EULER CHARACTERISTIC OF THE TRUNCATED ORDER COMPLEX OF GENERALIZED NONCROSSING PARTITIONS

D. ARMSTRONG^{†} AND C. KRATTENTHALER^{‡}

Dedicated to Anders Björner on the occasion of his 60th birthday

Abstract. The purpose of this note is to complete the study, begun in the first author’s PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

1. Introduction

We say that a partition of the set [n] := {1, 2, …, n} is noncrossing if, whenever we have {a, c} in block A and {b, d} in block B of the partition with a < b < c < d, it follows that A = B. For an introduction to the rich history of this subject, see [1, Chapter 4.1]. We say that a noncrossing partition of [mn] is m-divisible if each of its blocks has cardinality divisible by m. The collection of m-divisible noncrossing partitions of [mn] — which we will denote by NC^{(m)}(n) — forms a join-semilattice under the refinement partial order. This structure was first studied by Edelman in his PhD thesis; see [7].

Twenty-six years later, in his own PhD thesis [1], the first author defined a generalization of Edelman’s poset to all finite real reflection groups. (We refer the reader to [8] for all terminology related to real reflection groups.) Let W be a finite group generated by reflections in Euclidean space, and let T ⊆ W denote the set of all reflections in the group. Let ℓ_{T} : W → ℤ denote the word length in terms of the generators T. Now fix a Coxeter element c ∈ W and a positive integer m. We define the set of m-divisible noncrossing partitions as follows:

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

∑_{i=0}^{m} ℓ_{T} (w_{i}) = ℓ_{T} (c)}.  (1.1)

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

Key words and phrases. root systems, reflection groups, Coxeter groups, generalized non-crossing partitions, chain enumeration, Euler characteristics, Chu–Vandermonde summation.

†Research partially supported by the NSF grant DMS-0603567.

‡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”.

That is, NC^{(m)}(W) consists of the minimal factorizations of c into m+1 group elements. We define a partial order on NC^{(m)}(W) by setting

(w_{0}; w_{1}, …, w_{m}) ≤ (u_{0}; u_{1}, …, u_{m}) if and only if ℓ_{T} (u_{i}) + ℓ_{T} (u_{i}^{−1}w_{i}) = ℓ_{T} (w_{i}) for 1 ≤ i ≤ m. (1.2)

In other words, we set (w_{0}; w_{1}, …, w_{m}) ≤ (u_{0}; u_{1}, …, u_{m}) if for each 1 ≤ i ≤ m the element u_{i} lies on a geodesic from the identity to w_{i} in the Cayley graph (W, T). We place no a priori restriction on the elements w_{0}, u_{0}, however it follows from the other conditions that ℓ_{T} (w_{0}) + ℓ_{T} (w_{0}^{−1}u_{0}) = ℓ_{T} (u_{0}). We note that the poset is graded with rank function

rk(w_{0}; w_{1}, …, w_{m}) = ℓ_{T} (w_{0}),  (1.3)

hence the element (c; ε, …, ε) ∈ W^{m+1} — where ε ∈ W is the identity — is the unique maximum element. When m = 1 there is a unique minimum element (ε; c) but for m > 1 there are many minimal elements. It turns out that the isomorphism class of the poset NC^{(m)}(W) is independent of the choice of Coxeter element c. Furthermore, when W is the symmetric group 𝔖_{n} we recover Edelman’s poset NC^{(m)}(n).

In this note we are concerned with the topology of the order complex Δ(NC^{(m)}(W)), which is the abstract simplicial complex whose d-dimensional faces are the chains π_{0} < π_{1} < ⋯ < π_{d} in the poset NC^{(m)}(W). In particular, we would like to compute the homotopy type of this and some related complexes. The answers involve the following quantity, called the positive Fuß–Catalan number:

Cat_{+}^{(m)}(W) := ∏_{i=1}^{n} \frac{mh + d_{i} − 2}{d_{i}}.  (1.4)

Here n is the rank of the group W (the number of simple reflections generating W), h is the Coxeter number (the order of a Coxeter element), and the integers d_{1}, d_{2}, …, d_{n} are the degrees of W (the degrees of the fundamental W -invariant polynomials). The prototypical theorem we wish to generalize is the following result Athanasiadis, Brady and Watt.

