Séminaire Lotharingien de Combinatoire 54 (2006), Article B54Ar

NON-SYMMETRIC HALL–LITTLEWOOD POLYNOMIALS

FRANÇOIS DESCOUENS AND ALAIN LASCOUX

À Adriano Garsia, en toute amitié

Abstract. Using the action of the Yang–Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall–Littlewood polynomials are a subfamily of one of them. For q = 0, these bases specialize to the two families of classical Key polynomials (i.e., Demazure characters for type A). We give a scalar product for which the two bases are adjoint to each other.

1. Introduction

We define two linear bases of the ring of polynomials in x_{1}, …, x_{n}, with coefficients in q. These polynomials, which we call q-Key polynomials, and denote by U_{v}, Û_{v}, v ∈ ℕ^{n}, spe-

cialize at q = 0 into key polynomials K_{v}, K̂_{v}. The polynomials U_{v} are symmetric polynomials for v such that v_{1} ≤ ⋯ ≤ v_{n}. In that case, U_{v} is equal to the Hall–Littlewood polynomial P_{λ}, λ being the partition [v_{n}, …, v_{1}].

Our main tool is the Hecke algebra H_{n}(q) of the symmetric group, acting on polynomials by deformation of divided differences. This algebra contains two adjoint bases of Yang–Baxter elements (Theorem 2.1). The q-Key polynomials are the images of dominant monomials under these Yang–Baxter elements (Def. 3.1). These polynomials are clearly two linear bases of polynomials, since the transition matrix to monomials is uni-triangular. We show in the last section that {U_{v}} and {Û_{v}} are two adjoint bases with respect to a certain scalar product reminiscent of Weyl’s scalar product on symmetric functions. We have intensively used MuPAD (package MuPAD-Combinat [13]) and Maple (package ACE [12]).

When this article was written, the authors were not aware of the work of Bogdan Ion [3, 4], who shows how to obtain, from nonsymmetric Macdonald polynomials, Demazure characters and their adjoint basis for an affine Kac–Moody algebra. Hence our own work should be considered as the part of the theory of nonsymmetric Macdonald polynomials for type A which is accessible using only divided differences, and not requiring double affine Hecke algebras.

2. The Hecke algebra H_{n}(q)

Let H_{n}(q) be the Hecke algebra of the symmetric group 𝔖_{n}, with coefficients the rational functions in a parameter q. It has generators T_{1}, …, T_{n−1} satisfying the braid relations

\begin{cases}T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}, \\ T_{i}T_{j} = T_{j}T_{i} (|j − i| > 1) , \end{cases}  (1)

and the Hecke relations

(T_{i} + 1)(T_{i} − q) = 0 , 1 ≤ i ≤ n − 1  (2)

For a permutation σ in 𝔖_{n}, we denote by T_{σ} the element T_{σ} = T_{i_{1}} … T_{i_{p}} where (i_{1}, …, i_{p}) is any reduced decomposition of σ. The set {T_{σ} : σ ∈ 𝔖_{n}} is a linear basis of H_{n}(q).

2.1. Yang–Baxter bases. Let s_{1}, …, s_{n−1} denote the simple transpositions, ℓ(σ) denote the length of σ ∈ 𝔖_{n}, and let ω be the permutation of maximal length.

Given any set of indeterminates u = (u_{1}, …, u_{n}), let

H_{n}(q)[u_{1}, …, u_{n}] = H_{n}(q) ⊗ ℂ[u_{1}, …, u_{n}].

One defines recursively a Yang–Baxter basis (Y_{σ}^{u})_{σ∈𝔖_{n}}, depending on u, by

Y_{σs_{i}}^{u} = Y_{σ}^{u} (T_{i} + \frac{1 − q}{1 − u_{σ_{i+1}}/u_{σ_{i}}} ) , when ℓ(σs_{i}) > l(σ) ,  (3)

starting with Y_{id}^{u} = 1.

Let φ be the anti-automorphism of H_{n}(q)[u_{1}, …, u_{n}] such that

\{ \begin{matrix}φ(T_{σ}) = T_{σ^{−1}}, \\ φ(u_{i}) = u_{n−i+1}. \end{matrix}

We define a bilinear form < , > on H_{n}(q)[u_{1}, …, u_{n}] by

< h_{1} , h_{2} > := coefficient of T_{ω} in h_{1} · φ(h_{2}) .  (4)

The main result of [8, Th. 5.1] is the following duality property of Yang–Baxter bases.

Theorem 2.1. For any set of parameters u = (u_{1}, …, u_{n}), the basis adjoint to (Y_{σ}^{u})_{σ∈𝔖_{n}} with respect to < , > is the basis (Ŷ_{σ}^{u})_{σ∈𝔖_{n}} = (Y_{σ}^{φ(u)})_{σ∈𝔖_{n}}. More precisely, one has

< Y_{σ}^{u} , Ŷ_{ν}^{u} >= δ_{λ, νω} for all σ, ν ∈ 𝔖_{n}.

Let us fix from now on the parameters u to be u = (1, q, q^{2}, …, q^{n−1}). Write H_{n} for H_{n}(q)[1, q, …, q^{n−1}].

