Yang-Baxter graphs, Jack and Macdonald polynomials

Alain Lascoux CNRS, Institut Gaspard Monge, Université de Marne-la-Vallée 77454 Marne-la-Vallée Cedex, France Alain.Lascoux@univ-mlv.fr

Abstract

We describe properties of the affine graph underlying the recursions between the different varieties of non symmetric Macdonald and Jack polynomials. We use an arbitrary function of one variable in the definition of affine edges, and of Cherednik’s elements, to unify the different theories. We describe the symmetrizing operators furnishing the symmetric polynomials from the non symmetric ones.

Dédié à Gian-Carlo Rota, ami et maître à pensées

1 Introduction

There are several families of mathematical objects related to the symmetric group which are obtained by simple recursions involving “local” operators [43].

All these constructions can be summarized as follows : one has a directed graph, with an origin ▷. Edges are labelled by elements θ of an associative algebra. One also chooses an initial element Y▷ in a right module over this algebra. The evaluation of a path from the origin ▷ to a given vertex v is the product θ_{1}θ_{2} ⋯ θ_{k} of the labels encountered along the path.

We shall say that a labelling is consistent if all the paths from the origin to any vertex v give an element Y_{v} := Y▷ θ_{1} ⋯ θ_{k} depending only upon v.

For example, Young orthogonal idempotents for a given partition λ can be described this way. The graph is the partially ordered set of tableaux of shape the diagram of λ, with origin the “Yamanouchi tabeau”, and one needs only to know the idempotent corresponding to this tableau, which will play the rôle of Y▷ (and that Young denotes P NP), to generate all the other idempotents, the labels of edges being of the type c(s_{i} + 1/k), c ∈ ℂ, k ∈ ℤ. We shall recall later that matrices representing the symmetric group can be described in the same fashion.

Other examples are associated to the permutohedron (i.e. the Cayley graph of the symmetric group 𝔖_{n}, edges v ↦ vs_{i} corresponding to multiplication on the right by simple transposition, whenever length increases: ℓ(vs_{i}) = ℓ(v) + 1 ).

Labelling edges by divided differences (resp. isobaric divided differences, to be defined later), and starting with

Y▷ = x_{1}^{n−1} ⋯ x_{n}^{0} or Y▷ = ∏_{i+j≤n}(x_{i} − y_{j})

(resp. Y▷ = ∏_{i+j≤n}(1−y_{j}/x_{i})) one generates polynomials Y_{v} which are called Schubert polynomials [29], [34] (resp. Grothendieck polynomials, [29],[32]).

We shall call a data :

permutohedron + initial Y▷ + consistent labelling

a Yang-Baxter graph for the symmetric group, and the elements Y_{v}, v ∈ 𝔖_{n}, Yang-Baxter elements. This notion underlies Young’s constructions [45], consistency arising from Yang-Baxter relations.

For example, writing Y_{v} on the graph instead of v, here is the Yang-Baxter graph giving Schubert polynomials for 𝔖_{3} [32], [34] :

(x_{1}−y_{1})(x_{1}−y_{2})(x_{2}−y_{1})
∂_{1} ↙  ↘^{∂_{2}}
(x_{1}−y_{1})(x_{2}−y_{1})  (x_{1}−y_{1})(x_{1}−y_{2})
∂_{2} ↓  ↓ ∂_{1}
x_{1}−y_{1}  x_{1}+x_{2}−y_{1}−y_{2}
∂_{1} ↘  ↙ ∂_{2}

1

Our main graph will be associated to the affine symmetric group. In fact, we shall take only ”half” of the extended affine symmetric group, i.e. we shall use the monoid 𝔖̂_{n} generated by

{s_{1}, …, s_{n−1}, τ : 1 ≤ i ≤ n − 1}

satisfying relations

s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} , s_{i}s_{j} = s_{j}s_{i} |i − j| > 1  (1)
s_{i} τ = τ s_{i−1} , 2 ≤ i ≤ n − 1 ; s_{1} τ^{2} = τ^{2} s_{n−1} .  (2)

The elements of 𝔖̂_{n} can be thought as acting both on infinite vectors v = […, v_{i}, …], i ∈ ℤ and on an infinite set of indeterminates x_{i}, i ∈ ℤ : the generator s_{i} transposes in v components v_{i+kn}, v_{i+1+kn} for all k ∈ ℤ, and transposes all x_{i+kn}, x_{i+1+kn} simultaneously; the generator τ acts as a shift on v: (vτ)_{i} = v_{i+1} and x_{i} τ = x_{i−1} for all i ∈ ℤ.

Using also the inverse \overline{τ} := τ^{−1} would give a faithful representation of the extended affine symmetric group.