Theorem 1 ([2]). The order complex of NC^{(1)}(W) with its unique maximum and minimum elements deleted has reduced Euler characteristic (−1)^{n} Cat_{+}^{(1)}(W), and it is homotopy equivalent to a wedge of Cat_{+}^{(1)}(W) many (n − 2)-dimensional spheres.

The first author was able to prove the following theorem for general m.

Theorem 2 ([1], Theorem 3.7.7). The order complex of NC^{(m)}(W) with its unique maximum element deleted has reduced Euler characteristic (−1)^{n−1} Cat_{+}^{(m−1)}(W), and it is homotopy equivalent to a wedge of Cat_{+}^{(m−1)}(W) many (n−1)-dimensional spheres.

However, if we set m = 1 in Theorem 2, we find that the reduced Euler characteristic of NC^{(1)}(W) with its maximum element deleted is (−1)^{n−1} Cat_{+}^{(0)}(W) = 0, which is not surprising because NC^{(1)}(W) has a unique minimum element, which is a cone point for the order complex, and hence this complex is contractible. Thus, Theorem 2 is not a generalization of Theorem 1. To truly generalize Theorem 1, we must delete the maximum element and all minimal elements of NC^{(m)}(W). The first author made a conjecture in this case [1, Conjecture 3.7.9], and our main result settles this conjecture.

Theorem 3. Let W be a finite real reflection group of rank n and let m be a positive integer. The order complex of the poset NC^{(m)}(W) with maximal and minimal elements deleted has reduced Euler characteristic

(−1)^{n}( Cat_{+}^{(m)}(W) − Cat_{+}^{(m−1)}(W)),  (1.5)

and it is homotopy equivalent to a wedge of this many (n − 2)-dimensional spheres.

A different, independent proof of this theorem was found simultaneously by Tomie in [13]. While our proof proceeds by explicitly enumerating the faces of the order complex involved in the above theorem, Tomie’s proof is based on the EL-shellability of this order complex, a result due to Thomas and the first author [1, Cor. 3.7.3], which makes it possible to compute the Euler characteristic by enumerating certain chains in this order complex.

In Section 2 we will collect some auxiliary results and in Section 3 we will prove the main theorem.

In [3, 4], Bessis and Corran have shown that the notion of noncrossing partitions extends rather straightforwardly to well-generated complex reflection groups. It is not done explicitly in [1], but from [3, 4] it is obvious that the definition of generalized noncrossing partitions in [1] can be extended without any effort to well-generated complex reflection groups, the same being true for many (most?) of the results from [1] (cf. [1, Disclaimer 1.3.1]). In Section 4, we show that the assertion in Theorem 3 on the Euler characteristic of the truncated order complex of generalized noncrossing partitions continues to hold for well-generated complex reflection groups. We suspect that this is also true for the topology part of Theorem 3, but what is missing here is the extension to well-generated complex reflection groups of the result of Hugh Thomas and the first author [1, Cor. 3.7.3] that the poset of generalized noncrossing partitions associated to real reflection groups is shellable. This extension has so far not even been done for [2], the special case of the poset of noncrossing partitions.

2. Auxiliary results

In this section we record some results that are needed in the proof of the main theorem. The first result is Theorem 3.5.3 from [1].

Theorem 4. The cardinality of NC^{(m)}(W) is given by the Fuß–Catalan number for reflection groups

Cat^{(m)}(W) := ∏_{i=1}^{n} \frac{mh + d_{i}}{d_{i}} ,  (2.1)

where, as before, n is the rank, h is the Coxeter number and the d_{i} are the degrees of W. Equivalently, given a Coxeter element c, the number of minimal decompositions

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

is given by Cat^{(m)}(W).