In that case, the Yang–Baxter basis (Y_{σ})_{σ∈𝔖_{n}} and its adjoint basis (Ŷ_{σ})_{σ∈𝔖_{n}} are defined recursively, starting with Y_{id} = 1 = Ŷ_{id}, by

Y_{σs_{i}} = Y_{σ} (T_{i} + 1/[k]_{q}) and Ŷ_{σs_{i}} = Ŷ_{σ} (T_{i} + q^{k−1}/[k]_{q})) , ℓ(σs_{i}) > ℓ(σ) ,  (5)

with k = σ_{i+1} − σ_{i} and [k]_{q} = (1 − q^{k})/(1 − q).

Notice that the maximal Yang–Baxter elements have another expression [2]:

Y_{ω} = ∑_{σ∈𝔖_{n}} T_{σ} and Ŷ_{ω} = ∑_{σ∈𝔖_{n}} (−q)^{ℓ(σω)} T_{σ} .

Example 2.2. For H_{3}, the transition matrix between {Y_{σ}}_{σ∈𝔖_{3}} and {T_{σ}}_{σ∈𝔖_{3}} is

\begin{matrix}123 & 1 & 1 & 1 & \frac{1}{q+1} & \frac{1}{q+1} & 1 \\ 132 & · & 1 & · & 1 & \frac{1}{q+1} & 1 \\ 213 & · & · & 1 & \frac{1}{q+1} & 1 & 1 \\ 231 & · & · & · & 1 & · & 1 \\ 312 & · & · & · & · & 1 & 1 \\ 321 & · & · & · & · & · & 1 \end{matrix} ,

writing ‘ · ‘ for 0. Each column represents the expansion of some element Y_{σ}.

2.2. Action of H_{n} on polynomials. Let 𝔓𝔬𝔩 be the ring of polynomials in the variables x_{1}, …, x_{n} with coefficients the rational functions in q. We write monomials exponentially: x^{v} = x_{1}^{v_{1}} … x_{n}^{v_{n}}, v = (v_{1}, …, v_{n}) ∈ ℤ^{n}. A monomial x^{v} is dominant if v_{1} ≥ … ≥ v_{n}.

We extend the natural order on partitions to elements of ℤ^{n} by

u ≤ v if and only if ∑_{i=k}^{n}(v_{i} − u_{i}) ≥ 0 for all k > 0.

For any polynomial P in 𝔓𝔬𝔩, we call leading term of P all the monomials (multiplied by their coefficients) which are maximal with respect to this partial order. This order is compatible with the right-to-left lexicographic order, that we shall also use. We also use the classiccal notation 𝔫(v) = 0v_{1} + 1v_{2} + 2v_{3} + ⋯ + (n − 1)v_{n}.

Let i be an integer such that 1 ≤ i ≤ n−1. As an operator on 𝔓𝔬𝔩, the simple transposition s_{i} acts by switching x_{i} and x_{i+1}, and we denote this action by f → f^{s_{i}}. The i-th divided difference ∂_{i} and the i-th isobaric divided difference π_{i}, written on the right of the operand, are the following operators:

∂_{i} : f ⟼ f ∂_{i} := \frac{f − f^{s_{i}}}{x_{i} − x_{i+1}}  ,  π_{i} : f ⟼ f π_{i} := \frac{x_{i}f − x_{i+1}f^{s_{i}}}{x_{i} − x_{i+1}} .

The Hecke algebra H_{n} has a faithful representation as an algebra of operators on 𝔓𝔬𝔩 given by the following equivalent formulas [2, 10]:

\{ \begin{matrix}T_{i} & = & □_{i} − 1 & = & (x_{i} − qx_{i+1}) ∂_{i} − 1 & = & (1 − qx_{i+1}/x_{i})π_{i} − 1 , \\ Y_{s_{i}} & = & □_{i} & = & (x_{i} − qx_{i+1}) ∂_{i} & = & (1 − qx_{i+1}/x_{i})π_{i} , \\ Ŷ_{s_{i}} & = & ∇_{i} & = & □_{i} − (1 + q) & = & ∂_{i} (x_{i+1} − qx_{i}) . \end{matrix}

The Hecke relations imply that

□_{i}^{2} = (1 + q)□_{i} , ∇_{i}^{2} = −(1 + q)∇_{i} and □_{i}∇_{i} = ∇_{i}□_{i} = 0 .