Let us take the origin vector

▷ : ▷_{1} = 0 = ⋯ = ▷_{n} , ▷_{i±n} = ▷_{i} ± 1 ∀i ∈ ℤ .  (3)

Define Γ_{n} to be the following graph: vertices are the orbit of ▷ under 𝔖̂_{n}; the (oriented) edges, labelled s_{i} or τ, join pairs v, v s_{i} or v, vτ. The connected components obtained by erasing τ-edges will be called cells. Since the vectors v are ”periodic” in the sense that v_{i+n} = v_{i} + 1, ∀i ∈ ℤ, one will denote them by the restriction to their components v_{1}, …, v_{n}. Thus ▷ will also be written [0, …, 0], and v will be written [v_{1}, …, v_{n}].

Definition 1 An affine Yang-Baxter graph for 𝔖̂_{n} is a consistent labelling of the edges of Γ_{n} by elements of an associative algebra, together with a choice of an initial element Y▷ in a right module for this algebra. The elements Y_{v}, v ∈ 𝔖̂_{n}, are called Yang-Baxter elements.

By default, we shall suppose that the algebra has a unit 1 and that Y▷ = 1. To find consistent labelling of edges, we shall label vertices by a second

infinite vector c(v), called the vector of spectral parameters. One chooses an initial vector c(▷), and one takes its orbit under 𝔖̂_{n} : as before, the generator s_{i} simultaneously transpose components of c(v) of index i + kn, i^{+}1 + kn, ∀k ∈ ℤ, and the generator τ acts by translation : c_{i}(vτ) = c_{i+1}(v), ∀i ∈ ℤ.

For example, if one takes c(▷) such that c_{i}(▷) is the remainder of i^{−}1 mod n, then ϵ(v) := c(v) will be periodic: ϵ_{i+n}(v) = ϵ_{i}(v)∀i ∈ ℤ, and

ϵ_{1}(v), …, ϵ_{n}(v)

is the standardization of v_{1}, …, v_{n} (i.e. one replaces multiple occurrence of the same value α in [v_{1}, …, v_{n}] by different letters α, α′, α′′ …, α > α′ > α′′ > ⋯, reading from left to right; then one replaces these new letters decreasingly by 0, 1, …, n^{−}1; this is an operation much used in the theory of Young tableaux, since the article of Schensted). In the case where v_{1} ≥ v_{2} ≥ ⋯ ≥ v_{n}, then ϵ(v) = [0, 1, …, n^{−}1].

The vectors c(v) that we shall choose will simply be a combination of v and ϵ(v).

For example, introducing extra parameters α, q, t = −t_{2}/t_{1}, we shall take

c(▷) = … [ 0,^{−}1, …,^{−}n^{+}1 ], α, α^{−}1, a^{−}2, …, 2α; 2α^{−}1, …

i.e. c(▷) = [0,^{−}1, …,^{−}n^{+}1] with periodicity c_{i+n}(▷) = α + c_{i}(▷). Then

c(v) = αv − ϵ(v) i.e. c_{i}(v) = αv_{i} − ϵ_{i}(v) .  (4)

Taking on the other hand c(▷) = [t^{0}, t^{1}, …, t^{n−1}] with periodicity c_{i+n}(▷) = qc_{i}(▷) entails

c_{i}(v) = q^{v_{i}}t^{ϵ_{i}(v)} .  (5)

2 Cells and the group algebra of the symmetric group

Let us first detail Yang-Baxter graphs for the symmetric group, taking, as a first algebra to index edges, the group algebra of 𝔖_{n} itself.

In fact, already at the beginning of the century Young gave an example of such a graph to describe representations of the symmetric group (for a modern approach, cf. [38]). Young [45] defined matrices σ_{i} representing simple transpositions s_{i}. Relations σ_{i}σ_{j} = σ_{j}σ_{i} are immediate in his construction, and thus, he had essentially to check that his construction worked for 𝔖_{3}. In that case, he had three parameters c_{1}, c_{2}, c_{3} (the contents of three consecutive letters in an arbitrary Young tableau). For simplicity of notation, let us specialize c_{1} = 0, c_{2} = 2, c_{3} = 5. Young considered a vector space of dimension 6, with basis indexed by the permutations of c(v^{0}) = [0, 2, 5] :

