Séminaire Lotharingien de Combinatoire 87 (2023), Article B87b

HAGLUND’S POSITIVITY CONJECTURE FOR MULTIPLICITY ONE PAIRS

ARITRA BHATTACHARYA

Abstract. Haglund’s conjecture states that \frac{⟨J_{λ}(q, q^{k}), s_{μ}⟩}{(1 − q)^{|λ|}} ∈ ℤ_{≥0}[q] for all partitions λ, μ and all non-negative integers k, where J_{λ} is the integral form Macdonald symmetric function and s_{μ} is the Schur function. This paper proves Haglund’s conjecture in the cases when the pair (λ, μ) satisfies K_{λ,μ} = 1 or K_{μ′,λ′} = 1 where K denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.

1. Introduction

The Macdonald symmetric functions P_{λ}(q, t), λ ∈ P ar, are a remarkable family of symmetric functions depending on two parameters q and t, indexed by the set of partitions P ar. They simultaneously generalize many known bases of symmetric functions.

We denote by K_{λ,μ}^{(1)}(q, t) the coefficient of the monomial symmetric function m_{μ} in the monomial expansion of P_{λ}(q, t). This is a rational function in q and t which has the Kostka number K_{λ,μ} as its limit when q and t tend to 1. There are various expressions for K_{λ,μ}^{(1)}(q, t) in the literature: in [Mac95] a tableaux formula is given, in [HHL04], [HHL06] a formula in terms of nonattacking fillings is found, and in [RY08] a formula in terms of alcove walks is given. A nice survey of some of the monomial expansions can be found in [GR21] and [GR21supplement].

In contrast, very little is known about the Schur expansion of P_{λ}(q, t). Some particular Schur coefficients of the integral form Macdonald polynomials J_{λ}(q, t) were found in [Yoo12] and [Yoo15]. The integral form Macdonald polynomial J_{λ}(q, t) is a certain normalization of P_{λ}(q, t), whose monomial coefficients are in ℤ[q, t].

In [HHL04] it was shown that the coefficient ⟨J_{λ}(q, t), h_{μ}⟩ of m_{μ} in J_{λ}(q, t) has the following positivity property:

\frac{⟨J_{λ}(q, q^{k}), h_{μ}⟩}{(1 − q)^{|λ|}} ∈ ℤ_{≥0}[q] for all k ∈ ℤ_{≥0} and λ, μ ∈ P ar.  (1)

Haglund [Hag10] conjectured that the above equation holds true if h_{μ} is replaced by s_{μ}.

Conjecture A (Haglund). For partitions λ and μ,

\frac{⟨J_{λ}(q, q^{k}), s_{μ}⟩}{(1 − q)^{|λ|}} ∈ ℤ_{≥0}[q] for all k ∈ ℤ_{≥0}.  (Hag(λ, μ))

Yoo showed this is true in some special cases in [Yoo12] and [Yoo15] by obtaining explicit formulas for the coefficients. Among them, some particular cases are when λ has only one row, when λ is of hook shape with ℓ(λ) ≥ λ_{1} − 2, when λ has at most two columns, when μ is of hook shape, or when λ and μ both have length at most 2.

2020 Mathematics Subject Classification. 05E05.

This paper extends Yoo’s results in a different direction, showing Haglund’s conjecture to be true for all ‘multiplicity one pairs’, i.e., for (λ, μ) such that K_{λ,μ} = 1 or K_{μ′,λ′} = 1.

The following is the main result of this paper.

Theorem 1.1. Hag(λ, μ) holds true for all pairs λ, μ such that either K_{λ,μ} = 1 or K_{μ′,λ′} = 1.

In fact, what we establish is a dual version of Haglund’s conjecture which can be easily seen to be equivalent to the original. Namely, let k_{λ,μ}(q, t) = ⟨s_{λ}, J_{μ}(q, t)⟩_{q,t} where ⟨ , ⟩_{q,t} is the (q, t)-deformation of the Hall scalar product (cf. [Mac95]). Then we have the following.

Conjecture B (Dual Haglund’s Conjecture). For partitions λ, μ, we have

\frac{k_{λ,μ}(t^{k}, t)}{(1 − t)^{|λ|}} ∈ ℤ_{≥0}[t] for all k ∈ ℤ_{≥0}.  (Hag′(λ, μ))

We also define another q, t-analogue K_{λ,μ}^{(2)}(q, t) of the Kostka numbers by

s_{λ} = ∑_{μ∈P ar} K_{λ,μ}^{(2)}(q, t)P_{μ}(q, t)for all λ ∈ P ar.  (2)

K_{λ,μ}^{(2)}(q, t) coincides with k_{λ,μ}(q, t) up to a constant. Analogous to K_{λ,μ}^{(1)}(q, t), this is a rational function in q, t, and K_{λ,μ}^{(2)}(0, t) = K_{λ,μ}(t), the Kostka–Foulkes polynomial. The matrix (K_{λ,μ}^{(2)})_{λ,μ ⊢ n} has the interesting property that (cf. [Mac95, Chapter VI, (5.1′′)])

K_{λ,μ}^{(2)}(q, t) = (K^{(2)})_{μ′,λ′}^{−1}(t, q).  (3)

Since the entries of (K^{(2)})^{−1} are the Schur coefficients of the P_{λ}(q, t), finding K^{(2)} is roughly equivalent to finding these.

We use Macdonald’s description of K_{λ,μ}^{(1)}(q, t) and basic properties of P_{λ}(q, t) to find some general reduction principles. In particular, we show that (see Lemma 4.2), if for a pair of partitions (λ, μ) we have λ_{1} + ⋯ + λ_{r} = μ_{1} + ⋯ + μ_{r} for some r, then, for i = 1, 2, we have

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{1},μ^{1}}^{±1} · (K^{(i)})_{λ^{2},μ^{2}}^{±1} ,

where λ^{1} = (λ_{1}, …, λ_{r}), μ^{1} = (μ_{1}, …, μ_{r}), λ^{2} = (λ_{r+1}, …), μ^{2} = (μ_{r+1}, …).

A dual version of this is given in Lemma 4.3, where instead of breaking the Young diagrams across a row, we break across a column.

Let λ, μ ∈ P ar such that λ ≥ μ. We can always decompose λ = (λ^{1}, …, λ^{r}) and μ = (μ^{1}, …, μ^{r}) where λ^{1}, … λ^{r}, μ^{1}, …, μ^{r} are partitions such that, for i ∈ {1, … r}, λ^{i} ≥ μ^{i} and, for i ∈ {1, … r − 1}, ℓ(λ^{i}) = ℓ(μ^{i}). Then, for i = 1, 2, we have

(K^{(i)})_{λ,μ}^{±1}(q, t) = ∏_{j=1}^{r}(K^{(i)})_{λ^{j},μ^{j}}^{±1}(q, t).  (4)

For j ∈ {1, …, r −1} each pair (λ^{j}, μ^{j}) contains a common rectangle of row length at least λ_{1}^{j+1}. Proving the dual Haglund conjecture is equivalent to proving it for each pair (λ^{j}, μ^{j}) with these rectangles removed (see Corollary 4.1). We can repeat this process for each pair until no further decomposition is possible.

This allows us to only look at pairs of partitions (λ, μ) which are irreducible, i.e., which satisfy λ ≥ μ and λ_{1} + ⋯ + λ_{i} > μ_{1} + ⋯ + μ_{i} for 1 ≤ i ≤ ℓ(λ). Berenshteĭn and Zelevinskiĭ in [BZ90] gave a criterion for a pair of partitions (λ, μ) to satisfy K_{λ,μ} = 1. This can be restated as follows.

Theorem 1.2. Let λ, μ be an irreducible pair. Then K_{λ,μ} = 1 if and only if either

(1) λ = (m^{n}) for some m ∈ ℤ_{≥1}, n ∈ ℤ_{≥0} and ℓ(μ) = n + 1, or

(2) λ = (n) and μ ⊢ n, μ ≠ λ for some n ∈ ℤ_{≥1}.

We find (see Corollary 5.1) that, if λ, μ are partitions contained in a rectangle (m^{n+1}), then, for i = 1, 2, we have

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{c},μ^{c}}^{±1} ,  (5)

where λ^{c}, μ^{c} denote the complements of the partitions λ, μ inside the rectangle (m^{n+1}). Since the complement of the partition (m^{n}) is (m), this reduces all computation of K_{λ,μ}^{(2)} when K_{λ,μ} = 1 to the case where λ is a single row.