One easily checks that the operators R_{i}(a, b) and S_{i}(a, b) defined by

R_{i}(a, b) = □_{i} − q \frac{[b − a − 1]_{q}}{[b − a]_{q}} and S_{i}(a, b) = ∇_{i} + q \frac{[b − a − 1]_{q}}{[b − a]_{q}}

satisfy the Yang–Baxter equation

R_{i}(a, b) R_{i+1}(a, c) R_{i}(b, c) = R_{i+1}(c, b) R_{i}(a, c) R_{i+1}(a, b) .  (6)

We have implicitly used these equations in the recursive definition of Yang–Baxter elements

This realization comes from geometry [5], where the maximal Yang–Baxter elements are interpreted as Euler–Poincaré characteristic for the flag variety of GL_{n}(ℂ). This gives in particular another expression for the maximal Yang–Baxter elements:

Y_{ω} = ∏_{1≤i<j≤n}(x_{i} − qx_{j}) ∂_{ω}  ,  Ŷ_{ω} = ∂_{ω} ∏_{1≤i<j≤n}(x_{j} − qx_{i}) .  (7)

Example 2.3. Let σ = (3412) = s_{2}s_{3}s_{1}s_{2}. The elements Y_{3412} and Ŷ_{3412} can be written as

Y_{3412} = □_{2} (□_{3} − \frac{q}{1 + q} ) (□_{1} − \frac{q}{1 + q} ) (□_{2} − \frac{q + q^{2}}{1 + q + q^{2}} ) ,
Ŷ_{3412} = ∇_{2} (∇_{3} + \frac{q}{1 + q} ) (∇_{1} + \frac{q}{1 + q} ) (∇_{2} + \frac{q + q^{2}}{1 + q + q^{2}} ) .

We shall now identify the images of dominant monomials under the maximal Yang–Baxter operators with Hall–Littlewood polynomials. Recall that there are two proportional families {P_{λ}} and {Q_{λ}} of Hall–Littlewood polynomials. Given a partition λ = [λ_{1}, λ_{2}, …, λ_{r}] = (0^{m_{0}}, 1^{m_{1}}, …, n^{m_{n}}), with m_{0} = n − r = n − m_{1} − ⋯ − m_{n}, then

Q_{λ} = ∏_{1≤i≤n} ∏_{j=1}^{m_{i}}(1 − q^{j}) P_{λ} .

Let moreover d_{λ}(q) = ∏_{0≤i≤n} ∏_{j=1}^{m_{i}}[j]_{q}. The definition of Hall–Littlewood polynomials with raising operators [9], [11, III.2] can be rewritten, thanks to (7), as follows.

Proposition 2.4. Let λ be a partition of n. Then one has

x^{λ} Y_{ω} d_{λ}(q)^{−1} = P_{λ}(x_{1}, …, x_{n}; q).  (8)

The family of the Hall–Littlewood functions {Q_{λ}} indexed by partitions can be extended to a family {Q_{v} : v ∈ ℤ^{n}}, using the following relations due to Littlewood ([9], [11, III.2.Ex. 2]):

Q_{(...,u_{i},u_{i+1},...)} = −Q_{(...,u_{i+1}−1,u_{i}+1,...)}+q Q_{(...,u_{i+1},u_{i},...)}+q Q_{(...,u_{i}+1,u_{i+1}−1,...)} if u_{i} < u_{i+1}, (9)
Q_{(u_{1},...,u_{n})} = 0 if u_{n} < 0 .  (10)

By iteration of the first relation, one can write any Q_{u} in terms of Hall–Littlewood functions indexed by decreasing vectors v such that |v| = |u|. Consequently, for any u with |u| = 0, Q_{u} must be proportional to Q_{0...0} = 1, i.e., Q_{u} is a constant that one can obtain as the specialization Q_{u}(0) (i.e., the specialization of Q_{u} at x_{1} = ⋯ = x_{n} = 0).

The final expansion of Q_{u}, after iterating (9) many times, is not easy to predict. In particular, one needs to know whether Q_{u} ≠ 0. For that purpose, we shall isolate a distinguished term in the expansion of Q_{u}. Given a sum ∑_{λ∈𝔓𝔞𝔯𝔱} c_{λ}(t)Q_{λ}, call top term the image of the leading term ∑ c_{μ}(t)Q_{μ} after restricting each coefficient c_{μ}(t) to its term in highest degree in t.

Given u ∈ ℤ^{n}, define recursively 𝔭(u) ∈ 𝔓𝔞𝔯𝔱 ∪ {−∞} by • if u ≱ [0, …, 0] then 𝔭(u) = −∞;

• if u_{2} ≥ u_{3} ≥ ⋯ ≥ u_{n} > 0 then 𝔭(u) is the maximal partition of length ≤ n, of weight |u| (eventual zero terminal parts are suppressed);

• 𝔭(u) = 𝔭(u 𝔭([u_{2}, …, u_{n}])).

Lemma 2.5. Let u ∈ ℤ^{n}. Then • if u ≱ [0, …, 0] then Q_{u} = 0 ,

• if u ≥ [0, …, 0], let v = 𝔭(u). Then Q_{u} ≠ 0 and its leading term is q^{𝔫(u)−𝔫(v)}Q_{v} .

Proof. Given any decomposition u = u′.u′′, one can apply (9) to u′′ and write Q_{u} as a linear combination of terms Q_{u′v} with v decreasing, with |v| = |u′′|. Therefore, if |u′′| = 0, the last components of such v are negative, all Q_{u′v} are 0, and Q_{u} = 0.

