ζ = −1 or ζ = 1,  (9.12g)
lim_{q→ζ} Cat^{m}(H_{4}; q) = 0, otherwise.  (9.12h)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.12). The only cases not covered by Lemmas 35 and 36 are the ones in (9.12b), (9.12c), (9.12f), and (9.12h). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 15(m+1)/2, respectively p = 15(m + 1)/4, out of which the first belongs to (9.12f), while the second belongs to (9.12h).

In the case that p = 15(m + 1)/2, the action of ψ^{p} is the same as the one in (8.10). We have found eight solutions to the counting problem (6.80) in (6.83), each of them giving rise to (m + 1)/2 elements in Fix_{NC^{m}(H_{4})}(ψ^{p}) since the index i (in (6.80)) ranges from 0 to (m − 1)/2, and we have found 30 solutions to the counting problem (6.82) in (6.84), each of them giving rise to \binom{(m+1)/2}{2} elements in Fix_{NC^{m}(H_{4})}(ψ^{p}) since 0 ≤ i_{1} < i_{2} ≤ (m − 1)/2 (in (6.82)). Hence, we obtain 8\frac{m+1}{2} + 30\binom{(m+1)/2}{2} = \frac{(m+1)(15m+1)}{4} elements in Fix_{NC^{m}(H_{4})}(ψ^{p}), which agrees with the limit in (9.12f).

If p = 15(m + 1)/4, the computation at the end of Case H_{4} in Section 6 did not find any solutions, which is in agreement with the limit in (9.12h).

Case G_{32}. The degrees are 12, 18, 24, 30, and hence we have

Cat^{m}(G_{32}; q) = \frac{[30m + 30]_{q} [30m + 24]_{q} [30m + 18]_{q} [30m + 12]_{q}}{[30]_{q} [24]_{q} [18]_{q} [12]_{q}} .

Let ζ be a 30(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{32}; q) = m + 1, if ζ = ζ_{30}, ζ_{15}, ζ_{10}, ζ_{5},  (9.13a)
lim_{q→ζ} Cat^{m}(G_{32}; q) = \frac{5m+5}{4}, if ζ = ζ_{24}, ζ_{8}, 4 | (m + 1),  (9.13b)
lim_{q→ζ} Cat^{m}(G_{32}; q) = \frac{(5m+5)(5m+3)}{8}, if ζ = ζ_{12}, ζ_{4}, 2 | (m + 1),  (9.13c)
lim_{q→ζ} Cat^{m}(G_{32}; q) = Cat^{m}(G_{32}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (9.13d)
lim_{q→ζ} Cat^{m}(G_{32}; q) = 0, otherwise.  (9.13e)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.13). The only cases not covered by Lemmas 35 and 36 are the ones in (9.13b), (9.13c), and (9.13e). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 2 and m_{2} = 4, h_{2} = 6 and m_{2} = 4, h_{2} = m_{2} = 3, h_{2} = 6 and m_{2} = 3, h_{2} = m_{2} = 2, respectively h_{2} = 6 and m_{2} = 2. These correspond to the choices p = 15(m + 1)/4, p = 5(m + 1)/4, p = 10(m + 1)/3, p = 5(m+1)/3, p = 15(m+1)/2, respectively p = 5(m+1)/2, out of which the first two belong to (9.13b), the next two belong to (9.13e), and the last two belong to (9.13c).

In the case that p = 15(m + 1)/4 or p = 5(m + 1)/4, we have found five solutions to the counting problem (6.88) in (6.89), each of them giving rise to (m + 1)/4 elements in Fix_{NC^{m}(G_{32})}(ψ^{p}). Hence, we obtain 5\frac{m+1}{4} = \frac{5m+5}{4} elements in Fix_{NC^{m}(G_{32})}(ψ^{p}), which agrees with the limit in (9.13b).

In the case that p = 10(m + 1)/3 or p = 5(m + 1)/3, the relevant counting problem is (6.95). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{32})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 3, which is absurd. This is in agreement with the limit in (9.13e).

In the case that p = 15(m + 1)/2 or p = 5(m + 1)/2, we have found ten solutions to the counting problem (6.102) in (6.105), each of them giving rise to (m + 1)/2 elements in Fix_{NC^{m}(G_{32})}(ψ^{p}), and we have found 25 solutions to the counting problem (6.104) in (6.106), each of them giving rise to \binom{(m+1)/2}{2} elements in Fix_{NC^{m}(G_{32})}(ψ^{p}). Hence, we obtain 10\frac{m+1}{2} + 25\binom{(m+1)/2}{2} = \frac{(5m+5)(5m+3)}{8} elements in Fix_{NC^{m}(G_{32})}(ψ^{p}), which agrees with the limit in (9.13c).