In this case we calculate K_{λ,μ}^{(2)} by the use of the Cauchy identity and Macdonald’s principal specialization formula (see Proposition 3.2.)

Finally, in Section 6 we finish the proof of the dual version of Haglund’s conjecture when K_{λ,μ} = 1 by analyzing the normalization factors for the integral form Macdonald polynomials. By exploiting duality of Macdonald polynomials (see Lemma 2.1) we get a proof when K_{μ′,λ′} = 1.

Our reduction principles may also be applied to instances where the final reduction is to pairs (λ, μ) of the form studied by Yoo. Therefore explicit formulas for k_{λ,μ}(q, t) can be obtained in a wider set of cases.

An interesting feature of our formulas and many of Yoo’s formulas is that K_{λ,μ}^{(2)}(q, t) can be written as a sum of K_{λ,μ} many terms, where each term is a product with factors of the form (t^{a} − q^{b}t^{c}) in the numerator and (1 − q^{d}t^{e}) in the denominator with a, b, c, d, e ∈ ℤ_{≥0}. This is especially striking if K_{μ′,λ′} = 1, where K_{λ,μ} could be larger than 1.

Acknowledgments. We would like to thank R. Venkatesh for organizing the Workshop on Macdonald Polynomials 2021. We are very grateful to A. Ram for his inspiring lectures and discussions. Finally, a lot of thanks to S. Viswanath for numerous discussions, going through the whole paper and giving helpful suggestions.

2. Definitions

We first review some basic definitions in the theory of symmetric functions. The main reference for this section is [Mac95].

2.1. Partitions. By a partition we mean a sequence λ = (λ_{1}, λ_{2}, …) of non-negative integers which is weakly decreasing and has a finite sum. By P ar we denote the set of all partitions. A partition λ may be realized as a set of boxes upper left justified and arranged in rows so that there are λ_{i} boxes in row i. This is called the Young diagram of λ. For a partition λ the conjugate partition λ′ is the partition obtained by interchanging the rows and columns in the diagram of λ. The length of a partition λ is ℓ(λ) = λ′_{1}, and the size is |λ| = ∑ λ_{i}

The set of partitions is partially ordered by the dominance order defined by

λ ≥ μ ⟺ λ_{1} + ⋯ + λ_{i} ≥ μ_{1} + ⋯ + μ_{i} for all i ≥ 1, and |λ| = |μ|.  (6)

For a box x = (r, c) in the diagram of λ, where r and c are the row and column numbers of x with row numbering starting from 1 in the top row and column numbering starting from 1 in the leftmost column, we write x ∈ λ and define the arm length a_{λ}(x), coarm length a′_{λ}(x), leg length l_{λ}(x) and the coleg length l′_{λ}(x) of x, by

a_{λ}(x) = λ_{r} − c, a′_{λ}(x) = c − 1 ,  (7)
l_{λ}(x) = λ′_{c} − r, l′_{λ}(x) = r − 1 .  (8)
a a′ l′ lx a_{λ}(x) = 3, a′_{λ}(x) = 3 l_{λ}(x) = 2, l′_{λ}(x) = 1

Figure 1. Young diagram for λ = (8, 7, 4, 4, 1) and arm, coarm, leg, coleg lengths of x = (2, 4)

When the partition is understood, we drop λ from the notation and simply write a(x), a′(x), l(x), l′(x).

The content of a box x ∈ λ is c(x) = a′(x) − l′(x). The hook length of the box x is h(x) = a(x) + 1 + l(x). For a partition λ, define n(λ) = ∑_{x∈λ} l′(x).

2.2. Symmetric Functions. Let q and t be complex variables. We work in the ring Λ of symmetric functions in infinitely many variables x_{1}, x_{2}, … with coefficients from ℂ(q, t).

The space of symmetric functions has a few known bases indexed by λ ∈ P ar, such as the monomial basis (m_{λ}), the elementary basis (e_{λ}), the homogeneous basis (h_{λ}), the power sum basis (p_{λ}), and the Schur basis (s_{λ}). Their definitions can be found in [Mac95].

For a partition λ and for i ≥ 1, let m_{i}(λ) be the number of parts of λ equal to i, and let z_{λ} = ∏_{i=1}^{ℓ(λ)} i^{m_{i}(λ)}m_{i}(λ)!. The Hall scalar product on Λ, denoted by ⟨ , ⟩, is defined by ⟨p_{λ}, p_{μ}⟩ = z_{λ}δ_{λ,μ}. We also have ⟨s_{λ}, s_{μ}⟩ = δ_{λ,μ} and ⟨h_{λ}, m_{μ}⟩ = δ_{λ,μ}.

2.3. Plethysm. The power sums p_{k}(X) = ∑_{i≥0} x_{i}^{k}, k ≥ 1, form an algebraically independent generating set for Λ over ℂ(q, t). For a formal series of rational functions E = E(z_{1}, z_{2}, …) we denote by p_{k}[E] the plethystic substitution of E into p_{k}, defined to be E(z_{1}^{k}, z_{2}^{k}, …). In other words, p_{k}[E] is the result of replacing each variable in E by its k-th power, and the constant is kept as it is. Since any symmetric function f ∈ Λ is a polynomial in the p_{k}, we can define f[E] as the unique ℂ-algebra homomorphism extending p_{k} ↦ p_{k}[E]. Note that we are not extending as ℂ(q, t)-algebra homomorphism, i.e., q, t are also treated as variables. When using plethysm, we write alphabets X = {x_{1}, x_{2}, … }, Y = {y_{1}, y_{2}, … } as X = ∑ x_{i}, Y = ∑ y_{i}. So p_{k}[X] = ∑ x_{i}^{k} = p_{k}(X) for all k, and by extension, f[X] = f(X) for any symmetric function f.

In particular,

p_{k}[X \frac{1 − q}{1 − t} ] = p_{k}(X)\frac{1 − q^{k}}{1 − t^{k}} ,  (9)
p_{k}[\frac{1 − q}{1 − t} ] = \frac{1 − q^{k}}{1 − t^{k}} .  (10)

For more on plethysm see [Hai99] and [Hag06].

2.4. Tableaux. A semistandard Young tableau or a tableau T of shape λ is a filling of the diagram λ with entries from 1, 2, … such that the entries are weakly increasing left to right along the rows and strictly increasing top to bottom along columns. If for each i ≥ 1 there are μ_{i} many i’s in T then T is said to have content μ = (μ_{1}, μ_{2}, …). We denote the set of tableaux of shape λ, content μ by SSY T (λ, μ). The cardinality of this set is the Kostka number K_{λ,μ}.

We have

s_{λ} = ∑_{μ} K_{λ,μ}m_{μ} .  (11)

By a matrix indexed by partitions we always mean that the rows and columns are indexed by partitions, and the indexing is compatible with dominance ordering, i.e., the partitions are listed in a sequence such that if λ ≥ μ in dominance order then λ precedes μ in the sequence, and if |λ| < |μ| then λ precedes μ. These are infinite matrices. For a matrix M indexed by partitions, we call it upper-unitriangular if M_{λ,μ} = 0 unless μ ≤ λ, and if M_{λ,λ} = 1. The set of upper unitriangular matrices forms a group.

We denote by K the matrix indexed by partitions whose (λ, μ)-th entry is K_{λ,μ}. This matrix K is upper-unitriangular.

2.5. Macdonald Symmetric Functions. The scalar product ⟨ , ⟩_{q,t} on Λ is defined by

⟨f , g⟩_{q,t} = ⟨f[X], g[X \frac{1 − q}{1 − t} ]⟩,  (12)

where ⟨ , ⟩ on the right-hand side is the Hall scalar product.

The Macdonald P -functions (P_{λ}[X; q, t])_{λ∈P ar} are characterized by the following two properties:

P_{λ}[X; q, t] − m_{λ}[X] ∈ ∑_{μ<λ} ℂ(q, t)m_{μ}[X] ,  (13)
⟨P_{λ}[X; q, t], P_{μ}[X; q, t]⟩_{q,t} = 0 if λ ≠ μ ,  (14)

where ≤ is the dominance order on partitions.

When the variables are understood, we may drop X from the notation and just write P_{λ}(q, t).

Let (Q_{λ}(q, t)) denote the dual basis to (P_{λ}(q, t)) under the q, t scalar product ⟨ , ⟩_{q,t}. For a partition μ, let

c_{μ}(q, t) = ∏_{x∈μ}(1 − q^{a(x)} t^{l(x)+1}) ,  (15)
c′_{μ}(q, t) = ∏_{x∈μ}(1 − q^{a(x)+1} t^{l(x)}) ,  (16)

b_{μ}(q, t) = \frac{c_{μ}(q, t)}{c′_{μ}(q, t)} .  (17)

Then we have the following relation.