If u ≥ [0, …, 0] and u is not a partition, write u = […, a, b, …], with a, b the rightmost increase in u. We apply relation (9), assuming the validity of the lemma for the three terms on the right-hand side:

Q_{...,a,b,...} = −Q_{...,b−1,a+1,...} + qQ_{...,b,a,...} + qQ_{...,a+1,b−1,...}

Notice that the first two terms have not necessarily an index ≥ [0, …, 0], but that […, a + 1, b − 1, …] ≥ [0, …, 0].

In any case, it is clear that 𝔭([…, b − 1, a + 1, …]) = p_{1} ≤ v, 𝔭([…, b, a, …]) = p_{2} ≤ v, and 𝔭([…, a + 1, b − 1, …]) = v.

Restricted to top terms, the expansion of the right-hand side in the basis Q_{λ} becomes

−((q^{𝔫(u)+a+1−b−𝔫(v)} + ⋯) Q_{v}) + q((q^{𝔫(u)+a−b−𝔫(v)} + ⋯) Q_{v}) + q((q^{𝔫(u)−1−𝔫(v)} + ⋯) Q_{v}) ,

where one or two of the first two terms may be replaced by 0, depending on the value of p_{1}, or p_{2}. Finally, the top term of the right-hand side is q^{𝔫(u)−𝔫(v)}Q_{v}, as desired. □

Example 2.6. For v = [−2, 3, 2], we have

Q_{−2,3,2} = (q^{3} − q^{2})Q_{3} + (q^{5} + q^{4} − q^{3} − 2q^{2} + 1)Q_{21} + (q^{4} − q^{3} − q^{2} + q)Q_{111} ,

and the top term is q^{4}Q_{111}, since 4 = (0(−2) + 1(3) + 2(2)) − (0(1) + 1(1) + 2(1)) and [1, 1, 1] > [2, 1], [1, 1, 1] > [3]. Notice that the coefficient of Q_{21} is of higher degree.

3. q-Key Polynomials

In this section, we show that the images of dominant monomials under the Yang–Baxter elements Y_{σ} (respectively Ŷ_{σ}), σ ∈ 𝔖_{n} constitute two bases of 𝔓𝔬𝔩, which specialize to the two families of Demazure characters.

We have already identified in the preceding section the images of dominant monomials under Y_{ω} as Hall–Littlewood polynomial, using the relation between Y_{ω} and ∂_{ω}. The other polynomials are new.

3.1. Two bases. The dimension of the linear span of the image of a monomial x^{v} under all permutations depends upon the stabilizer of v. We meet the same phenomenon when taking the images of a monomial under Yang–Baxter elements.

Let λ = [λ_{1}, …, λ_{n}], λ_{1} ≥ … ≥ λ_{n} ≥ 0, be a partition (adding eventual parts equal to 0). Denote its orbit under permutations of components by O(λ). Given any v in O(λ), let ζ(v) be the permutation of maximal length such that λ ζ(v) = v, and let η(v) be the permutation of minimal length such that λ η(v) = v. These two permutations are representatives of the same coset of 𝔖_{n} modulo the stabilizer of λ.

[figure: U_{210} = x^{210}; □_{1}; □_{2}; U_{120} = x^{120} + x^{210}; □_{2}−q/(1+q); U_{201} = x^{201} + x^{210}; □_{1}−q/(1+q); U_{102} = x^{102} + (1 − q)x^{111} + \frac{1}{1+q} x^{120} + x^{201} + \frac{1}{1+q} x^{210}; □_{1}; U_{021} = x^{021} + (1 − q)x^{111} + \frac{1}{1+q} x^{201} + x^{120} + \frac{1}{1+q} x^{210}; □_{2}; U_{012} = +x^{102} + (−q − q + 2)x^{111} + x^{120} + x^{201} + x^{210}]

Figure 1. q-Key polynomials generated from x^{210}.

Definition 3.1. For all v in ℕ^{n}, the q-Key polynomials U_{v} and Û_{v} are the following polynomials:

U_{v}(x; q) = (\frac{1}{d_{λ}(q)}x^{λ}) Y_{ζ(v)}  ,  Û_{v}(x; q) = x^{λ}Ŷ_{η(v)} ,

where λ is the dominant reordering of v.

In particular, if v is (weakly) increasing, then ζ(v) = ω and U_{v} is a Hall–Littlewood polynomial.

Lemma 3.2. The leading term of U_{v} and Û_{v} is x^{v}. Consequently, the transition matrix between the U_{v} (respectively the Û_{v}) and the monomials is upper unitriangular with respect to the right-to-left lexicographic order.

Proof. Let k be an integer and u be a weight such that u_{k} > u_{k+1}. Suppose by induction that x^{u} is the leading term of U_{u}. Recall the the explicit action of □_{k} is (concentrating only on the two variables x_{k}, x_{k+1})

x^{βα} □_{k} = x^{βα} + (1 − t)(x^{β−1,α+1} + ⋯ + x^{α+1,β−1}) + x^{αβ} , β > α
x^{ββ} □_{k} = (1 + t)x^{ββ}
x^{αβ} □_{k} = tx^{βα} + (t − 1)(x^{β−1,α+1} + ⋯ + x^{α+1,β−1}) + tx^{αβ} , α < β .