[0, 2, 5], [0, 5, 2], [2, 0, 5], [2, 5, 0], [5, 0, 2], [5, 2, 0] ,

and wrote the two matrices

σ = [\begin{matrix}−\frac{1}{2} & 0 & \frac{\sqrt{3}}{2} & 0 & 0 & 0 \\ 0 & −\frac{1}{5} & 0 & 0 & \frac{2\sqrt{6}}{5} & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & −\frac{1}{3} & 0 & \frac{2\sqrt{2}}{3} \\ 0 & \frac{2\sqrt{6}}{5} & 0 & 0 & \frac{1}{5} & 0 \\ 0 & 0 & 0 & \frac{2\sqrt{2}}{3} & 0 & \frac{1}{3} \end{matrix} ] , σ′ = [\begin{matrix}−\frac{1}{3} & \frac{2\sqrt{2}}{3} & 0 & 0 & 0 & 0 \\ \frac{2\sqrt{2}}{3} & \frac{1}{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & −\frac{1}{5} & \frac{2\sqrt{6}}{5} & 0 & 0 \\ 0 & 0 & \frac{2\sqrt{6}}{5} & \frac{1}{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & −\frac{1}{2} & \frac{\sqrt{3}}{2} \\ 0 & 0 & 0 & 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix} ]

These two matrices are made of 2 × 2 sub-matrices,

[\begin{matrix}−ρ & \sqrt{1 − ρ^{2}} \\ \sqrt{1 − ρ^{2}} & ρ \end{matrix}] ,

each of which depends upon a single parameter ρ (= 1/2, 1/3 or 1/5) and codes a representation of 𝔖_{2} on a space of dimension 2. Now the matrix σ is a patch of three elementary matrices and can be encoded by the arrows:

[0, 2, 5] ⟶^{s+ \frac{1}{2}} [2, 0, 5] , [0, 5, 2] ⟶^{s+ \frac{1}{5}} [5, 0, 2] , [2, 5, 0] ⟶^{s+ \frac{1}{3}} [5, 2, 0] ,

and the matrix σ′ can similarly be encoded by:

[0, 2, 5] ⟶^{s′+ \frac{1}{3}} [0, 5, 2] , [2, 0, 5] ⟶^{s′+ \frac{1}{5}} [2, 5, 0] , [5, 0, 2] ⟶^{s′+ \frac{1}{2}} [5, 2, 0] .

Thus the two matrices can be encoded at the same time by the graph

[0, 2, 5]
s + \frac{1}{2} ↙  ↘ s′ + \frac{1}{3}
[2, 0, 5]  [2, 5, 0]
s′ + \frac{1}{5} ↓  ↓ s + \frac{1}{5}
[0, 5, 2]  [5, 0, 2]
s + \frac{1}{3} ↘  ↙ s′ + \frac{1}{2}
[5, 2, 0]

which describes their action on the six-dimensional vector space with basis [0, 2, 5], …, [5, 2, 0].

Relation σσ′σ = σ′σσ′ becomes

(s_{1} + \frac{1}{2})\frac{1}{\sqrt{1 − \frac{1}{2^{2}}}} (s_{2} + \frac{1}{5})\frac{1}{\sqrt{1 − \frac{1}{5^{2}}}} (s_{1} + \frac{1}{3})\frac{1}{\sqrt{1 − \frac{1}{3^{2}}}} =

= (s_{2} + \frac{1}{3})\frac{1}{\sqrt{1 − \frac{1}{3^{2}}}} (s_{1} + \frac{1}{5})\frac{1}{\sqrt{1 − \frac{1}{5^{2}}}} (s_{2} + \frac{1}{2})\frac{1}{\sqrt{1 − \frac{1}{2^{2}}}} ,

but now, it is an identity in the group algebra of 𝔖_{3}, instead of being a matrix identity, and the graph itself, with Y_{v^{0}} = 1, is a Yang-Baxter graph for 𝔖_{3}.

Yang [44] solved the same problem as Young did, but formulated it directly in terms of group algebra instead of matrices of representation: he wanted to label the edges of the permutohedron by elements of the group algebra of the symmetric group in such a way that the product of elements from the origin to a given permutation v be independent of the path. He found the solution of choosing a vector of ”generic” parameters for the identity permutation (the mutual differences of these parameters are only required to be different from 0 and 1), generating vectors c(v) by simple transpositions of components, and labelling the edge [v → v s_{i}] by

s_{i} + \frac{1}{c_{i+1}(v) − c_{i}(v)} .  (6)

Since the permutohedron is paved by squares s_{i}s_{j} = s_{j}s_{i}, |i − j| > 1, and hexagons s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}, consistency of Yang’s solution lies upon Yang-Baxter equation

(s_{i} + \frac{1}{α})(s_{i+1} + \frac{1}{α^{+}β})(s_{i} + \frac{1}{β}) = (s_{i+1} + \frac{1}{β})(s_{i} + \frac{1}{α^{+}β})(s_{i+1} + \frac{1}{α}) , (7)

the commutations

(s_{i} + \frac{1}{α})(s_{j} + \frac{1}{β}) = (s_{j} + \frac{1}{β})(s_{i} + \frac{1}{α}) , |i − j| > 1  (8)

being evident.