Theorem 2.1 ([Mac95, Chapter VI, (6.19)]).

Q_{μ}(q, t) = b_{μ}(q, t)P_{μ}(q, t) .

The integral form Macdonald polynomials are

J_{μ}(q, t) = c_{μ}(q, t)P_{μ}(q, t) = c′_{μ}(q, t)Q_{μ}(q, t) .  (18)

The monomial coefficients of J_{μ}(q, t) are in ℤ[q, t].

2.6. q, t-Kostka Functions. We define two q, t-analogues of the Kostka numbers.

Definition 2.1. For λ, μ ∈ P ar, let K_{λ,μ}^{(1)}(q, t) and K_{λ,μ}^{(2)}(q, t) be defined as the following change of basis coefficients:

P_{λ}(q, t) = ∑_{μ} K_{λ,μ}^{(1)}(q, t) m_{μ} ,  (19)
s_{λ} = ∑_{μ} K_{λ,μ}^{(2)}(q, t) P_{μ}(q, t) .  (20)

In other words, K_{λ,μ}^{(1)}(q, t) = ⟨P_{λ}(q, t), h_{μ}⟩ and K_{λ,μ}^{(2)}(q, t) = ⟨s_{λ}, Q_{μ}⟩_{q,t} . Note that both the change of basis matrices are upper- unitriangular, i.e., for i ∈ {1, 2}, we have K_{λ,μ}^{(i)}(q, t) = 0 unless μ ≤ λ and, for all λ ∈ P ar, we have K_{λ,λ}^{(i)}(q, t) = 1. For K^{(1)} this follows from the definition of Macdonald polynomials, and for K^{(2)} this is because

K = K^{(2)}K^{(1)} ,  (21)

where K is the matrix of Kostka numbers, which is known to be upper-unitriangular.

2.7. Macdonald’s Formula for K_{λ,μ}^{(1)}(q, t). Macdonald gave a formula for K_{λ,μ}^{(1)}(q, t) in [Mac95], which we now describe. For two partitions λ, μ with μ_{i} ≤ λ_{i} for all i, λ/μ is called a horizontal strip if λ′_{j} − μ′_{j} ∈ {0, 1} for all j ∈ ℤ_{>0}. If λ/μ is a horizontal strip, then define

ψ_{λ/μ}(q, t) = ∏_{\begin{matrix}x=(r,c)∈μ \\ λ_{r}≠μ_{r} \\ λ′_{c}=μ′_{c} \end{matrix}} \frac{(1 − q^{a_{μ}(x)} t^{l_{μ}(x)+1})}{(1 − q^{a_{μ}(x)+1} t^{l_{μ}(x)})} \frac{(1 − q^{a_{λ}(x)+1} t^{l_{λ}(x)})}{(1 − q^{a_{λ}(x)} t^{l_{λ}(x)+1})} .  (22)

Given a tableau T, let T_{≤i} denote the shape obtained by the boxes with content 1, …, i. In a tableau T, T_{≤i}/T_{≤(i−1)} is by definition a horizontal strip for each i. For a tableau T, define

ψ_{T} = ∏_{i≥1} ψ_{T_{≤i}/T_{≤(i−1)}} .  (23)

Then we have the following formula.

Theorem 2.2 ([Mac95, Chapter VI, (7.13′)]). For partitions λ, μ,

K_{λ,μ}^{(1)}(q, t) = ∑_{T ∈SSY T (λ,μ)} ψ_{T} (q, t) .  (24)

2.8. Specializations.

(1) At q = 0, K_{λ,μ}^{(2)}(0, t) is the Kostka–Foulkes polynomial K_{λ,μ}(t). It is a polynomial in t with non-negative integer coefficients. A combinatorial formula for the Kostka–Foulkes polynomial was given by Lascoux and Schützenberger. There is a function ch : SSY T (λ, μ) → ℤ_{≥0} called charge, such that

K_{λ,μ}(t) = ∑_{T ∈SSY T (λ,μ)} t^{ch(T)} .  (25)

(2) At t = 1, K_{λ,μ}^{(2)}(q, 1) = K_{λ,μ} for arbitrary q.

(3) At q = 1, the matrix K^{(2)}(1, t) = J(K^{−1})^{tr}J, where J is the matrix indexed by partitions with J_{λ,μ} = δ_{λ,μ′}. So the matrix K^{(2)} contains information on both the matrix K and its inverse. A general version is Lemma 2.1 below.

2.9. Duality. By (K^{(i)})_{λ,μ}^{−1} we mean taking the (λ, μ)-entry of the inverse matrix.

The following lemma says that expanding the Macdonald functions in the Schur basis is roughly equivalent to its inverse problem, namely expanding the Schur functions in the Macdonald basis.

Lemma 2.1 ([Mac95, Chapter VI, (5.1′′)]).

K_{λ,μ}^{(2)}(q, t) = (K^{(2)})_{μ′,λ′}^{−1}(t, q) .  (26)

Since c_{λ}(q, t) = c′_{λ′}(t, q) (cf. [Mac95, Chapter VI, (8.2)]) from (18),(26) and Definition 2.1 we get

⟨J_{λ}(t, q), s_{μ}⟩ = ⟨J_{λ′}(q, t), s_{μ′}⟩_{q,t} .  (27)

2.10. Haglund’s Conjecture. In [HHL06] it was shown that the monomial coefficients ⟨J_{λ}(q, t), h_{μ}⟩ of J_{λ}(q, t) have the following positivity property:

\frac{⟨J_{λ}(q, q^{k}), h_{μ}⟩}{(1 − q)^{|λ|}} ∈ ℤ_{≥0}[q] for all k ∈ ℤ_{≥0} .  (28)

Haglund made the following conjecture in [Hag10] about the Schur coefficients ⟨J_{λ}(q, t), s_{μ}⟩ of J_{λ}(q, t).

Conjecture 1 (Haglund). For partitions λ and μ,

\frac{⟨J_{λ}(q, q^{k}), s_{μ}⟩}{(1 − q)^{|λ|}} ∈ ℤ_{≥0}[q] for all k ∈ ℤ_{≥0} .  (Hag(λ, μ))

Yoo [Yoo12,Yoo15] showed that this is true in some special cases. Let

k_{λ,μ}(q, t) = ⟨s_{λ}, J_{μ}(q, t)⟩_{q,t} .  (29)

Then

k_{λ,μ}(q, t) = K_{λ,μ}^{(2)}(q, t)c′_{μ}(q, t) = (K^{(2)})_{μ′,λ′}^{−1}(t, q)c′_{μ}(q, t) .  (30)

Note that c′_{μ}(0, t) = 1, so

k_{λ,μ}(0, t) = K_{λ,μ}(t) ,  (31)

where the right-hand side is the Kostka–Foulkes polynomial as in Subsection 2.8. By (27), we have the following dual version of Haglund’s conjecture.

Conjecture 2. For partitions λ, μ,

\frac{k_{λ,μ}(t^{k}, t)}{(1 − t)^{|λ|}} ∈ ℤ_{≥0}[t] for all k ∈ ℤ_{≥0} .  (Hag′(λ, μ))

3. The Case Where λ Is a Single Row or μ Is a Single Column

In this section we derive formulas for K_{(n),μ}^{(2)}(q, t) and K_{λ,1^{n}}^{(2)}(q, t), and use them to show Hag′(λ, μ) is true in these cases.

We begin by reviewing Cauchy formulas and principal specializations.

3.1. Cauchy Identity. For a series of rational functions E we let Ω[E] = ∑_{n≥0} h_{n}[E]. Let (u_{λ})_{λ∈Par}, (v_{λ})_{λ∈Par} be Hall-dual bases. Then the Cauchy identity says that

Ω[XY] = ∑_{λ∈Par} u_{λ}[X]v_{λ}[Y] .  (32)

If (u_{λ})_{λ∈Par}, (v_{λ})_{λ∈Par} are q, t-dual bases, then by definition of ⟨, ⟩_{q,t}, the bases

(u_{λ}[X \frac{1 − q}{1 − t} ])_{λ∈P ar} and (v_{λ}[X])_{λ∈P ar} are Hall-dual, hence,

Ω[XY] = ∑_{λ∈P ar} u_{λ}[X \frac{1 − q}{1 − t} ]v_{λ}[Y] .  (33)

Equivalently, by replacing X by X \frac{1 − t}{1 − q},

we get the (q, t)-Cauchy identity

Ω[XY \frac{1 − t}{1 − q} ] = ∑_{λ∈P ar} u_{λ}[X]v_{λ}[Y] .  (34)

Since (h_{λ})_{λ∈P ar} and (m_{λ})_{λ∈P ar} are Hall-dual, (h_{λ}[X \frac{1 − t}{1 − q} ])_{λ∈P ar} is q, t-dual to

(m_{λ}[X])_{λ∈P ar}.