From these formulas, it is clear that for any constant c, the leading term of x^{u} (□_{k} + c) is (x^{u})^{s_{k}}, and, for any v such that v < u, all the monomials in x^{v} (□_{k} + c) are strictly less (with respect to the partial order) than (x^{u})^{s_{k}}. □

Example 3.3. For n = 3, Figures 1 and 2 show the case of a regular dominant weight x^{210}, and Figures 3 and 4 correspond to a case, x^{200}, where the stabilizer is not trivial. In this

[figure: Û_{210} = x^{210}; ∇_{1}; ∇_{2}; Û_{120} = x^{120} − qx^{210}; ∇_{2}+ \frac{q}{1+q}; Û_{201} = x^{201} − qx^{210}; ∇_{1}+ \frac{q}{1+q}; Û_{102} = x^{102} + (1 − q)x^{111} − qx^{201} − \frac{q^{2}}{q+1} x^{120} + \frac{q^{3}}{q+1} x^{210}; ∇_{1}; Û_{021} = x^{021} + (1 − q)x^{111} − qx^{120} − \frac{q^{2}}{q+1} x^{201} + \frac{q^{3}}{q+1} x^{210}; ∇_{2}; Û_{012} = (1 + q)B − qK̂_{120}]

Figure 2. Dual q-Key polynomials generated from x^{210}.

[figure: x^{200}/(1 + q); □_{1}; □_{2}; A; □_{2}−q/(1+q); U_{200} = x^{200}; □_{1}−q/(1+q); B; □_{1}; U_{020} = x^{020} − qx^{200} + (1 − q)x^{110}; □_{2}; U_{002} = (1 − q)x^{011} + (1 − q)x^{101} + x^{002} + \frac{q^{2}(q − 1)}{q + 1} x^{110} − \frac{q^{2}}{q + 1} x^{020} − \frac{q^{2}}{1 + q} x^{200}]

Figure 3. q-Key polynomials generated from x^{200}/(1 + q).

[figure: Û_{200} = x^{200}; ∇_{1}; ∇_{2}; Û_{020} = (1 − q)x^{110} + x^{020} + \frac{1}{q + 1} x^{200}; ∇_{2}+q/(1+q); 0; ∇_{1}+q/(1+q); Û_{002} = (1 − q)x^{011} + (1 − q)x^{101} + x^{002} + (1 − q)x^{110} + x^{020} + x^{200}; ∇_{1}; 0; ∇_{2}; 0]

Figure 4. Dual q-Key polynomials generated from x^{200}.

last case, the polynomials belonging to the family are framed, the extra polynomials denoted A, B do not belong to the basis.

3.2. Specialization at q = 0. The specialization at q = 0 of the Hecke algebra is called the 0-Hecke algebra. The elementary Yang–Baxter elements specialize in that case to

Y_{s_{i}} = T_{i} + 1 = □_{i} → x_{i}∂_{i} = π_{i} ,  (11)
Ŷ_{s_{i}} = T_{i} = ∇_{i} → ∂_{i}x_{i+1} = π̂_{i}.  (12)

Definition 3.4 (Key polynomials). Let v ∈ ℕ^{n}. The Key polynomials K_{v} and K̂_{v} are defined recursively, starting with K_{v} = x^{v} = K̂_{v} if x^{v} dominant, by

K_{vs_{i}} = K_{v} π_{i}  ,  K̂_{vs_{i}} = K̂_{v} π̂_{i} , for i such that v_{i} > v_{i+1} .

In particular, the subfamily (K_{v}) for v increasing is the family of Schur functions in x_{1}, …, x_{n}. Demazure [1] defined Key polynomials (using another terminology) for all the classical groups, and not only for the type A_{n−1} which is our case.

Lemma 3.2 specializes to the following lemma.

Lemma 3.5. The transition matrix between the U_{v} and the K_{v} (respectively from Û_{v} to K̂_{v}) is upper unitriangular with respect to the lexicographic order.

Example 3.6. For n = 3, the transition matrix between {U_{v}} and {K_{v}} in weight 3 is (reading a column as the expansion of some U_{v})

\begin{matrix}300 & 1 & · & · & · & · & · & \frac{−q}{(q+1)} & · & · & · \\ 210 & · & 1 & · & · & · & · & · & · & · & · \\ 201 & · & · & 1 & · & · & · & · & \frac{−q}{(q+1)} & · & · \\ 120 & · & · & · & 1 & · & \frac{−q}{(q+1)} & −q & · & · & · \\ 111 & · & · & · & · & 1 & −q & · & −q & −q (q + 1) & q^{2} \\ 102 & · & · & · & · & · & 1 & · & · & · & · \\ 030 & · & · & · & · & · & · & 1 & · & · & · \\ 021 & · & · & · & · & · & · & · & 1 & · & · \\ 012 & · & · & · & · & · & · & · & · & 1 & −q \\ 003 & · & · & · & · & · & · & · & · & · & 1 \end{matrix} ,