Thus, consistency arises from additivity :

c_{i+2} − c_{i} = (c_{i+2} − c_{i+1}) + (c_{i+1} − c_{i}) ,

as in the case of Young where parameters were axial distances (= contents).

Writing on the vertices of the graph c[v] under v, and taking c([1, 2, 3]) = [0, α, α^{+}β], then a cell with edges in the group algebra of 𝔖_{3} looks like

{[1,2,3] \atop [0,α,α+β]}

s + \frac{1}{α} ↙  ↘ s′ + \frac{1}{β}
{[2,1,3] \atop [α,0,α+β]}  {[1,3,2] \atop [0,α+β,α]}
s′ + \frac{1}{α+β} ↓  ↓ s + \frac{1}{α+β}
{[2,3,1] \atop [α,α+β,0]}  {[3,1,2] \atop [α+β,0,α]}
s + \frac{1}{β} ↘  ↙ s′ + \frac{1}{α}

{[3,2,1] \atop [α+β,α,0]}

3 Cells and the degenerate affine Hecke algebra

We have some freedom for what concerns choosing the Y_{v} as operators on polynomials in the x_{i}’s. Indeed, any family of operators satisfying the braid relations can be used.

We can keep the relations s_{i}^{2} = 1, but deform the transpositions of variables. For example, we can take

σ_{i} = s_{i} + ∂_{i} , 1 ≤ i ≤ n − 1 ,  (9)

where ∂_{i} is the divided difference [4]

f ↦ f ∂_{i} := \frac{f − f^{s_{i}}}{x_{i} − x_{i+1}}  (10)

(one could introduce an extra parameter γ and take σ_{i} := s_{i} + γ ∂_{i}; this is the homogeneous version of the preceding one, and thus not more general).

The σ_{i} still represent the symmetric group (they satisfy (1)) and generate an algebra with linear basis {σ_{w} : w ∈ 𝔖_{n}} (cf. [29]).

Labelling of edges uses the spectral parameters in the same way as in the preceding section, since the σ_{i}’s satisfy the same relations as the s_{i}’s : the edge [v ↦ v σ_{i}] is labelled

σ_{i} + \frac{1}{c_{i+1}(v) − c_{i}(v)} = s_{i} + ∂_{i} + \frac{1}{c_{i+1}(v) − c_{i}(v)}  (11)

Relations between indeterminates x_{i}, 1 ≤ i ≤ n (considered as “multiplication by x_{i}”) and the σ_{i} or ∂_{i} are given by Leibniz rule:

x_{i}σ_{i} = σ_{i}x_{i+1} + 1 , x_{i+1}σ_{i} = σ_{i}x_{i} − 1 , x_{j}σ_{i} = σ_{i}x_{j}, j ≠ i, i + 1  (12)

The algebra generated by the σ_{i}, 1 ≤ i ≤ n − 1 and x_{i}, 1 ≤ i ≤ n is called the affine degenerate Hecke algebra.

The limit case σ_{i} = ∂_{i} gives also rise to a Yang-Baxter graph because the ∂_{i}’s themselves satisfy the braid relations. Thus for any permutation v exists a well defined divided difference ∂_{w}, as well as a σ_{w}.

In fact, since 1 − s_{i} = ∂_{i} (x_{i} − x_{i+1}) the affine Hecke algebra is a subalgebra of the algebra of divided differences generated by the ∂_{i}, and rational functions in the x_{j} (see [29],[30]). This last algebra is a free module over the ring of rational functions in the x_{i}’s, with basis the divided differences ∂_{w}, w ∈ 𝔖_{n}. Expansions of the σ_{w} or w in the basis of ∂_{w} involve Schubert polynomials (cf. [27], [28], [11]).

4 Cells and the affine Hecke algebra

One has more general operators on polynomials satisfying the braid relations (see [29], [30]).

In particular, let q be an extra parameter, and let x_{i+n} = qx_{i}, ∀i ∈ ℤ. Then one defines isobaric divided differences \overline{π}_{i}, i = 1, …, n − 1 acting on the quotient ring ℤ[x_{i} : i ∈ ℤ]/(x_{i+n} = qx_{i}) as twisted derivations:

fg\overline{π}_{i} = f(g\overline{π}_{i}) + f\overline{π}_{i}(gs_{i})

satisfying the conditions

x_{j}\overline{π}_{i} = 0 , j ≢ i, i^{+}1 mod (n) ,
x_{j}\overline{π}_{i} = x_{j+1} , j ≡ i ,  (13)
\overline{π}_{i}^{2} = −\overline{π}_{i} .

Lemma 2 The \overline{π}_{i}, 1 ≤ i ≤ n − 1, satisfy the braid relations (1).