We give a quick proof of the following well-known result (cf. [Mac95, Chapter VI, (5.5)]) to illustrate the power of plethystic methods.

Proposition 3.1. We have

Q_{(n)}[X; q, t] = h_{n}[X \frac{1 − t}{1 − q} ] .  (35)

Proof. The Cauchy identity gives

∑_{λ∈P ar} h_{λ}[X \frac{1 − t}{1 − q} ]m_{λ}[Y] = ∑_{λ∈P ar} Q_{λ}[X; q, t]P_{λ}[Y; q, t] ,  (36)

Substitute Y = y, where y is a variable. Since

m_{λ}[y] = m_{λ}(y, 0, 0, …) = \begin{cases}y^{n}, & if λ = (n) for some n ≥ 0, \\ 0, & otherwise, \end{cases}  (37)

and by unitriangularity for P_{λ}, we get

P_{λ}[y] = \begin{cases}y^{n}, & if λ = (n) for some n ≥ 0, \\ 0, & otherwise. \end{cases}  (38)

Hence, by (36) above,

∑_{n≥0} h_{(n)}[X \frac{1 − t}{1 − q} ]y^{n} = ∑_{n≥0} Q_{(n)}[X; q, t]y^{n} .  (39)

Comparing coefficient of y^{n} we get the result. □

3.2. Principal Specializations. We can use the Cauchy identity and plethystic substitution to calculate the first row of various transition matrices. The next lemma and the corollaries illustrate that. We will need Macdonald’s evaluation identity stated below.

Theorem 3.1 ([Mac95, Chapter VI, (6.17)]). Let z be a complex variable. Then

P_{λ}[\frac{1 − z}{1 − t}; q, t] = ∏_{s∈λ} \frac{t^{l′(s)} − q^{a′(s)}z}{1 − q^{a(s)}t^{l(s)+1}} ,  (40)

Q_{λ}[\frac{1 − z}{1 − t}; q, t] = ∏_{s∈λ} \frac{t^{l′(s)} − q^{a′(s)}z}{1 − q^{a(s)+1}t^{l(s)}} .  (41)

3.3. The Case Where λ Is a Row or μ Is a Column. We now calculate k_{λ,μ}(q, t)

in these special cases. Replacing Y by \frac{1 − z}{1 − t} in the Cauchy identity (34) and using Theorem 3.1 we get the following lemma.

Lemma 3.1. Let z be a complex variable. For a pair (u_{λ})_{λ∈Par} and (v_{λ})_{λ∈Par} of q, t-dual bases of Λ we have

h_{n}[X \frac{1 − z}{1 − q} ] = ∑_{λ ⊢n} v_{λ}[\frac{1 − z}{1 − t} ]u_{λ}[X] .  (42)

In particular,

h_{n}[X \frac{1 − z}{1 − q} ] = ∑_{λ ⊢n} ∏_{x∈λ} \frac{t^{l′(x)} − q^{a′(x)}z}{1 − q^{a(x)+1}t^{l(x)}} P_{λ}[X; q, t] .  (43)

At t = q,

h_{n}[X \frac{1 − z}{1 − q} ] = ∑_{λ ⊢n} q^{n(λ)} ∏_{x∈λ} \frac{1 − q^{c(x)}z}{1 − q^{h(x)}} s_{λ}[X] .  (44)

Proposition 3.2. We have

k_{(n),μ}(q, t) = t^{n(μ)} ∏_{x∈μ}(1 − q^{a′(x)+1}t^{−l′(x)}) .  (45)

Proof. We substitute z = q in (43), use (18), (29) and the fact that s_{(n)} = h_{n}. □

We can use Lemma 3.1 to get another proof of the following result of [Yoo12] in the dual formalism.

Proposition 3.3. We have

k_{λ,1^{n}}(q, t) = \frac{t^{n(λ′)}[n]_{t}!}{∏_{x∈λ}[h(x)]_{t}} ∏_{x∈λ}(1 − t^{−c(x)}q)  (46)
= K_{λ,1^{n}}(t) ∏_{x∈λ}(1 − t^{−c(x)}q) ,  (47)

where [j]_{t} = \frac{1 − t^{j}}{1 − t} for j ∈ ℤ_{≥1} and [n]_{t}! = [n]_{t}[n − 1]_{t} … [1]_{t} .

Proof. Substituting z = t in (44), we get

h_{n}[X \frac{1 − t}{1 − q} ] = ∑_{λ⊢n} ∏_{x∈λ} \frac{q^{l′(x)} − q^{a′(x)}t}{1 − q^{h(x)}}s_{λ}[X] .  (48)

By Proposition 3.1, the left-hand side is Q_{n}(q, t), and so the product in front of s_{λ}[X] on the right-hand side is ⟨Q_{n}(q, t), s_{λ}⟩. Observe that c′_{(n)}(q, t) = (q; q)_{n}, where by (a; x)_{n} we mean the product ∏_{i=0}^{n−1}(1 − ax^{i}). Since c′_{(n)}Q_{n} = J_{n}, Equation (27) gives

k_{λ′,1^{n}}(q, t) = (t; t)_{n} ∏_{x∈λ} \frac{t^{l′(x)} − t^{a′(x)}q}{1 − t^{h(x)}} .  (49)

Since conjugation interchanges coleg lengths and coarm lengths, and the set of hook lengths remains unchanged, we have

k_{λ,1^{n}}(q, t) = (t; t)_{n} ∏_{x∈λ} \frac{t^{a′(x)} − t^{l′(x)}q}{1 − t^{h(x)}}  (50)

= \frac{t^{n(λ′)}(t; t)_{n}}{∏_{x∈λ}(1 − t^{h(x)})} ∏_{x∈λ}(1 − t^{−c(x)}q) .  (51)

Since k_{λ,μ}(0, t) = K_{λ,μ}(t), the result follows. □

Corollary 3.1. Hag′(λ, μ) holds when λ is a row or when μ is a column.

Proof. When λ = (n) and μ is arbitrary, from (45) we have

k_{(n),μ}(t^{k}, t) = t^{n(μ)} ∏_{x∈μ}(1 − t^{k(a′(x)+1)−l′(x)}) for all k ∈ ℤ_{≥0} .  (52)

Since the quantity k(a′(x) + 1) − l′(x) decreases by 1 down a column and it is nonnegative in the top row, if it is negative for some x ∈ μ then it must also attain the value 0 above x in the same column. Hence

\frac{k_{(n),μ}(t^{k}, t)}{(1 − t)^{n}} = \begin{cases}t^{n(μ)} ∏_{x∈μ}[k(a′(x) + 1) − l′(x)]_{t}, & if ℓ(μ) ≤ k, \\ 0, & otherwise, \end{cases}  (53)

where [m]_{t} = 1 + t + ⋯ + t^{m−1}. In both cases \frac{k_{(n),μ}(t^{k}, t)}{(1 − t)^{n}} ∈ ℤ_{≥0}[t].

Next, when μ = 1^{n} and λ is arbitrary, we get

k_{λ,1^{n}}(t^{k}, t) = K_{λ,1^{n}}(t) ∏_{x∈λ}(1 − t^{k−c(x)}) .  (54)

Along a row the quantity k − c(x) decreases by 1 as one moves from left to right, and for the first box in the row (in the first column) k −c(x) ≥ 0. So if k −c(x) < 0 for some x ∈ λ then in that row there is a box y such that k − c(y) = 0, implying k_{λ,1^{n}}(t^{k}, t) = 0. Otherwise k − c(x) ≥ 0 for all x ∈ λ, and hence

\frac{k_{λ,1^{n}}(t^{k}, t)}{(1 − t)^{n}} = \begin{cases}K_{λ,1^{n}}(t) ∏_{x∈λ}[k − c(x)]_{t}, & if λ_{1} ≤ k, \\ 0, & otherwise. \end{cases}  (55)

Since K_{λ,μ}(t) ∈ ℤ_{≥0}[t], we obtain \frac{k_{λ,1^{n}}(t^{k}, t)}{(1 − t)^{n}} ∈ ℤ_{≥0}[t]. □

4. Multiplication

We state a general lemma on posets that will be useful in this section.