and the transition matrix between {Û_{v}} and {K̂_{v}} is

\begin{matrix}300 & 1 & · & · & · & · & · & −q & · & · & \frac{−q^{2}}{(q+1)} \\ 210 & · & 1 & −q & −q & · & \frac{q^{3}}{(q+1)} & −q & \frac{q^{3}}{(q+1)} & −q^{3} & \frac{q^{3}}{(q+ 1)} \\ 201 & · & · & 1 & · & · & −q & · & \frac{−q^{2}}{(q+1)} & q^{2} & −q \\ 120 & · & · & · & 1 & · & \frac{−q^{2}}{(q+1)} & −q & −q & q^{2} & \frac{q^{3}}{(q+1)} \\ 111 & · & · & · & · & 1 & −q & · & −q & q (q + 1) & q^{2} \\ 102 & · & · & · & · & · & 1 & · & · & −q & −q \\ 030 & · & · & · & · & · & · & 1 & · & · & \frac{−q^{2}}{(q+1)} \\ 021 & · & · & · & · & · & · & · & 1 & −q & −q \\ 012 & · & · & · & · & · & · & · & · & 1 & −q \\ 003 & · & · & · & · & · & · & · & · & · & 1 \end{matrix} .

4. Orthogonality properties for the q-Key polynomials

We show in this section that the q-Key polynomials U_{v} and Û_{v} are two adjoint bases with respect to a certain scalar product.

4.1. A scalar product on 𝔓𝔬𝔩. For any Laurent series f = ∑_{i=k}^{∞} f_{i}x^{i}, we denote by CT_{x}(f) the coefficient f_{0}.

Let

Θ := ∏_{1≤i<j≤n} \frac{1 − x_{i}/x_{j}}{1 − qx_{i}/x_{j}} .

Therefore, for any Laurent polynomial f(x_{1}, …, x_{n}), the expression

CT (f Θ) := CT_{x_{n}} (CT_{x_{n−1}} (… (CT_{x_{1}} (f Θ)) …))

is well defined. Let us use it to define a bilinear form ( , )_{q} on 𝔓𝔬𝔩 by

(f , g)_{q} = CT (f g^{♣} ∏_{1≤i<j≤n} \frac{1 − x_{i}/x_{j}}{1 − qx_{i}/x_{j}} ) ,  (13)

where ♣ is the automorphism defined by x_{i} ⟼ 1/x_{n+1−i} for 1 ≤ i ≤ n.

Since Θ is invariant under ♣, the form ( , )_{q} is symmetric. Under the specialization q = 0, the previous scalar product becomes

(f , g) := (f , g)|_{q=0} = CT (f g^{♣} ∏_{1≤i<j≤n}(1 − x_{i}/x_{j})) .  (14)

We can also write (f , g)_{q} = (f , gΩ) with Ω = ∏_{1≤i<j≤n}(1 − qx_{i}/x_{j})^{−1}.

Notice that, interpreting Schur functions as characters of unitary groups, Weyl defined the scalar product of two symmetric functions f, g in n variables as the constant term of

\frac{1}{n!} f g^{♣} ∏_{i,j: i≠j}(1 − x_{i}/x_{j}) .

Essentially, Weyl takes the square of the Vandermonde, while we are taking the quotient of the Vandermonde by the q-Vandermonde.

We now examine the compatibility of □_{i} and ∇_{i} with the scalar product.

Lemma 4.1. For i such that 1 ≤ i ≤ n − 1, □_{i} (respectively ∇_{i}) is adjoint to □_{n−i} (respectively ∇_{n−i}) with respect to ( , )_{q}.

Proof. Since π_{i} (respectively π̂_{i}) is adjoint to π_{n−i} (respectively π̂_{n−i}) with respect to ( , ) (see [7] for more details), we have

(f□_{i} , g)_{q} = (f , g Ω π_{n−i}(1 − qx_{n−i+1}/x_{n−i}))

= (f , g \frac{(1 − qx_{n−i+1}/x_{n−i})}{(1 − qx_{n−i+1}/x_{n−i})} Ω π_{n−i}(1 − qx_{n−i+1}/x_{n−i})).

Since the polynomial Ω/(1 − qx_{n−i+1}/x_{n−i}) is symmetric in the indeterminates x_{n−i} and x_{n−i+1}, it commutes with the action of π_{n−i}. Therefore

(f□_{i} , g)_{q} = (f , g (1 − qx_{n−i+1}/x_{n−i}) π_{n−i} Ω) = (f , g □_{n−i})_{q} .

This proves that □_{i} is adjoint to □_{n−i}, and, equivalently, that ∇_{i} is adjoint to ∇_{n−i}. □

We shall need to characterize whether the scalar product of two monomials vanishes or not. Notice that, by definition,

(x^{u} , x^{v}) = (x^{u−vω} , 1) ,

so that one of the two monomials can be taken equal to 1.

Lemma 4.2. For any u ∈ ℤ^{n}, we have (x^{u}, 1)_{q} ≠ 0 if and only if |u| = 0 and u ≥ [0, … 0]. In that case, (x^{u}, 1)_{q} = Q_{u}(0).