Indeed, the image of \overline{π}_{i} under the isomorphism of the quotient ring onto the space of polynomials in x_{1}, …, x_{n}, is the usual isobaric divided difference (that we shall denote by the same letter)

\overline{π}_{i} : f ↦ (f − f^{s_{i}})\frac{x_{i+1}}{x_{i} − x_{i+1}} , 1 ≤ i ≤ n − 1 .  (14)

These operators commute with multiplication by symmetric functions in x_{1}, …, x_{n} and braid relations can directly be checked on an appropriate basis of the space of polynomials (as a module over symmetric polynomials in x_{1}, …, x_{n}).

Demazure [8] defined such operators for any root system.

Let us notice that the operators π_{i} := \overline{π}_{i}+1 also satisfy the braid relations together with π_{i}^{2} = π_{i}.

It has been noticed in [29]( where, more generally one uses D_{i} := t_{1}π_{i} + t_{2}∂_{i} + t_{3}s_{i}) and [33] that the operators

T_{i} = T_{x_{i},x_{i+1}} := π_{i}(t_{1} + t_{2}) − t_{2} s_{i}  (15)

represent the Hecke algebra H_{n} of the symmetric group 𝔖_{n}. The generators T_{i} satisfy the braid relations (1) together with

(T_{i} − t_{1}) (T_{i} − t_{2}) = 0 .  (16)

Reversing the order on x_{1}, …, x_{n} give other generators

\overline{T}_{i} := −t_{1}t_{2}T_{i}^{−1} = \overline{π}_{i} (t_{1} + t_{2}) − t_{2} s_{i} = T_{i} − (t_{1} + t_{2}) , 1 ≤ i ≤ n − 1 , (17)

the Hecke relation becoming in that case

(\overline{T}_{i} + t_{1}) (\overline{T}_{i} + t_{2}) = 0 .  (18)

We prefer keeping two parameters t_{1}, t_{2}, because all cases [t_{1}, t_{2}] = [t, −1], [−1, t], [t, 1/t], [1, t], [t, 1] occur in the literature. We shall also have to use the involutions [t_{1}, t_{2}] ↔ [±t_{2}, ±t_{1}].

Leibniz formula (12) implies the following commutations

x_{i} \overline{T}_{i} = T_{i} x_{i+1} , 1 ≤ i ≤ n^{−}1  (19)
x_{j}T_{i} = T_{i} x_{j} , j ≠ i, i^{+}1 , 1 ≤ j ≤ n .

One still has a Yang-Baxter equation,

(T_{i} + \frac{t_{1}^{+}t_{2}}{α^{−}1} ) (T_{i+1} + \frac{t_{1}^{+}t_{2}}{αβ^{−}1}) (T_{i} + \frac{t_{1}^{+}t_{2}}{β^{−}1} ) =

(T_{i+1} + \frac{t_{1}^{+}t_{2}}{β^{−}1} ) (T_{i} + \frac{t_{1}^{+}t_{2}}{αβ^{−}1}) (T_{i+1} + \frac{t_{1}^{+}t_{2}}{α^{−}1} ) . (20)

As in the preceding section, one chooses for the identity permutation v^{0} a vector of parameters c(v^{0}) = [c_{1}, …, c_{n}] (all different, and different from 0) and define c(v), v ∈ 𝔖_{n}, to be [c_{v_{1}}, …, c_{v_{n}}].

Thanks to Yang-Baxter relation (20), consistency will be now insured by taking, on an edge v ↦ v s_{i}, the label

T_{i} + \frac{t_{1} + t_{2}}{c_{i+1}/c_{i}(v) − 1} .  (21)

In particular, given two positive integers a, b, taking the spectral parameters [(−t_{2})^{a+b}, t_{1}^{a}(−t_{2})^{b}, t_{1}^{a+b}], that is α = (−t_{1}/t_{2})^{a}, β = (−t_{1}/t_{2})^{b}, the left hand side of (20) becomes

(T_{i} + \frac{(−t_{2})^{a}}{t_{1}^{a−1} − t_{1}^{a−2}t_{2} + ⋯ + (−t_{2})^{a−1}} )

(T_{i+1} + \frac{(−t_{2})^{a+b}}{t_{1}^{a+b−1} − t_{1}^{a+b−2}t_{2} + ⋯ + (−t_{2})^{a+b−1}} )

(T_{i} + \frac{(−t_{2})^{b}}{t_{1}^{b−1} − t_{1}^{b−2}t_{2} + ⋯ + (−t_{2})^{b−1}} )

When dealing with reduced decompositions, it is convenient to use planar displays, Young tableaux, and jeu de taquin, to handle more easily the braid relations (cf. [31] for the nilplactic monoid). All the displays that we shall write will have the property that the words obtained by reading them columnwise, from left to right, are congruent modulo the braid relations to the words obtained by reading them row-wise (from top to bottom). For example, writing i for T_{i}, then the planar object 2 3 4 1 2 3 can be read (T_{2}T_{3}T_{4})(T_{1}T_{2}T_{3}) as well as (T_{2}T_{1})(T_{3}T_{2})(T_{4}T_{3}).