Lemma 4.1. Let M be an upper unitriangular matrix indexed by a partially ordered set P, i.e., for all a ∈ P, M_{a,a} = 1 and M_{a,b} = 0 unless b ≤ a. For an interval I in P let M|_{I} = (M_{a,b})_{a,b∈I} denote the matrix restricted to the interval I. Then, for any interval J containing a, b, we have (M^{−1})_{a,b} = (M|_{J})_{a,b}^{−1}.

Proof. Let N = M^{−1}. The lemma follows since one can simply write down the solution to the equations

δ_{a,b} = ∑_{a≤c≤b} N_{a,c}M_{c,b}  (56)

recursively, starting from b = a. □

Let λ^{1}, λ^{2} be two partitions such that the least non-zero part of λ^{1} is at least the largest part of λ^{2}. In this case by (λ^{1}, λ^{2}) we mean the partition (λ_{1}^{1}, …, λ_{ℓ(λ^{1})}^{1}, λ_{1}^{2}, …).

Lemma 4.2. Let λ = (λ^{1}, λ^{2}) and μ = (μ^{1}, μ^{2}) where λ^{j} ≥ μ^{j} (in particular, |λ^{j}| = |μ^{j}|) for j = 1, 2 and ℓ(λ^{1}) = ℓ(μ^{1}).

Then for i = 1, 2,

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{1},μ^{1}}^{±1}(K^{(i)})_{λ^{2},μ^{2}}^{±1} .  (57)

Proof. Let T ∈ SSY T (λ, μ). Since ℓ(λ^{1}) = ℓ(μ^{1}) and |λ^{1}| = |μ^{1}|, in the first ℓ(λ^{1}) rows of T the content must be μ^{1}. We can break the tableau T into two pieces; let T^{1} comprise the first ℓ(λ^{1}) rows, and T^{2} be the tableau formed by the remaining rows of T in which we subtract ℓ(λ^{1}) from each entry. It is clear that we get a bijection

SSY T (λ, μ) → SSY T (λ^{1}, μ^{1}) × SSY T (λ^{2}, μ^{2}),
T ↦ (T^{1}, T^{2}).

In particular, K_{λ,μ} = K_{λ^{1},μ^{1}}K_{λ^{2},μ^{2}}.

For 1 ≤ i ≤ ℓ(λ^{1}) the horizontal strips T_{≤i}/T_{≤i−1} and T_{≤i}^{1}/T_{≤i−1}^{1} are the same, so ψ_{T_{≤i}/T_{≤i−1}} = ψ_{T_{≤i}^{1}/T_{≤i−1}^{1}}.

For i > ℓ(λ^{1}) the horizontal strips T_{≤i}/T_{≤i−1} differ from T_{≤i−ℓ(λ^{1})}^{2}/T_{≤i−ℓ(λ^{1})−1}^{2} in the first ℓ(λ^{1}) rows, but none of those boxes contribute to ψ_{T_{≤i}/T_{≤i−1}}, and the contribution from the remaining boxes is the same since the arms and legs are the same. So ψ_{T_{≤i}/T_{≤i−1}} = ψ_{T_{≤i−ℓ(λ^{1})}^{2}/T_{≤i−ℓ(λ^{1})−1}^{2}} .

Therefore the above bijection preserves ψ, that is,

ψ_{T} = ψ_{T^{1}}ψ_{T^{2}} .  (58)

By (24), we get

K_{λ,μ}^{(1)} = K_{λ^{1},μ^{1}}^{(1)}K_{λ^{2},μ^{2}}^{(1)} .  (59)

Let λ ≥ ν ≥ μ. Setting ν^{1} = (ν_{1}, …, ν_{ℓ(λ^{1})}) and ν^{2} = (ν_{ℓ(λ^{1})+1}, …), we have λ^{i} ≥ ν^{i} ≥ μ^{i} for i = 1, 2 and ℓ(λ^{1}) = ℓ(ν^{1}) = ℓ(μ^{1}). So the interval [λ, μ] in the dominance order becomes [λ, μ] = [λ^{1}, μ^{1}] × [λ^{2}, μ^{2}]. The matrix K^{(1)} restricted to [λ, μ] is then the tensor product

K^{(1)}|_{[λ,μ]} = K^{(1)}|_{[λ^{1},μ^{1}]} ⊗ K^{(1)}|_{[λ^{2},μ^{2}]} .  (60)

In particular, K|_{[λ,μ]} = K|_{[λ^{1},μ^{1}]} ⊗ K|_{[λ^{2},μ^{2}]}.

Since the inverse of the tensor product of two matrices is the tensor product of their inverses, we get

(K^{(1)})^{−1}|_{[λ,μ]} = (K^{(1)})^{−1}|_{[λ^{1},μ^{1}]} ⊗ (K^{(1)})^{−1}|_{[λ^{2},μ^{2}]} .  (61)

By (21), we have

K^{(2)}|_{[λ,μ]} = K^{(2)}|_{[λ^{1},μ^{1}]} ⊗ K^{(2)}|_{[λ^{2},μ^{2}]} .  (62)

Once again, taking inverses we get

(K^{(2)})^{−1}|_{[λ,μ]} = (K^{(2)})^{−1}|_{[λ^{1},μ^{1}]} ⊗ (K^{(2)})^{−1}|_{[λ^{2},μ^{2}]} .  (63)

By Lemma 4.1, we conclude that, for i = 1, 2,

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{1},μ^{1}}^{±1}(K^{(i)})_{λ^{2},μ^{2}}^{±1} .  □

Let λ^{1}, λ^{2} be two partitions such that the last non-zero column length of λ^{1} is greater than or equal to the first column length of λ^{2}. Then by λ^{1} + λ^{2} we denote the partition (λ_{1}^{1} + λ_{1}^{2}, λ_{2}^{1} + λ_{2}^{2}, …). Note that, if λ = λ^{1} + λ^{2}, then λ′ = (λ^{1′}, λ^{2′}) in the notation of Lemma 4.2. By λ − λ^{1} we mean the partition λ^{2}.

HAGLUND’S POSITIVITY CONJECTURE FOR MULTIPLICITY ONE PAIRS 13

λ^{1} λ^{2} λ^{1} = (3, 3, 3, 2, 1) λ^{2} = (4, 3, 2) λ^{1} + λ^{2} = (7, 6, 5, 2, 1)

Figure 2. Example of λ^{1} + λ^{2}

Lemma 4.3. Let λ, μ ∈ P ar such that λ = λ^{1} + λ^{2} and μ = μ^{1} + μ^{2} with λ^{1} ≥ μ^{1}, λ^{2} ≥ μ^{2}, and λ_{1}^{1} = μ_{1}^{1}. Then, for i = 1, 2, we have

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{1},μ^{1}}^{±1}(K^{(i)})_{λ^{2},μ^{2}}^{±1} .  (64)

Proof. Note that λ′ = (λ^{1′}, λ^{2′}) and μ′ = (μ^{1′}, μ^{2′}), μ^{1′} ≥ λ^{1′} and ℓ(μ^{1′}) = μ_{1}^{1} = λ_{1}^{1} = ℓ(λ^{1′}). Therefore we can apply Lemma 4.2 to μ′, λ′. By Lemma 2.1,

K_{λ,μ}^{(2)}(q, t) = (K^{(2)})_{μ′,λ′}^{−1}(t, q)  (65)
= (K^{(2)})_{μ^{1′},λ^{1′}}^{−1}(t, q)(K^{(2)})_{μ^{2′},λ^{2′}}^{−1}(t, q)  (66)
= K_{λ^{1},μ^{1}}^{(2)}(q, t)K_{λ^{2},μ^{2}}^{(2)}(q, t) .  (67)

In particular,

K_{λ,μ} = K_{λ^{1},μ^{1}}K_{λ^{2},μ^{2}} .  (68)

If a partition γ satisfies λ ≥ γ ≥ μ then μ′ ≥ γ′ ≥ λ′. By our earlier argument, [μ′, λ′] = [μ^{1′}, λ^{1′}] × [μ^{2′}, λ^{2′}]. Then [λ, μ] = [λ^{1}, μ^{1}] + [λ^{2}, μ^{2}], and any γ^{1} ∈ [λ^{1}, μ^{1}] satisfies λ_{1}^{1} = γ_{1}^{1} = μ_{1}^{1}. Thus,

K^{(2)}|_{[λ,μ]} = K^{(2)}|_{[λ^{1},μ^{1}]} ⊗ K^{(2)}|_{[λ^{2},μ^{2}]} .  (69)

As before, we take inverses and use (21) to prove the statement. □

Let λ^{1}, λ^{2}, μ^{1}, μ^{2} be as in Lemma 4.2. Note that, since ℓ(λ^{1}) = ℓ(μ^{1}) and λ^{1} ≥ μ^{1}, we have μ_{ℓ(λ^{1})}^{1} ≥ λ_{ℓ(λ^{1})}^{1}. Thus both λ^{1} and μ^{1} contain the rectangle R of row length λ_{ℓ(λ^{1})}^{1} and column length ℓ(λ^{1}).