Case G_{33}. The degrees are 4, 6, 10, 12, 18, and hence we have

Cat^{m}(G_{33}; q) = \frac{[18m + 18]_{q} [18m + 12]_{q} [18m + 10]_{q} [18m + 6]_{q} [18m + 4]_{q}}{[18]_{q} [12]_{q} [10]_{q} [6]_{q} [4]_{q}} .

Let ζ be a 18(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{33}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (9.14a)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{9m+9}{5}, if ζ = ζ_{10}, ζ_{5}, 5 | (m + 1),  (9.14b)
lim_{q→ζ} Cat^{m}(G_{33}; q) = \frac{(m+1)(3m+2)(3m+1)}{2}, if ζ = ζ_{6}, ζ_{3},  (9.14c)
lim_{q→ζ} Cat^{m}(G_{33}; q) = Cat^{m}(G_{33}), if ζ = −1 or ζ = 1,  (9.14d)
lim_{q→ζ} Cat^{m}(G_{33}; q) = 0, otherwise.  (9.14e)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.14). The only cases not covered by Lemmas 35 and 36 are the ones in (9.14b) and (9.14e). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 18(m + 1)/5, p = 9(m + 1)/5, p = 9(m + 1)/4, respectively p = 9(m + 1)/2, out of which the first two belong to (9.14b), while the others belong to (9.14e).

In the case that p = 18(m + 1)/5 or p = 9(m + 1)/5, we have found nine solutions to the counting problem (6.115) in (6.116). Hence, we obtain 9\frac{m+1}{5} = \frac{9m+9}{5} elements in Fix_{NC^{m}(G_{33})}(ψ^{p}), which agrees with the limit in (9.14b).

If p = 9(m + 1)/4, the computation at the end of Case G_{33} in Section 6 did not find any solutions, which is again in agreement with the limit in (9.13e).

In the case that p = 9(m + 1)/2, the relevant counting problems are (6.122) and (6.124). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{33})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0})+ℓ_{T} (w_{1})+⋯+ ℓ_{T} (w_{m}) be even, which is absurd. This is in agreement with the limit in (9.14e).

Case G_{34}. The degrees are 6, 12, 18, 24, 30, 42, and hence we have

Cat^{m}(G_{34}; q) = \frac{[42m + 42]_{q} [42m + 30]_{q} [42m + 24]_{q}}{[42]_{q} [30]_{q} [24]_{q}}

× \frac{[42m + 18]_{q} [42m + 12]_{q} [42m + 6]_{q}}{[18]_{q} [12]_{q} [6]_{q}} .

Let ζ be a 42(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(G_{34}; q) = m + 1, if ζ = ζ_{42}, ζ_{21}, ζ_{14}, ζ_{7},  (9.15a)
lim_{q→ζ} Cat^{m}(G_{34}; q) = Cat^{m}(G_{34}), if ζ = ζ_{6}, ζ_{3}, −1, 1,  (9.15b)
lim_{q→ζ} Cat^{m}(G_{34}; q) = 0, otherwise.  (9.15c)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.15). The only case not covered by Lemmas 35 and 36 is the one in (9.15c). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 5, h_{2} = 3 and m_{2} = 5, h_{2} = 6 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, h_{2} = 6 and m_{2} = 4, h_{2} = m_{2} = 3, h_{2} = 6 and m_{2} = 3, h_{2} = m_{2} = 2, h_{2} = 6 and m_{2} = 2, respectively h_{2} = 6 and m_{2} = 6. These correspond to the choices p = 42(m + 1)/5, p = 21(m + 1)/5, p = 14(m + 1)/5, p = 7(m + 1)/5, p = 21(m + 1)/4, p = 7(m + 1)/4, p = 14(m + 1)/3, p = 7(m + 1)/3, p = 21(m + 1)/2, p = 7(m + 1)/2, respectively p = 7(m + 1)/6, all of which belong to (9.15c).

In the case that p = 42(m+1)/5, p = 21(m+1)/5, p = 14(m+1)/5, or p = 7(m+1)/5, the relevant counting problem is (6.128). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{34})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 5, which is absurd. This is in agreement with the limit in (9.15c).