Since the numbers h − d_{i} + 2 are a permutation of the degrees [8, Lemma 3.16], we have an alternate formula for the positive Fuß–Catalan number:

Cat_{+}^{(m)}(W) = ∏_{i=1}^{n} \frac{mh + d_{i} − 2}{d_{i}} = (−1)^{n} Cat^{(−m−1)}(W).

Our next result is Theorem 3.6.9(1) from [1].

Theorem 5. The total number of (multi-)chains

π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l}

in NC^{(m)}(W) is equal to Cat^{(ml)}(W).

And, moreover, we have the following.

Lemma 6. The number of (multi-)chains π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l} in NC^{(m)}(W) with rk(π_{1}) = 0 is equal to Cat^{(ml−1)}(W).

Proof. If π_{1} = (w_{0}^{(1)}; w_{1}^{(1)}, …, w_{m}^{(1)}) then the condition rk(π_{1}) = 0 is equivalent to w_{0}^{(1)} = ε. We note that Theorem 3.6.7 of [1], together with the fundamental map between multichains and minimal factorizations [1, Definition 3.2.3], establishes a bijection between multichains π_{1} ≤ ⋯ ≤ π_{l} in NC^{(m)}(W) and elements (u_{0}; u_{1}, …, u_{ml}) of NC^{(ml)}(W) for which u_{0} = w_{0}^{(1)}. Since ℓ_{T} (ε) = 0, we wish to count factorizations u_{1}u_{2} ⋯ u_{ml} = c in which ℓ_{T} (u_{1}) + ⋯ + ℓ_{T} (u_{ml}) = ℓ_{T} (c). By the second part of Theorem 4, this number is equal to Cat^{(ml−1)}(W), as desired. □

A stronger version of rank-selected chain enumeration will be important in the proof of our main theorem in Section 3. Given a finite reflection group W of rank n, let R_{W} (s_{1}, s_{2}, …, s_{l}) denote the number of (multi-)chains

π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l−1}

in NC^{(m)}(W), such that rk(π_{i}) = s_{1}+s_{2}+⋯+s_{i}, i = 1, 2, …, l−1, and s_{1}+s_{2}+⋯+s_{l} = n. The following lemma says that zeroes in the argument of R_{W} ( ·) can be suppressed except for a zero in the first argument.

Lemma 7. Let W be a finite real reflection group of rank n and let s_{1}, s_{2}, …, s_{l} be non-negative integers with s_{1} + s_{2} + ⋯ + s_{l} = n. Then

R_{W} (s_{1}, …, s_{i}, 0, s_{i+1}, …, s_{l}) = R_{W} (s_{1}, …, s_{i}, s_{i+1}, …, s_{l})  (2.2)

for i = 1, 2, …, l. If i = l, equation (2.2) must be interpreted as

R_{W} (s_{1}, …, s_{l}, 0) = R_{W} (s_{1}, …, s_{l}).

Proof. This is obvious as long as i < l. If i = l, then, by definition, R_{W} (s_{1}, …, s_{l}, 0) counts all multi-chains π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l} with rk(π_{i}) = s_{1} +s_{2} +⋯+s_{i}, i = 1, 2, …, l. In particular, rk(π_{l}) = s_{1} + s_{2} + ⋯ + s_{l} = n, so that π_{l} must be the unique maximal element (c; ε, …, ε) of NC^{(m)}(W). Thus we are counting multi-chains π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l−1} with rk(π_{i}) = s_{1} + s_{2} + ⋯ + s_{i}, i = 1, 2, …, l − 1, and, again by definition, this number is given by R_{W} (s_{1}, …, s_{l}). □

Finally we quote the version of inclusion-exclusion given in [12, Sec. 2.1, Eq. (4)] that will be relevant to us.