With these conventions, a recursive application of braid relations leads to the following identities.

Lemma 3 Let i, j, k be three positive integers, i < j, i < k. Then the following two words are congruent modulo the braid relations:

k ⋯ j^{+}k^{−}i ; ⋮  ⋮ ; i ⋯ j ; j^{−}1 ; ⋮ ; i ≡ k^{+}j^{−}i ; ⋮ ; k^{+}1 ; k ⋯ j^{+}k^{−}i ; ⋮  ⋮ ; i ⋯ j .  (22)

Moreover

\begin{matrix}k & ⋯ & j^{+}k^{−}i \\ ⋮ & & ⋮ \\ i & ⋯ & j \end{matrix} \overline{T}_{i} ⋯ \overline{T}_{j−1} ≡ \overline{T}_{k+1} ⋯ \overline{T}_{j+k−i} \begin{matrix}k & ⋯ & j^{+}k^{−}i \\ ⋮ & & ⋮ \\ i & ⋯ & j \end{matrix} .  (23)

For example,

2 3 4 1 2 3 2 1 ≡ 4 3 2 3 4 1 2 3  and  2 3 4 1 2 3 \overline{T}_{1}\overline{T}_{2} ≡ \overline{T}_{3}\overline{T}_{4} 2 3 4 1 2 3 .

5 Symmetrizations

Each of the algebras considered in the preceding sections contains elements (that we shall call symmetrizing operators) sending polynomials in x_{1}, …, x_{n} onto symmetric polynomials, or to products of symmetric polynomials by a fixed factor of Vandermonde type.

The first, and most fundamental, of symmetrizing operators, is due to Cauchy and Jacobi:

f ↦ ∑_{w∈𝔖_{n}}(−1)^{ℓ(w)}w \frac{1}{Δ} ,  (24)

where Δ := ∏_{i<j≤n}(x_{i} − x_{j}) is the Vandermonde. It sends monomials to ± a Schur function, or to 0. It can be written as a product of divided differences. In fact, it is equal to ∂_{ω}, where ω := [n, …, 1] is the permutation of maximal length in 𝔖_{n}. Its multiple ∂_{ω}/n! can be interpreted as the 1-dimensional idempotent corresponding to the alternating representation of the symmetric group.

In the Hecke algebra, there are two 1-dimensional (quasi)-idempotents which similarly can be expressed as a sum of linear generators, or expressed with divided differences.

Define

∇_{ω} := ∑_{w∈𝔖_{n}}(−t_{1})^{ℓ(wω)}T_{w} ,  (25)
□_{ω} := ∑_{w∈𝔖_{n}}(−t_{2})^{ℓ(wω)}T_{w} .  (26)

Theorem 4 1) The element ∇_{ω}, as an operator on polynomials, factorizes:

∇_{ω} = ∂_{ω} ∏_{1≤i<j≤n}(t_{2}x_{i} + t_{1}x_{j}) .

2) It factorizes inside the Hecke algebra:

∇_{ω} = ∇_{ω′}(T_{n−1} ⋯ T_{1} − t_{2}T_{n−1} ⋯ T_{2} + ⋯ + (−t_{2})^{n−2}T_{n−1} + (−t_{2})^{n−1}) ,

where ω′ = [n^{−}1, …, 1, n].

3) It is a Yang-Baxter element Y_{ω}, for the symmetric group 𝔖_{n}, for the initial vector of parameters [t_{1}^{n−1}, −t_{1}^{n−2}t_{2}, …, (−t_{2})^{n−1}].

4) It is equal to another sum

∇_{ω} := ∑_{w∈𝔖_{n}} t_{2}^{ℓ(wω)}\overline{T}_{w} .

5) The element □_{ω} is obtained from ∇_{ω} by interchange of t_{1} and t_{2} in the expressions 2,3,4. The involution t_{1} ↦ −t_{2}, t_{2} ↦ −t_{1}, T_{i} ↔ \overline{T}_{i},i = 1, …, n^{−}1, preserves ∇_{ω} and □_{ω}. Moreover,

□_{ω} := ∏_{1≤i<j≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω} .

A proof of this theorem is given in [9]. Since all the operators appearing in theorem 4 commute with multiplication by symmetric functions, theorem 4 essentially reduces to check that all elements, but one, of the Schubert basis for 𝔖_{n} are sent to 0 by ∇_{ω}.

The last factorization appear in geometry, □_{ω} being interpreted as an Euler-Poincaré characteristic for a relative flag manifold (cf. [12], first chapter of [21], [22]).