In the case that p = 21(m + 1)/4 or p = 7(m + 1)/4, the relevant counting problem is (6.140). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(G_{34})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 4, which is absurd. This is in agreement with the limit in (9.15c).

In the case that p = 14(m + 1)/3 or p = 7(m + 1)/3, the relevant counting problems are (6.146) and (6.148). However, the computations with the help of CHEVIE performed in Case G_{34} in Section 6 did not find any solutions for (6.146) or (6.148). This is in agreement with the limit in (9.15c).

In the case that p = 21(m+1)/2, the relevant counting problems are (6.157), (6.158), and (6.159). However, the computations with the help of CHEVIE performed in Case G_{34} in Section 6 found no w_{i} with ℓ_{T} (w_{i}) = 3 in (6.157), and hence no solutions for (w_{i_{1}}, w_{i_{2}}) with ℓ_{T} (w_{i_{1}})+ℓ_{T} (w_{i_{2}}) = 3 in (6.158), and no solutions for (w_{i_{1}}, w_{i_{2}}, w_{i_{3}}) in (6.159). This is in agreement with the limit in (9.15c).

If p = 7(m + 1)/6, the computation at the end of Case G_{34} in Section 6 did not find any solutions, which is also in agreement with the limit in (9.15c).

Case G_{35} = E_{6}. The degrees are 2, 5, 6, 8, 9, 12, and hence we have

Cat^{m}(E_{6}; q) = \frac{[12m + 12]_{q} [12m + 9]_{q} [12m + 8]_{q} [12m + 6]_{q} [12m + 5]_{q} [12m + 2]_{q}}{[12]_{q} [9]_{q} [8]_{q} [6]_{q} [5]_{q} [2]_{q}} .

Let ζ be a 12(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(E_{6}; q) = m + 1, if ζ = ζ_{12},  (9.16a)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{4m+4}{3}, if ζ = ζ_{9}, 3 | (m + 1),  (9.16b)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{3m+3}{2}, if ζ = ζ_{8}, 2 | (m + 1),  (9.16c)
lim_{q→ζ} Cat^{m}(E_{6}; q) = (m + 1)(2m + 1), if ζ = ζ_{6},  (9.16d)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{4},  (9.16e)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(4m+3)(2m+1)}{3}, if ζ = ζ_{3},  (9.16f)
lim_{q→ζ} Cat^{m}(E_{6}; q) = \frac{(m+1)(3m+2)(2m+1)(6m+1)}{2}, if ζ = −1,  (9.16g)
lim_{q→ζ} Cat^{m}(E_{6}; q) = Cat^{m}(E_{6}), if ζ = 1,  (9.16h)
lim_{q→ζ} Cat^{m}(E_{6}; q) = 0, otherwise.  (9.16i)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.16). The only cases not covered by Lemmas 35 and 36 are the ones in (9.16b), (9.16c), and (9.16i). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 5, respectively h_{2} = 2 and m_{2} = 5. These correspond to the choices p = 12(m + 1)/5, respectively p = 6(m + 1)/5, both of which belong to (9.16i).

In the case that p = 12(m + 1)/5, the relevant counting problem is (6.172). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(E_{6})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0})+ℓ_{T} (w_{1})+⋯+ℓ_{T} (w_{m}) be divisible by 5, which is absurd. This is in agreement with the limit in (9.16i).

If p = 6(m + 1)/5, the computation at the end of Case E_{6} in Section 6 did not find any solutions, which is also in agreement with the limit in (9.16i).

Case G_{36} = E_{7}. The degrees are 2, 6, 8, 10, 12, 14, 18, and hence we have

Cat^{m}(E_{7}; q) = \frac{[18m + 18]_{q} [18m + 14]_{q} [18m + 12]_{q}}{[18]_{q} [14]_{q} [12]_{q}}

× \frac{[18m + 10]_{q} [18m + 8]_{q} [18m + 6]_{q} [18m + 2]_{q}}{[10]_{q} [8]_{q} [6]_{q} [2]_{q}} .