Corollary 4.1. Let λ^{1}, λ^{2}, μ^{1}, μ^{2} be as in Lemma 4.2, and let R be the rectangular partition of row length λ_{ℓ(λ^{1})}^{1} and column length ℓ(λ^{1}). If Hag′(λ, μ) is true for λ^{1} − R, μ^{1} − R and for λ^{2}, μ^{2} then it is true for λ, μ.

Proof. By applying Lemma 4.3 to λ^{1} = R +(λ^{1} −R) and μ^{1} = R +(μ^{1} −R), for i = 1, 2 we get

K_{λ^{1},μ^{1}}^{(i)} = K_{R,R}^{(i)}K_{λ^{1}−R,μ^{1}−R}^{(i)} = K_{λ^{1}−R,μ^{1}−R}^{(i)} .  (70)

So, by Lemma 4.2,

K_{λ,μ}^{(i)} = K_{λ^{1},μ^{1}}^{(i)}K_{λ^{2},μ^{2}}^{(i)} = K_{λ^{1}−R,μ^{1}−R}^{(i)}K_{λ^{2},μ^{2}}^{(i)} .  (71)

Since the arms and legs of boxes in μ^{1} − R have no intersection with those of μ^{2}, it follows that

\frac{c′_{μ}}{c′_{μ^{1}−R}c′_{μ^{2}}} = ∏_{x∈R}(1 − q^{a_{μ}(x)+1}t^{l_{μ}(x)}) .  (72)

From (30), we get

k_{λ,μ}(q, t) = k_{λ^{1}−R,μ^{1}−R}(q, t) k_{λ^{2},μ^{2}}(q, t) ∏_{x∈R}(1 − q^{a_{μ}(x)+1}t^{l_{μ}(x)}) .  (73)

So,

\frac{k_{λ,μ}(t^{k}, t)}{(1 − t)^{|λ|}} = \frac{k_{λ^{1}−R,μ^{1}−R}(t^{k}, t)}{(1 − t)^{|λ^{1}|−|R|}} \frac{k_{λ^{2},μ^{2}}(t^{k}, t)}{(1 − t)^{|λ^{2}|}} \frac{∏_{x∈R}(1 − t^{k(a_{μ}(x)+1)+l_{μ}(x)})}{(1 − t)^{|R|}} . (74)

The last term on the right-hand side is a product of t-numbers and hence in ℤ_{≥0}[t]. □

As particular cases we get the next corollary.

Corollary 4.2. If Hag′(λ, μ) is true for λ, μ then it is true for (R, λ), (R, μ), and for (S + λ), (S + μ), where R, S are rectangular partitions, with row-length of R at least as big as λ_{1} and column length of S at least as big as ℓ(μ).

Remark 4.1. We deduce [Yoo12, Theorem 3.1.1] in the dual setup by (73) and using Proposition 3.2. In particular this proves that Hag′(λ, μ) holds for λ = (a+k, b−k) and μ = (a, b). Similarly, we obtain [Yoo15, Proposition 3.4] in the dual setup by applying (73) and using Proposition 3.3.

Example 4.1. We demonstrate the two principles of Lemma 4.2 and Lemma 4.3 in the following example (see (71)):

K_{533,44111}^{(2)}(q, t) = K_{53,44}^{(2)}(q, t)K_{3,111}^{(2)}(q, t) = K_{2,11}^{(2)}(q, t)K_{3,111}^{(2)}(q, t) .

5. Complementation

In this section we switch to type GL_{n} Macdonald polynomials and then come back to symmetric functions. The reference for this section is [Mac95, Chapter VI, Section 9].

Let n ∈ ℤ_{>0} be fixed, and let q and t be two complex numbers with t = q^{k} for some fixed k ∈ ℤ_{≥0}.

Let P = ℤ^{n}, and P^{+} = {α ∈ P : α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{n}}. For α = (α, …, α_{n}) ∈ ℤ^{n} let x^{α} = (x_{1}^{α_{1}}, …, x_{n}^{α_{n}}). Let W = S_{n}. By ℂ[P] we denote the ring of Laurent polynomials ℂ[x_{1}^{±1}, x_{2}^{±1}, …, x_{n}^{±1}]. There is a ring homomorphism Λ → ℂ[P]^{W} defined by sending x_{n+i} ↦ 0 for i ≥ 1. We denote the image of a symmetric function f by f(X_{n}). For any λ ∈ P^{+} (not necessarily with non-negative coordinates) we can define P_{λ}(x; q, t) ∈ ℂ[P]^{W} by the relations

P_{λ}(x; q, t) = P_{λ}(X_{n}; q, t), if λ_{n} ≥ 0,  (75)
P_{λ+(1^{n})}(x; q, t) = (x_{1} ⋯ x_{n})P_{λ}(x; q, t), for all λ ∈ P^{+}.  (76)

We also define, for all λ ∈ P^{+}, m_{λ}(x) = P_{λ}(x; q, 1) and s_{λ}(x) = P_{λ}(x; q, q). Furthermore, P^{+} carries the dominance partial order ≥ defined by

λ ≥ μ if and only if λ − μ ∈ ℤ_{≥0}{e_{i} − e_{j} : 1 ≤ i < j ≤ n},  (77)

where e_{i} denotes the standard basis vector (0, …, 1, …, 0) with 1 in the i-th position and 0 everywhere else.

Let ¯ : ℂ[P] → ℂ[P] be the ℂ-algebra involution defined by x_{i} ↦ x_{i}^{−1} for i ∈ {1, …, n}. Then there is a scalar product on ℂ[P] defined by

⟨f, g⟩′ = \frac{1}{n!}ct(f\overline{g}Δ) ,  (78)

where ct denotes the constant term map on the Laurent polynomial ring, i.e., ct(f) = coefficient of x^{0} in f, and

Δ = ∏_{i≠j} ∏_{r=0}^{k−1}(1 − q^{r} \frac{x_{i}}{x_{j}}) .  (79)

The next theorem characterizes the P_{λ}(x; q, t) : λ ∈ P^{+}.

Theorem 5.1. The set {P_{λ}(x; q, t) : λ ∈ P^{+}} is the unique family of elements in ℂ[P]^{W} satisfying

P_{λ}(x; q, t) = m_{λ}(x; q, t) + ∑_{μ<λ} a_{λ,μ}(q, t)m_{μ}(x) ,  (80)
⟨P_{λ}(x; q, t), P_{μ}(x; q, t)⟩′ = 0 for λ ≠ μ ,  (81)

for some a_{λ,μ}(q, t) ∈ ℂ(q, t) ⊂ ℂ .

Proof. If λ ∈ P ar with ℓ(λ) ≤ n, then P_{λ}(x; q, t) is a homogeneous polynomial of degree |λ|. By (75), P_{λ}(X_{n}; q, t) is a homogeneous Laurent polynomial with degree λ_{1}+⋯+λ_{n} for all λ ∈ P^{+} . By comparing degrees, it is clear that, if λ_{1} + ⋯ + λ_{n} ≠ μ_{1} + ⋯ + μ_{n}, then ⟨P_{λ}(x; q, t), P_{μ}(x; q, t)⟩′ = 0. By (75) and the definition of ⟨ , ⟩′ it is enough to check orthogonality of the set {P_{λ}(X_{n}; q, t) : λ ∈ P^{+} with λ_{n} ≥ 0}. This is shown in [Mac95, Chapter VI, (9.5)]. Since {m_{λ}(x) : λ ∈ P^{+}} is a basis of ℂ[P]^{W}, so is {P_{λ}(x; q, t) : λ ∈ P^{+}}.

Any two bases of ℂ[P]^{W} satisfying the above conditions will be related by a triangular orthogonal matrix and any such matrix is necessarily diagonal, so the uniqueness follows.