One has similar factorizations in the degenerate affine Hecke algebra generated by the σ_{i} = s_{i} + γ∂_{i} , as shows the following theorem (cf. [27]):

Theorem 5 1) The sum of all the linear generators of the degenerate affine Hecke algebra factorizes (as operators on polynomials)

˜□_{ω} := ∑_{w∈𝔖_{n}} σ_{w} = ∏_{i<j≤n}(x_{i} − x_{j} + γ) ∂_{ω} .  (27)

2) It is a Yang-Baxter element for the parameters [1, 2, …, n]:

˜□_{ω} = (σ_{1} + 1)((σ_{2} + \frac{1}{2})(σ_{1} + 1)) ⋯ ((σ_{n−1} + \frac{1}{n^{−}1}) ⋯ (σ_{2} + \frac{1}{2})(σ_{1} + 1)) .

6 Affine edges

We have shown that labelling of edges of a given cell are determined by a single vector of spectral parameters. We have only retained three algebras, the group algebra of the symmetric group, that we shall see as the specialization γ = 0 of the degenerate affine Hecke algebra, and the affine Hecke algebra.

It remains to define the labels of the affine edges connecting cells and to check the compatibility between the labels of different cells. We shall see that it amounts to introduce “periodicity” conditions on the x_{i}’s and on the initial vector c(▷) of spectral parameters.

To treat the different varieties of Jack and Macdonald polynomials at the same time, we shall use an arbitrary function G := G_{n} of x_{n}, and define the label of all affine edges of the Yang-Baxter graph to be

Φ := τ G

(recall that τ is the shift: x_{i} τ = x_{i−1} τ).

We could even be more general and take for example G to be a differential operator on functions of x_{n}. At this stage, we need only the commutation of G with x_{1}, …, x_{n−1}, s_{1}, …, s_{n−2}.

Let us write

G_{i} := s_{i} ⋯ s_{n−1}G_{n}s_{n−1} ⋯ s_{i} , 1 ≤ i ≤ n − 1 .

In the case of the degenerate affine Hecke algebra with generators σ_{i} = s_{i} + γ∂_{i}, we used differences c_{i+1} − c_{i} and x_{i+1} − x_{i}, and consistency requires them to be invariant under i ↦ i + n. For that purpose, we introduce two more parameters α, β:

x_{i+n} = x_{i} + β , c_{i+n}(▷) = c_{i}(▷) + α ∀i ∈ ℤ .  (28)

The subgraph \begin{aligned}v & → & vs_{i+1} \\ ↓ & & ↓ \\ vτ & → & vτs_{i} & = vs_{i+1}τ \end{aligned} give rise to the following commutative

diagram (writing c(v) instead of v, and 1,2,3 instead of i, i + 1, i + 2):

\begin{matrix}[… c_{1}, c_{2}, c_{3} …] & ⟶^{s_{2}+γ∂_{2}+ \frac{1}{c_{3}−c_{2}}} & [… c_{1}, c_{3}, c_{2} …] \\ Φ ↓ & & Φ ↓ \\ [… c_{2}, c_{3}, c_{4} …] & ⟶^{s_{1}+γ∂_{1}+ \frac{1}{c_{3}−c_{2}}}i & [… c_{3}, c_{2}, c_{4} …] \end{matrix}  (29)

which says nothing more than x_{2}τ = x_{1}, x_{3}τ = x_{2}, ∂_{2}τ = τ∂_{1}, s_{2}τ = τs_{1}. The commutativity of

[… c_{n−1}, c_{n}, c_{n+1}, c_{n+2} …] ⟶^{s_{1}+γ∂_{1}+ \frac{1}{c_{2}−c_{1}}} [… c_{n−1}, c_{n}, c_{n+2}, c_{n+1} …]
ΦΦ ↓  ΦΦ ↓
[… c_{n+1}, c_{n+2}, c_{n+3}, c_{n+4} …] ⟶^{s_{n−1}+γ∂_{n−1}+ \frac{1}{c_{2}−c_{1}}} [… c_{n+2}, c_{n+1}, c_{n+3}, c_{n+4} …]

(30)

is also immediate, once noticing that τG_{n}τG_{n} = τ^{2}G_{n−1}G_{n}, and that G_{n−1}G_{n}, being symmetrical in x_{n−1} and x_{n}, commutes with s_{n−1}, as well as with ∂_{n−1}.

Braid relations, together with these two types of diagrams allow to deform any path ▷ ↦ v into any other one with the same end points. Thus conditions (11) and (28) furnish a Yang-Baxter graph with edges in the degenerate Hecke algebra.