Let ζ be a 18(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(E_{7}; q) = m + 1, if ζ = ζ_{18}, ζ_{9},  (9.17a)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{9m+9}{7}, if ζ = ζ_{14}, ζ_{7}, 7 | (m + 1),  (9.17b)
lim_{q→ζ} Cat^{m}(E_{7}; q) = \frac{(m + 1)(3m + 2)(3m + 1)}{2}, if ζ = ζ_{6}, ζ_{3},  (9.17c)
lim_{q→ζ} Cat^{m}(E_{7}; q) = Cat^{m}(E_{7}), if ζ = −1 or ζ = 1,  (9.17d)
lim_{q→ζ} Cat^{m}(E_{7}; q) = 0, otherwise.  (9.17e)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.17). The only cases not covered by Lemmas 35 and 36 are the ones in (9.17b) and (9.17e). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 1 and m_{2} = 7, h_{2} = 2 and m_{2} = 7, h_{2} = 1 and m_{2} = 5, h_{2} = 2 and m_{2} = 5, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 18(m + 1)/7, p = 9(m + 1)/7, p = 18(m + 1)/5, p = 9(m + 1)/5, p = 9(m + 1)/4, respectively p = 9(m + 1)/2, out of which the first two belong to (9.16b), and all others belong to (9.16i).

In the case that p = 18(m + 1)/7 or p = 9(m + 1)/7, we have found nine solutions to the counting problem (6.176) in (6.177). Hence, we obtain 9\frac{m+1}{7} = \frac{9m+9}{7} elements in Fix_{NC^{m}(E_{7})}(ψ^{p}), which agrees with the limit in (9.17b).

In the case that p = 18(m + 1)/5 or p = 9(m + 1)/5, the relevant counting problem is (6.186). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(E_{7})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 5, which is absurd. This is in agreement with the limit in (9.17e).

In the case that p = 9(m + 1)/4, the relevant counting problem is (6.192). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(E_{7})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0})+ℓ_{T} (w_{1})+⋯+ℓ_{T} (w_{m}) be divisible by 4, which is absurd. This is again in agreement with the limit in (9.17e).

In the case that p = 9(m + 1)/2, the relevant counting problems are (6.195), (6.196), and (6.197). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(E_{7})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be even, which is absurd. This is also in agreement with the limit in (9.17e).

Case G_{37} = E_{8}. The degrees are 2, 8, 12, 14, 18, 20, 24, 30, and hence we have

Cat^{m}(E_{8}; q) = \frac{[30m + 30]_{q} [30m + 24]_{q} [30m + 20]_{q} [30m + 18]_{q}}{[30]_{q} [24]_{q} [20]_{q} [18]_{q}}

× \frac{[30m + 14]_{q} [30m + 12]_{q} [30m + 8]_{q} [30m + 2]_{q}}{[14]_{q} [12]_{q} [8]_{q} [2]_{q}} .

Let ζ be a 30(m + 1)-th root of unity. The following cases on the right-hand side of (7.2) occur:

lim_{q→ζ} Cat^{m}(E_{8}; q) = m + 1, if ζ = ζ_{30}, ζ_{15},  (9.18a)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{5m+5}{4}, if ζ = ζ_{24}, 4 | (m + 1),  (9.18b)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{3m+3}{2}, if ζ = ζ_{20}, 2 | (m + 1),  (9.18c)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(5m+5)(5m+3)}{8}, if ζ = ζ_{12}, 2 | (m + 1),  (9.18d)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(m+1)(3m+2)}{2}, if ζ = ζ_{10}, ζ_{5},  (9.18e)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(5m + 5)(15m + 7)}{16}, if ζ = ζ_{8}, 4 | (m + 1),  (9.18f)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(m + 1)(5m + 4)(5m + 3)(5m + 2)}{24}, if ζ = ζ_{6}, ζ_{3},  (9.18g)
lim_{q→ζ} Cat^{m}(E_{8}; q) = \frac{(m + 1)(5m + 3)(15m + 7)(15m + 1)}{64}, if ζ = ζ_{4}, 2 | (m + 1), (9.18h)

lim_{q→ζ} Cat^{m}(E_{8}; q) = Cat^{m}(E_{8}), if ζ = −1 or ζ = 1,  (9.18i)
lim_{q→ζ} Cat^{m}(E_{8}; q) = 0, otherwise.  (9.18j)

We must now prove that the left-hand side of (7.2) in each case agrees with the values exhibited in (9.18). The only cases not covered by Lemmas 35 and 36 are the ones in (9.18b), (9.18c), (9.18d), (9.18f), (9.18h), and (9.18j). On the other hand, the only cases left to consider according to Remark 4 are the cases where h_{2} = 2 and m_{2} = 8, h_{2} = 1 and m_{2} = 7, h_{2} = 2 and m_{2} = 7, h_{2} = 2 and m_{2} = 4, respectively h_{2} = m_{2} = 2. These correspond to the choices p = 15(m + 1)/8, p = 30(m + 1)/7, p = 15(m + 1)/7, p = 15(m+1)/4, respectively 15(m+1)/2, out of which the first three belong to (9.18j), the fourth belongs to (9.18f), and the last belongs to (9.18h).