Proof. Let us first show that the scalar products (x^{u} , 1)_{q} satisfy the same relations (9) as the Hall–Littlewood functions Q_{u}.

Let k be a positive integer less than n. Write x_{k} = y, x_{k+1} = z. Any monomial x^{v} can be written as x^{t} y^{a}z^{b}, with x^{t} of degree 0 in x_{k}, x_{k+1}. The product

x^{t}(y^{a}z^{b} + y^{b}z^{a})(z − qy) ∏_{1≤i<j≤n} \frac{1 − x_{i}/x_{j}}{1 − qx_{i}/x_{j}}

is equal to

(y^{a}z^{b} + y^{b}z^{a})(z − qy)\frac{1 − y/z}{1 − qy/z} F_{1} = (y^{a}z^{b} + y^{b}z^{a})(z − y)F_{1},

with F_{1} symmetric in y, z. The constant term CT_{x_{k−1}} … CT_{x_{1}}(x^{t}(y^{a}z^{b} + y^{b}z^{a})F_{1}) = F_{2} is still symmetric in x_{k}, x_{k+1}. Therefore

CT_{y}(CT_{z}((z − y)F_{2}))

is null, and finally

CT (x^{t}(y^{a}z^{b} + y^{b}z^{a})(z − qy) ∏_{1≤i<j≤n} \frac{1 − x_{i}/x_{j}}{1 − qx_{i}/x_{j}} ) = 0 .

This relation can be rewritten as

(y^{a}z^{b+1}x^{t}, 1)_{q} + (y^{b+1}z^{a+1}x^{t}, 1)_{q} − q(y^{b+1}z^{a}x^{t}, 1)_{q} − q(y^{a+1}z^{b}x^{t}, 1)_{q} = 0 ,

which is indeed relation (9).

On the other hand, if u_{n} < 0, then there is no term of degree 0 in x_{n} in

x^{u} ∏_{1≤i<j≤n}(1 − x_{i}/x_{j})(1 − qx_{i}/x_{j})^{−1},

and (x^{u}, 1) = 0, so that rule (10) is also satisfied.

As a consequence, the function u ∈ ℤ^{n} → (x^{u}, 1) is determined by the values (x^{λ}, 1), λ a partition, as the function u ∈ ℤ^{n} → Q_{u} is determined by its restriction to partitions. However, for degree reasons, (x^{λ}, 1) = 0 if λ ≠ 0. Since (x^{0}, 1) = 1, one has finally that (x^{u}, 1) = Q_{u}(0). □

Example 4.3. For u = [1, 0, 3] and v = [0, 1, 3],

(x^{103} , x^{013})_{q} = (x^{−2,−1,3} , 1)_{q} = Q_{−2,−1,3}(0) = q^{2}(1 − q)(1 − q^{2}) .

4.2. Duality between (U_{v})_{v∈ℕ^{n}} and (Û_{v})_{v∈ℕ^{n}}. Using that □_{i} is adjoint to □_{n−i}, we are going to prove in this section that U_{v} and Û_{v} are two adjoint bases of 𝔓𝔬𝔩 with respect to the scalar product ( , )_{q}.

We first need some technical lemmas, to allow an induction on the q-Key polynomials, starting with dominant weights.

Lemma 4.4. Let i be an integer such that 1 ≤ i ≤ n − 1, let f_{1}, f_{2}, g_{1} be three polynomials and b be a constant such that

f_{2} = f_{1} (□_{i} + b) , (f_{1} , g_{1})_{q} = 0 and (f_{2} , g_{1})_{q} = 1 .

Then the polynomial g_{2} = g_{1}(∇_{n−i} − b) is such that

(f_{1} , g_{2})_{q} = 1 , (f_{2} , g_{2})_{q} = 0 .

Proof. Using that ∇_{n−i} is adjoint to □_{i} and that □_{i}∇_{i} = 0, one has

(f_{2} , g_{2})_{q} = (f_{1} (□_{i} + b) , g_{1} (∇_{n−i} − b))_{q} = (f_{1} (□_{i} + b) (∇_{i} − b) , g_{1})_{q}

= (f_{1}(−b(1 + q) − b^{2}), g_{1})_{q} = 0 .

Similarly, we have

(f_{1} , g_{2})_{q} = (f_{1} , g_{1} (∇_{n−i} − b))_{q}
= (f_{1} , g_{1} (□_{n−i} − 1 − q − b))_{q}
= (f_{1} (□_{i} + b − 1 − q − 2b) , g_{1})_{q} = (f_{2} , g_{1})_{q} = 1.
□

Corollary 4.5. Let i be an integer such that 1 ≤ i ≤ n − 1, let V be a vector space such that V = V′⊕ < f_{1}, f_{2} > with f_{2} = f_{1}(□_{i} + b) and V′ stable under □_{i}, and let g_{1} be an element with

(f_{1} , g_{1})_{q} = 0 and (f_{2} , g_{1})_{q} = 1 and (v , g_{1})_{q} = 0, for all v ∈ V′.