Proposition 8. Let A be a finite set and w : A → ℂ a weight function on A. Furthermore, let S be a set of properties an element of A may or may not have. Given a subset Y of S, we define the functions f_{=}(Y) and f_{≥}(Y) by

f_{=}(Y) := ∑_{a} ′w(a),

where ∑ ′ is taken over all a ∈ A which have exactly the properties Y, and by

f_{≥}(Y) = ∑_{X⊇Y} f_{=}(X).

Then

f_{=}(∅) = ∑_{Y ⊆S}(−1)^{|Y |}f_{≥}(Y).  (2.3)

3. Proof of Main Theorem

Let c denote the unique maximum element (c; ε, …, ε) of NC^{(m)}(W) and let mins denote its set of minimal elements, the cardinality of which is Cat^{(m−1)}(W). The truncated poset

NC^{(m)}(W)\({c} ∪ mins)

is a rank-selected subposet of NC^{(m)}(W), the latter being shellable due to [1, Cor. 3.7.3]. If we combine this observation with the fact (see [5, Theorem 4.1]) that rank-selected subposets of shellable posets are also shellable, we conclude that NC^{(m)}(W)\({c} ∪ mins) is shellable. Since it is known that a pure d-dimensional shellable simplicial complex Δ is homotopy equivalent to a wedge of χ̃(Δ) d-dimensional spheres (this follows from Fact 9.19 in [6] and the fact that shellability implies the property of being homotopy-Cohen-Macaulay [6, Sections 11.2, 11.5]), it remains only to compute the reduced Euler characteristic χ̃( ·) of (the order complex of) NC^{(m)}(W)\({c} ∪ mins).

For a finite real reflection group W of rank n, let us again write R_{W} (s_{1}, s_{2}, …, s_{l}) for the number of (multi-)chains

π_{1} ≤ π_{2} ≤ ⋯ ≤ π_{l−1}

in NC^{(m)}(W) with rk(π_{i}) = s_{1}+s_{2}+⋯+s_{i}, i = 1, 2, …, l−1, and s_{1}+s_{2}+⋯+s_{l} = n. By definition, the reduced Euler characteristic is

−1 + ∑_{l=2}^{n}(−1)^{l}_{∑_{s_{1}+···+s_{l}=n} ; s_{1},...,s_{l}>0}R_{W} (s_{1}, s_{2}, …, s_{l}).  (3.1)

The sum over s_{1}, s_{2}, …, s_{l} in (3.1) could be easily calculated from Theorem 5, if there were not the restriction s_{1}, s_{2}, …, s_{l} > 0. In order to overcome this difficulty, we appeal to the principle of inclusion-exclusion. More precisely, for a fixed l, in Proposition 8 choose A = {(s_{1}, s_{2}, …, s_{l}) : s_{1} + s_{2} + ⋯ + s_{l} = n}, w((s_{1}, s_{2}, …, s_{l})) = R_{W} (s_{1}, s_{2}, …, s_{l}), and S = {S_{i} : i = 1, 2, …, l}, where S_{i} is the property of an element (s_{1}, s_{2}, …, s_{l}) ∈ A to satisfy s_{i} = 0. Then (2.3) becomes

∑_{s_{1}+···+s_{l}=n} ; s_{1},...,s_{l}>0R_{W} (s_{1}, s_{2}, …, s_{l}) = ∑_{I⊆{1,...,l}}(−1)^{|I|}_{∑_{s_{1}+···+s_{l}=n} ; s_{1},...,s_{l}≥0 ; s_{i}=0 for i∈I}R_{W} (s_{1}, s_{2}, …, s_{l}).