□

Let w_{0} ∈ S_{n} be the permutation that sends i to n + 1 − i for i ∈ {1, …, n}. Let ϕ : ℂ[P] → ℂ[P] be the ℂ-algebra homomorphism defined by ϕ(x^{α}) = x^{−w_{0}α}. So ϕ(f)(x) = w_{0}\overline{f}(x).

Proposition 5.1. For λ ∈ P^{+}, we have ϕ(P_{λ}(x; q, t)) = P_{−w_{0}λ}(x; q, t). In particular, ϕ(s_{λ}(x)) = s_{−w_{0}λ}(x).

Proof. Since ϕ(m_{μ}(x)) = m_{−w_{0}μ}(x) for all μ ∈ P^{+}, we have

ϕ(P_{λ}(x; q, t)) = m_{−w_{0}λ}(x) + ∑_{μ<λ} a_{λ,μ}(q, t)m_{−w_{0}μ}(x) .  (82)

Note that μ ≤ λ if and only if −w_{0}μ ≤ −w_{0}λ, so ϕ(P_{λ}) satisfies the unitriangularity property. To prove ϕ(P_{λ}(x; q, t)) = P_{−w_{0}λ}(x; q, t), we only need to prove that, for λ ≠ μ, we have ⟨ϕ(P_{λ}(x; q, t)), ϕ(P_{μ}(x; q, t))⟩′ = 0. This follows from the fact that the constant term of a Laurent polynomial is S_{n}-invariant, w_{0} is a homomorphism, and that Δ is S_{n}-invariant. □

Corollary 5.1. Let λ, μ ⊂ (m^{n}) be partitions. Let λ^{c} = (m − λ_{n}, m − λ_{n−1}, …, m − λ_{1}) and μ^{c} = (m − μ_{n}, m − μ_{n−1}, …, m − μ_{1}). Then, for i = 1, 2, we have

(K^{(i)})_{λ,μ}^{±1} = (K^{(i)})_{λ^{c},μ^{c}}^{±1} .  (83)

λ λ^{c} m = 9 n = 5 λ = (8, 7, 4, 4, 1) λ^{c} = (8, 5, 5, 2, 1)

Figure 3. Example of λ, λ^{c} when m = 9, n = 5

Proof. Let w_{0} ∈ S_{n} be as before. First note that, if λ ⊂ (m^{n}) is a partition, then λ^{c} = (m−λ_{n}, …, m−λ_{1}) is also a partition. So it makes sense to talk about K_{λ^{c},μ^{c}}^{(i)}(q, t) in this situation.

In the ring Λ, we have

P_{λ}(q, t) = ∑_{ν≤λ} K_{λ,ν}^{(1)}(q, t) m_{ν} .  (84)

Specializing to n variables, we have

P_{λ}(X_{n}; q, t) = ∑_{ν≤λ} K_{λ,ν}^{(1)}(q, t) m_{ν}(X_{n}) .  (85)

On the right-hand side, if ℓ(ν) > n, then the corresponding term vanishes. Now applying the map ϕ to this equation, we get

P_{−w_{0}λ}(X_{n}; q, t) = ∑_{ν≤λ} K_{λ,ν}^{(1)}(q, t)m_{−w_{0}ν}(X_{n}) .  (86)

Multiplying by (x_{1} ⋯ x_{n})^{m} and using (75), we get

P_{(m^{n})−w_{0}λ}(X_{n}; q, t) = ∑_{ν≤λ} K_{λ,ν}^{(1)}(q, t)m_{(m^{n})−w_{0}ν}(X_{n}) .  (87)

Again, consider the equation in Λ

P_{λ^{c}}(q, t) = ∑_{γ≤λ^{c}} K_{λ^{c},γ}^{(1)}(q, t)m_{γ} ,  (88)

and specialize to n variables. We see that the coefficient of m_{μ^{c}}(X_{n}) is K_{λ^{c},μ^{c}}^{(1)}(q, t). Here we have used the fact that m_{μ^{c}}(X_{n}) ≠ 0 since μ ⊂ (m^{n}). Comparing with (87), we get the desired statement for K^{(1)}. By considering

m_{λ} = ∑_{ν≤λ}(K^{(1)})_{λ,ν}^{−1}(q, t)P_{ν}(q, t) ,  (89)

we get the inverse statement for K^{(1)}. The proof for (K^{(2)})^{±1} is obtained by replacing the m_{μ}’s in the above argument by s_{μ}’s everywhere. □

6. Multiplicity-one pairs

In this section we finish the proof that Hag′(λ, μ) holds for all pairs (λ, μ) such that either K_{λ,μ} = 1 or K_{μ′,λ′} = 1. We recall the reduction principle given in the introduction.

Let λ, μ ∈ P ar such that λ ≥ μ. We can always decompose λ = (λ^{1}, …, λ^{r}) and μ = (μ^{1}, …, μ^{r}) where λ^{1}, … λ^{r}, μ^{1}, …, μ^{r} are partitions such that, for i ∈ {1, … r}, λ^{i} ≥ μ^{i} and, for i ∈ {1, … r − 1}, ℓ(λ^{i}) = ℓ(μ^{i}). Then, for i = 1, 2, by Lemma 4.2 we have

(K^{(i)})_{λ,μ}^{±1}(q, t) = ∏_{j=1}^{r}(K^{(i)})_{λ^{j},μ^{j}}^{±1}(q, t) .  (90)

For j ∈ {1, …, r − 1}, each pair (λ^{j}, μ^{j}) contains a common rectangle of row length at least λ_{1}^{j+1}. By Corollary 4.1, proving the dual Haglund conjecture is equivalent to proving it for each pair (λ^{j}, μ^{j}) with these rectangles removed. We can repeat this process for each pair until no further decomposition is possible.

Therefore, to prove the dual Haglund’s conjecture we only need to prove Hag′(λ, μ) for pairs of partitions (λ, μ) which are irreducible, i.e., which satisfy λ ≥ μ and λ_{1}+⋯+λ_{i} > μ_{1} + ⋯ + μ_{i} for 1 ≤ i ≤ ℓ(λ).

6.1. The case where K_{λ,μ} = 1. Recall the characterization of irreducible pairs (λ, μ) with K_{λ,μ} = 1 given in Theorem 1.2. Case 2 of Theorem 1.2 is already done by Corollary 3.1. Now consider Case 1 of Theorem 1.2. Let λ = (m^{n}) ≥ μ and ℓ(μ) = n+1. Since λ and μ are both contained in (m^{n+1}), by Corollary 5.1 we have

(K^{(i)})_{λ,μ}^{±1}(q, t) = (K^{(i)})_{(m),μ^{c}}^{±1}(q, t), for i = 1, 2.  (91)

Here μ^{c} = (m − μ_{n+1}, ⋯ − μ_{1}) is the complement of μ in the rectangle (m^{n+1}), and (m) is the complement of λ.

In particular, K_{λ,μ}^{(2)} = K_{(m),μ^{c}}^{(2)} implies

k_{λ,μ}(q, t) = k_{(m),μ^{c}}(q, t)\frac{c′_{μ}(q, t)}{c′_{μ^{c}}(q, t)} .  (92)

Note that c′_{μ} is a product of |μ| = mn many terms, c′_{μ^{c}} is a product of m(n + 1) − |μ| = m many terms. If all terms in the denominator cancel off with terms in c′_{μ} then

\frac{c′_{μ}(q, t)}{c′_{μ^{c}}(q, t)} will be a product of mn − m terms each of the form (1 − q^{α}t^{β}) for α, β ∈ ℤ_{≥0}.

Substituting q = t^{k} and dividing by (1 − t), we get \frac{1 − t^{kα+β}}{1 − t} ∈ ℤ_{≥0}[t]. Since we know

\frac{k_{(m),μ^{c}}(t^{k}, t)}{(1 − t)^{m}} ∈ ℤ_{≥0} by Corollary 3.1, this would imply that \frac{k_{λ,μ}(t^{k}, t)}{(1 − t)^{mn}} ∈ ℤ_{≥0}[t]. To prove

that \frac{c′_{μ}(q, t)}{c′_{μ^{c}}(q, t)} is a product of mn − m terms, each of the form (1 − q^{α}t^{β}) for α, β ∈ ℤ_{≥0},

we consider f_{μ}(q, t) = ∑_{x∈μ} q^{a(x)}t^{l(x)} and show that f_{μ}(q, t) − f_{μ^{c}}(q, t) ∈ ℤ_{≥0}[q, t]. Since each monomial q^{α}t^{β} (counted with multiplicity) that occurs in f_{μ}(q, t) − f_{μ^{c}}(q, t)

corresponds to a term (1 − q^{α+1}t^{β}) in \frac{c′_{μ}}{c′_{μ^{c}}}, this is clearly sufficient.