Then the element g_{2} = g_{1}(∇_{n−i} − b) satisfies

(f_{2} , g_{2})_{q} = 0 and (f_{1} , g_{2})_{q} = 1 and (v , g_{2})_{q} = 0, for all v ∈ V′.

Lemma 4.6. Let u and λ be two dominant weights, and let v and μ be two permutations of u and λ, respectively. If (x^{v} , x^{λ}) ≠ 0 and (x^{u} , x^{μ}) ≠ 0 then

u = λ , v = λω and μ = uω .

Proof. Using Lemma 4.2, the conditions (x^{v}, x^{λ})_{q} ≠ 0 and (x^{u}, x^{μ})_{q} ≠ 0 imply two systems of inequalities:

\{ \begin{matrix}v_{n} & ≥ & λ_{1} , \\ v_{n} + v_{n−1} & ≥ & λ_{1} + λ_{2} , \\ ⋮ & ⋮ & ⋮ \\ v_{n} + … + v_{1} & ≥ & λ_{1} + … + λ_{n} . \end{matrix}  and  \{ \begin{matrix}μ_{n} & ≥ & u_{1} , \\ μ_{n} + μ_{n−1} & ≥ & u_{1} + u_{2} , \\ ⋮ & ⋮ & ⋮ \\ μ_{n} + … + μ_{1} & ≥ & u_{1} + … + u_{n} . \end{matrix}

The first inequalities of the systems give v_{n} ≥ λ_{1} ≥ μ_{n} ≥ u_{1} ≥ v_{n}. Consequently u_{1} = λ_{1} = v_{n} = u_{n}. By induction, using the other inequalities, one gets the lemma. □

Corollary 4.7. Let v be a weight and λ be a dominant weight. Then,

(U_{v}, x^{λ})_{q} = δ_{v,λω} .

Proof. Let u be the decreasing reordering of v, and let σ be the permutation such that U_{v} = x^{u}Y_{σ}. By Lemma 4.6 and the fact that the leading term of U_{v} is x^{v}, the condition (x^{u}Y_{σ} , x^{λ})_{q} ≠ 0 implies (x^{v} , x^{λ})_{q} ≠ 0. By writing △_{σ} for the adjoint of Y_{σ} with respect to ( , )_{q}, we have (x^{u}Y_{σ} , x^{λ})_{q} = (x^{u} , x^{λ}△_{σ})_{q} ≠ 0. As the leading term of x^{λ}△_{σ} is x^{λσ′}, where λσ′ is a permutation of λ, we obtain that (x^{u} , x^{λσ′})_{q} ≠ 0. Using Lemma 4.6 we conclude that v = λω. □

Our main result is the following duality property between U_{v} and Û_{v}.

Theorem 4.8. The two sets of polynomials (U_{v})_{v∈ℕ^{n}} and (Û_{v})_{v∈ℕ^{n}} are two adjoint bases of 𝔓𝔬𝔩 with respect to the scalar product ( , )_{q}. More precisely, they satisfy

(U_{v} , Û_{uω})_{q} = δ_{v,u} .

Proof. Let λ be a dominant weight, and let V be the vector space spanned by the U_{v} for v in O(λ). The idea of the proof is to build by iteration the elements (Û_{v})_{v∈O(λ)} , starting with x^{λ} = Û_{λ}. By definition of the q-Key polynomials, there exists a constant b such that U_{λω} = U_{λωs_{1}} (□_{1} + b). One can write the decomposition V = V′⊕ < U_{λω} , U_{λωs_{1}} >, with V′ invariant under the action of □_{1}. Using the previous lemma, we have that (U_{λω} , x^{λ})_{q} = (U_{λω} , Û_{λ})_{q} = 1 and (U_{λωσ_{1}} , x^{λ})_{q} = (U_{λωσ_{1}} , Û_{λ})_{q} = 0. Consequently, by Lemma 4.5, the function x^{λ}(∇_{n−1} − b) = Û_{λs_{1}} satisfies the duality conditions

(U_{λω} , Û_{λs_{1}})_{q} = 0 , (U_{λωσ_{1}} , Û_{λs_{1}})_{q} = 1 and (v , Û_{λs_{1}})_{q} = 0 for all v ∈ V′.

By iteration, this proves that for all u, v, one has (U_{v} , Û_{uω})_{q} = δ_{v,u} . □

This theorem implies that the space of symmetric functions and the linear span of dominant monomials are dual of each other, the Hall–Littlewood functions being the basis dual to dominant monomials.

We finally mention that in the case q = 0, one has a reproducing kernel, as stated by the following theorem of [6], which gives another implicit definition of the scalar product ( , ).

Theorem 4.9. The two families of polynomials (K_{v})_{v∈ℕ^{n}} and (K̂_{v})_{v∈ℕ^{n}} satisfy the Cauchy formula

∑_{u∈ℕ^{n}} K_{u}(x)K̂_{uω}(y) = ∏_{i+j≤n+1} \frac{1}{1 − x_{i}y_{j}} .  (15)

Acknowledgment

We thank Bogdan Ion for pointing out in detail the connection with the theory of nonsymmetric Macdonald polynomials.

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