In view of Lemma 7, the right-hand side may be simplified, so that we obtain the equation

∑_{s_{1}+···+s_{l}=n} ; s_{1},...,s_{l}>0R_{W} (s_{1}, s_{2}, …, s_{l}) = _{∑_{I⊆{1,...,l}} ; 1∈I}(−1)^{|I|}_{∑_{s_{2}+···+s_{l−|I|+1}=n} ; s_{2},...,s_{l−|I|+1}≥0}R_{W} (0, s_{2}, …, s_{l−|I|+1})

+_{∑_{I⊆{1,...,l}} ; 1∉I}(−1)^{|I|}_{∑_{s_{1}+···+s_{l−|I|}=n} ; s_{1},...,s_{l−|I|}≥0}R_{W} (s_{1}, s_{2}, …, s_{l−|I|})

= ∑_{j=1}^{l}(−1)^{j}\binom{l − 1}{j − 1}_{∑_{s_{2}+···+s_{l−j+1}=n} ; s_{2},...,s_{l−j+1}≥0}R_{W} (0, s_{2}, …, s_{l−j+1})

+ ∑_{j=0}^{l}(−1)^{j}\binom{l − 1}{j}_{∑_{s_{1}+···+s_{l−j}=n} ; s_{1},...,s_{l−j}≥0}R_{W} (s_{1}, s_{2}, …, s_{l−j}).

By Lemma 6, the sum over s_{2}, …, s_{l−j+1} on the right-hand side is equal to Cat^{((l−j)m−1)}(W), while by Theorem 5 the sum over s_{1}, …, s_{l−j} is equal to Cat^{((l−j−1)m)}(W). If we substitute all this in (3.1), we arrive at the expression

− 1 + ∑_{l=2}^{n}(−1)^{l}( ∑_{j=1}^{l}(−1)^{j}\binom{l − 1}{j − 1} Cat^{((l−j)m−1)}(W)

+ ∑_{j=0}^{l−1}(−1)^{j}\binom{l − 1}{j} Cat^{((l−j−1)m)}(W)) (3.2)

for the reduced Euler characteristic that we want to compute. We now perform the replacement l = j + k in both sums. Thereby we obtain the expression

− 1 − Cat^{(−1)}(W) − Cat^{(−m)}(W) + Cat^{(0)}(W)

+ ∑_{k=0}^{n}(−1)^{k}( ∑_{j=1}^{n−k} \binom{j + k − 1}{j − 1} Cat^{(km−1)}(W) + ∑_{j=0}^{n−k} \binom{j + k − 1}{j} Cat^{((k−1)m)}(W)), (3.3)

the various terms in the first line being correction terms that cancel terms in the sums in the second line violating the condition l = j + k ≥ 2, which is present in (3.2). Since we shall make use of it below, the reader should observe that, by the definition (2.1) of Fuß–Catalan numbers, both Cat^{(km−1)}(W) and Cat^{((k−1)m)}(W) are polynomials in k of degree n with leading coefficient (mh)^{n}.

Again by (2.1), we have Cat^{(−1)}(W) = 0 and Cat^{(0)}(W) = 1. Therefore, if we evaluate the sums over j in (3.3) (this is a special instance of the Chu–Vandermonde summation), then we obtain the expression

− Cat^{(−m)}(W) + ∑_{k=0}^{n}(−1)^{k}(\binom{n}{k + 1} Cat^{(km−1)}(W) + \binom{n}{k} Cat^{((k−1)m)}(W))
= − Cat^{(−m)}(W) + Cat^{(−m−1)}(W)

− ∑_{k=0}^{n}(−1)^{k}\binom{n}{k} Cat^{((k−1)m−1)}(W) + ∑_{k=0}^{n}(−1)^{k}\binom{n}{k} Cat^{((k−1)m)}(W)

= − Cat^{(−m)}(W) + Cat^{(−m−1)}(W) − (−1)^{n}n!(mh)^{n} + (−1)^{n}n!(mh)^{n}
= − Cat^{(−m)}(W) + Cat^{(−m−1)}(W)
= −(−1)^{n} Cat_{+}^{(m−1)}(W) + (−1)^{n} Cat_{+}^{(m)}(W),

where, to go from the second to the third line, we used the well-known fact from finite difference calculus (cf. [12, Sec. 1.4, Eq. (26) and Prop. 1.4.2]), that, for any polynomial p(k) in k of degree n and leading coefficient p_{n}, we have

∑_{k=0}^{n}(−1)^{k}\binom{n}{k}p(k) = (−1)^{n}n!p_{n}.