Lemma 6.1. Let μ ∈ P ar. Then

f_{μ}(q, t) = ∑_{j=0}^{ℓ(μ)−1} t^{j}( ∑_{i=1}^{ℓ(μ)−j} q^{μ_{i}−μ_{i+j}}[μ_{i+j} − μ_{i+j+1}]_{q}) .  (93)

Proof. For i ∈ {1, … ℓ(μ)}, in the i-th row of μ the rightmost μ_{i} − μ_{i+1} many boxes have leg length 0 and arm lengths 0, …, μ_{i} −μ_{i+1} −1. The next μ_{i+1} −μ_{i+2} many boxes from the right have leg length 1 and arm lengths μ_{i} − μ_{i+1}, …, μ_{i} − μ_{i+2} − 1 and so on. Hence the contribution of the i-th row to f_{μ}(q, t) is

∑_{j=0}^{ℓ(μ)−i} t^{j}q^{μ_{i}−μ_{i+j}}[μ_{i+j} − μ_{i+j+1}]_{q} .  (94)

So,

f_{μ}(q, t) = ∑_{i=1}^{ℓ(μ)} ∑_{j=0}^{ℓ(μ)−i} t^{j}q^{μ_{i}−μ_{i+j}}[μ_{i+j} − μ_{i+j+1}]_{q}  (95)

= ∑_{j=0}^{ℓ(μ)−1} t^{j}( ∑_{i=1}^{ℓ(μ)−j} q^{μ_{i}−μ_{i+j}}[μ_{i+j} − μ_{i+j+1}]_{q}) .  □

Corollary 6.1. Let μ ⊂ (m^{n+1}), with μ_{1} < m and ℓ(μ) = n + 1. Let μ^{c} denote the complement of μ in (m^{n+1}). Then

f_{μ}(q, t) − f_{μ^{c}}(q, t) = \frac{1}{1 − q} ∑_{j=0}^{n} t^{j}(q^{μ_{n+1−j}^{c}} − q^{μ_{n+1−j}}) .  (96)

Proof. We have

f_{μ}(q, t) = ∑_{j=0}^{n} t^{j}( ∑_{i=1}^{n−j} q^{μ_{i}−μ_{i+j}}[μ_{i+j} − μ_{i+j+1}]_{q} + q^{μ_{n+1−j}−μ_{n+1}}[μ_{n+1}]_{q}) .  (97)

Since (μ^{c})_{i} = m − μ_{n+2−i}, we have ℓ(μ^{c}) = n + 1 since μ_{1} < m. Thus

f_{μ^{c}}(q, t) = ∑_{j=0}^{n+1} t^{j}( ∑_{i=1}^{n−j} q^{μ_{n+2−i−j}−μ_{n+2−i}}[μ_{n+1−i−j} − μ_{n+2−i−j}]_{q}

+ q^{μ_{1}−μ_{n+2−n−1+j}}[m − μ_{1}]_{q})

= ∑_{j=0}^{n+1} t^{j}( ∑_{i′=1}^{n−j} q^{μ_{i′+1}−μ_{i′+j+1}}[μ_{i′} − μ_{i′+1}]_{q} + q^{μ_{1}−μ_{j+1}}[m − μ_{1}]_{q}) .

Therefore the coefficient of t^{j} in f_{μ}(q, t) − f_{μ^{c}}(q, t) is

\frac{1}{1 − q} ( ∑_{i=1}^{n−j}(q^{μ_{i}−μ_{i+j}} − q^{μ_{i}−μ_{i+j+1}}) − ∑_{i′=1}^{n−j}(q^{μ_{i′+1}−μ_{i′+j+1}} − q^{μ_{i′}−μ_{i′+j+1}})

ker n3cm + q^{μ_{n+1−j}−μ_{n+1}} − q^{μ_{n+1−j}} − q^{μ_{1}−μ_{j+1}} + q^{m−μ_{j+1}})

= \frac{1}{1 − q} (q^{μ_{1}−μ_{1+j}} − q^{μ_{n+1−j}−μ_{n+1}} + q^{μ_{n+1−j}−μ_{n+1}} − q^{μ_{n+1−j}} − q^{μ_{1}−μ_{j+1}} + q^{m−μ_{j+1}})

= \frac{1}{1 − q} (q^{m−μ_{j+1}} − q^{μ_{n+1−j}}) = \frac{1}{1 − q} (q^{μ_{n+1−j}^{c}} − q^{μ_{n+1−j}}) .  □

Lemma 6.2. Suppose μ ⊢ mn, ℓ(μ) = n + 1, and μ ⊂ (m^{n+1}). Then, for all i ∈ {1, …, n + 1}, we have μ_{i}^{c} ≤ μ_{i}.

Proof. Suppose μ_{i}^{c} = m − μ_{n+2−i} > μ_{i} for some i ∈ {1, …, n + 1}.

If i ≠ n + 2 − i, then

(n − 1)μ_{1} ≥ ∑_{j∉{i,n+2−i}} μ_{j} = |μ| − (μ_{i} + μ_{n+2−i}) > (n − 1)m,  (98)

which implies μ_{1} > m, a contradiction.

If i = n + 2 − i, then m − μ_{n+2−i} > μ_{i} implies 2μ_{i} < m. Consequently,

2(n − 1)μ_{1} ≥ 2 ∑_{j≠i} μ_{j} = 2|μ| − 2μ_{i} > m(2n − 1).  (99)

Since m ≥ μ_{1}, we get 2(n − 1) > 2n − 1, a contradiction. □

With the assumptions of Corollary 6.1, by Lemma 6.2 we have

f_{μ}(q, t) − f_{μ^{c}}(q, t) ∈ ℤ_{≥0}[q, t].  (100)

Corollary 6.2. Hag′(λ, μ) is true whenever K_{λ,μ} = 1.

Proof. We showed that, if λ, μ is an irreducible pair with K_{λ,μ} = 1, then Hag′(λ, μ) is true. Corollary 4.1 shows that Hag′(λ, μ) remains true when patching together irreducible pairs. □

6.2. The case where K_{μ′,λ′} = 1.

Proposition 6.1. Hag′(λ, μ) is true whenever K_{μ′,λ′} = 1.

Proof. Suppose λ, μ ∈ P ar such that K_{μ′,λ′} = 1. To compute k_{λ,μ}(q, t), in the earlier case we used K_{λ,μ}^{(2)}(q, t). Here we use (K^{(2)})_{μ′,λ′}^{−1}(t, q). This amounts to applying the equality between the far left-hand side and the far right-hand side in (30). As before, by use of Lemma 4.2 and Lemma 4.3, we reduce the calculation of (K^{(2)})_{μ′,λ′}^{−1}(t, q) to the case where μ′, λ′ is an irreducible pair. The case where μ′ = (m) for some m is done by Corollary 3.1. Let μ′ = (m^{n}) and ℓ(λ′) = n + 1. By Corollary 5.1, we have

(K^{(2)})_{μ′,λ′}^{−1}(t, q) = (K^{(2)})_{(m),λ^{′c}}^{−1}(t, q).  (101)

Hence, by (30),

k_{λ,μ}(q, t) = k_{λ^{c},(1^{m})}(q, t)\frac{c_{(m^{n})}(q, t)}{c_{(m)}(q, t)} .  (102)

In \frac{c_{(m^{n})}(q, t)}{c_{(m)}(q, t)}, the contribution from the last row of (m^{n}) cancels with the contribution

from (m), so it is a product of mn − 1 terms each of the form 1 − q^{α}t^{β} for some

α, β ∈ ℤ_{≥0}. Since, by Corollary 3.1, \frac{k_{λ^{c},1^{m}}(t^{k}, t)}{(1 − t)^{m}} ∈ ℤ_{≥0}[t], this implies Hag′(λ, μ) for

λ, μ, and hence for all pairs λ, μ with K_{μ′,λ′} = 1 when μ′, λ′ is irreducible. Applying Corollary 4.1, we conclude the statement for general λ, μ with K_{μ′,λ′} = 1. □

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The Institute of Mathematical Sciences, A CI of Homi Bhabha National Institute, Chennai 600113, India

Email address: baritra@imsc.res.in