If p = 15(m + 1)/8, the relevant counting problem is (6.242). However, the computation at the end of Case E_{8} in Section 6 did not find any solutions, which is in agreement with the limit in (9.18j). Hence, the left-hand side of (7.2) is equal to 0, as required.

In the case that p = 30(m + 1)/7 or p = 15(m + 1)/7, the relevant counting problem is (6.213). However, no element (w_{0}; w_{1}, …, w_{m}) ∈ Fix_{NC^{m}(E_{8})}(ψ^{p}) can be produced in this way since the counting problem imposes the restriction that ℓ_{T} (w_{0}) + ℓ_{T} (w_{1}) + ⋯ + ℓ_{T} (w_{m}) be divisible by 7, which is absurd. This is also in agreement with the limit in (9.18j).

In the case that p = 15(m + 1)/4, the relevant counting problems are (6.227) and (6.228). We have found 45 solutions w_{i} to (6.227) of type A_{1}^{2} in (6.229), and we have found 20 solutions w_{i} to (6.227) of type A_{2} in (6.230), which implied 150 solutions for (w_{i_{1}}, w_{i_{2}}) to (6.228). The first two give rise to to (45 + 20)\frac{m+1}{4} = 65\frac{m+1}{4} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}), while the third give rise to 150\binom{(m+1)/4}{2} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}). Hence, we obtain 65\frac{m+1}{4} + 150\binom{(m+1)/4}{2} = \frac{(5m+5)(15m+7)}{16} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}), which agrees with the limit in (9.18f).

In the case that p = 15(m+1)/2, the relevant counting problems are (6.233), (6.234), (6.235), and (6.236). We have found 15 solutions w_{i} to (6.233) of type A_{1}^{2}∗A_{2} in (6.237), we have found 45 solutions w_{i} to (6.233) of type A_{1} ∗ A_{3} in (6.238), we have found 5 solutions w_{i} to (6.233) of type A_{2}^{2} in (6.239), we have found 18 solutions w_{i} to (6.233) of type A_{4} in (6.240), we have found 5 solutions w_{i} to (6.233) of type D_{4} in (6.241), each giving rise to (m + 1)/2 elements in Fix_{NC^{m}(E_{8})}(ψ^{p}). Using the notation from there, these imply 2n_{3,1} + n_{2,2} = 2 · 660 + 1195 = 2515 solutions for (w_{i_{1}}, w_{i_{2}}) to (6.234) with ℓ_{T} (w_{i_{1}}) + ℓ_{T} (w_{i_{2}}) = 4, each giving rise to \binom{(m+1)/2}{2} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}). They also imply 3n_{2,1,1} = 3 · 2850 = 8550 solutions for (w_{i_{1}}, w_{i_{2}}, w_{i_{3}}) to (6.235) with ℓ_{T} (w_{i_{1}})+ℓ_{T} (w_{i_{2}})+ℓ_{T} (w_{i_{3}}) = 4, each giving rise to \binom{(m+1)/2}{3} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}). Finally, they imply as well n_{1,1,1,1} = 6750 solutions for (w_{i_{1}}, w_{i_{2}}, w_{i_{3}}, w_{i_{4}}) to (6.236), each giving rise to \binom{(m+1)/2}{4} elements in Fix_{NC^{m}(E_{8})}(ψ^{p}).

In total, we obtain

(15+45+5+18+5)\frac{m + 1}{2}+2515\binom{(m + 1)/2}{2}+8550\binom{(m + 1)/2}{3}+6750\binom{(m + 1)/2}{4}

= \frac{(m + 1)(5m + 3)(15m + 7)(15m + 1)}{64}

elements in Fix_{NC^{m}(E_{8})}(ψ^{p}), which agrees with the limit in (9.18h).

Acknowledgements

The authors thank an anonymous referee for a very careful reading of the paper [21], and for the many pertinent suggestions which helped to improve that paper and also this manuscript considerably.

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