4. The case of well-generated complex reflection groups

We conclude the paper by pointing out that our result in Theorem 3 on the Euler characteristic of the truncated poset of generalized noncrossing partitions extends naturally to well-generated complex reflection groups. We refer the reader to [10, 11] for all terminology related to complex reflection groups.

Let W be a finite group generated by (complex) reflections in ℂ^{n}, and let T ⊆ W denote the set of all reflections in the group. (Here, a reflection is a non-trivial element of GL(ℂ^{n}) which fixes a hyperplane pointwise and which has finite order.) As in Section 1, let ℓ_{T} : W → ℤ denote the word length in terms of the generators T. Now fix a regular element c ∈ W in the sense of Springer [11] and a positive integer m. (If W is a real reflection group, that is, if all generators in T have order 2, then the notion of “regular element” reduces to that of a “Coxeter element.”) As in the case of Coxeter elements, it can be shown that any two regular elements are conjugate to each other. A further assumption that we need is that W is well-generated, that is, that it is generated by n reflections given that n is minimal such that W can be realized as reflection group on ℂ^{n}. A complex reflection group 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. If V is the geometric representation of W, the degrees arise from the W -invariant polynomials on V, and the codegrees arise from the W -invariant polynomials in the dual representation V^{∗}. Orlik and Solomon [9] observed, using case-by-case checking, that W is well-generated if and only if its degrees and codegrees satisfy

d_{i} + d_{i}^{∗} = d_{n}

for all 1 ≤ i ≤ n. Together with the classification of Shephard and Todd [10], this constitutes a classification of well-generated complex reflection groups.

Given these extended definitions of ℓ_{T} and c, we define the set of m-divisible noncrossing partitions by (1.1), and its partial order by (1.2), as before. In the extension of Theorem 3 to well-generated complex reflection groups, we need the Fuß–Catalan number for W, which is again defined by (2.1), where the d_{i}’s are the degrees of (homogeneous polynomial generators of the invariants of) W, and where h is the largest of the degrees.

Theorem 9. Let W be a finite well-generated (complex) reflection group of rank n and let m be a positive integer. The order complex of the poset NC^{(m)}(W) with maximal and minimal elements deleted has reduced Euler characteristic

Cat^{(−m−1)}(W) − Cat^{(−m)}(W).  (4.1)

In order to prove this theorem, we may use the proof of Theorem 3 given in Sections 2 and 3 essentially verbatim. The only difference is that all notions (such as the reflections T or the order ℓ_{T}, for example), have to be interpreted in the extended sense explained above, and that “Coxeter element” has to be replaced by “regular element” everywhere. In particular, the extension of Theorem 4 to well-generated complex reflection groups is Proposition 13.1 in [3], and the proofs of Theorems 3.6.7 and Theorems 3.6.9(1) in [1] (which we used in order to establish Lemma 6 respectively Theorem 5) carry over essentially verbatim to the case of well-generated complex reflection groups.

References

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[8] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.

[9] P. Orlik and L. Solomon, Unitary reflection groups and cohomology, Invent. Math. 59 (1980), 77–94.

[10] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

[11] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198.

[12] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, California, 1986; reprinted by Cambridge University Press, Cambridge, 1998.

[13] M. Tomie, Möbius numbers of some modified generalized noncrossing partitions, preprint;

   arχiv:0905.1660.

Department of Mathematics, University of Miami, Coral Gables, Florida 33146,

USA. WWW: http://www.math.miami.edu/~armstrong.

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

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