In the case of the affine Hecke algebra, we used c_{i+1}/c_{i}−1 and 1−x_{i+1}/x_{i} in the labels of edges of cells, and we need them to be invariant under i ↦ i+n. We thus introduce two extra parameters q, q′ (in the following sections, we shall restrict to q = q′) and put

x_{i+n} = qx_{i} , c_{i+n}(▷) = q′c_{i}(▷) , ∀i ∈ ℤ  (31)

Now, since π_{i}τ = τπ_{i−1}, s_{i}τ = τs_{i−1}, 1 < i < n, and since G_{n−1}G_{n} commutes with T_{n−1}, one still has commutative diagrams (where T_{1}, T_{2} stand for T_{i}, T_{i+1}, 1 ≤ i ≤ n − 2 in the first diagram) :

\begin{matrix}[… c_{1}, c_{2}, c_{3} …] & ⟶^{T_{2}+ \frac{t_{1}+t_{2}}{c_{3}/c_{2}−1}} & [… c_{1}, c_{3}, c_{2} …] \\ Φ ↓ & & Φ ↓ \\ [… c_{2}, c_{3}, c_{4} …] & ⟶^{T_{1}+ \frac{t_{1}+t_{2}}{c_{3}/c_{2}−1}} & [… c_{3}, c_{2}, c_{4} …] \end{matrix}  (32)

and

[… c_{n−1}, c_{n}, c_{n+1}, c_{n+2} …] ⟶^{T_{1}+ \frac{t_{1}+t_{2}}{c_{2}/c_{1}−1}} [… c_{n−1}, c_{n}, c_{n+2}, c_{n+1} …]
ΦΦ ↓  ΦΦ ↓
[… c_{n+1}, c_{n+2}, c_{n+3}, c_{n+4} …] ⟶^{T_{n−1}+ \frac{t_{1}+t_{2}}{c_{2}/c_{1}−1}} [… c_{n+2}, c_{n+1}, c_{n+3}, c_{n+4} …]

(33)

Conditions (21) and (31) provide a Yang-Baxter graph with edges in the Hecke algebra.

7 Cherednik’s Algebras

The double affine Hecke algebra ℂ[t_{1}, t_{2}, q][T_{1}, …, T_{n−1}, τ^{±1}, x_{1}^{±1}, … x_{n}^{±1}] admits some maximal commutative subalgebra which was found by Cherednik in his study of Macdonald’s polynomials. We need a more general algebra than the original Cherednik’s algebra, and for that purpose, we shall use the element G_{n} introduced in the preceding section, supposing only for the moment that it is an invertible morphism of the space of functions in x_{n}.

Let us write \overline{G} := G^{−1}, \overline{τ} = τ^{−1}, \overline{T}_{i} := −t_{1}t_{2} T_{i}^{−1}, 1 ≤ i ≤ n − 1. Recall that x_{i} \overline{τ} = \overline{τ} x_{i+1}, and that Φ = τG_{n} label the affine edges of the Yang-Baxter graph..

For 1 ≤ i ≤ n, define the Cherednik element ξ_{i} by

ξ_{i} := T_{i} … T_{n−1} x_{n} \overline{G}_{n} \overline{τ} \overline{T}_{1} ⋯ \overline{T}_{i−1} .  (34)

Proposition 6 The ξ_{i} generate a commutative algebra and satisfy the relations

ξ_{i+1}T_{i} = \overline{T}_{i} ξ_{i} & (ξ_{i} + ξ_{i+1})T_{i} = T_{i}(ξ_{i} + ξ_{i+1}) , 1 ≤ i ≤ n − 1 ,

ξ_{j}T_{i} = T_{i}ξ_{j} , j ≠ i, i + 1  (35)
ξ_{i+1} Φ = Φ ξ_{i} , i = 1, …, n − 1 .

Moreover, if G_{n} commutes with multiplication by x_{n}, then

qξ_{1} Φ = Φ ξ_{n} .  (36)

Proof. One checks without difficulty the relations between the ξ_{i} and T_{j} and the commutativity of the ξ_{i}.

Thanks to these relations, the compatibilities between Φ and the ξ_{i} is reduced to show that ξ_{2} Φ = Φ ξ_{1}, i.e.

T_{2} ⋯ T_{n−1}x_{n}\overline{G}_{n}\overline{τ} \overline{T}_{1}τG_{n} = τG_{n}T_{1} ⋯ T_{n−2}T_{n−1}x_{n}\overline{G}_{n} \overline{τ} .

Now the factor T_{1} ⋯ T_{n−2} commutes with G_{n} and cancels with the factor T_{2} ⋯ T_{n−1} on the left hand side; we are left to show that

x_{n}\overline{G}_{n}\overline{τ} \overline{T}_{1}τG_{n} = τG_{n} T_{n−1}x_{n}\overline{G}_{n} \overline{τ} .

Let us take the indeterminates x_{2}, …, x_{n}, x_{n+1} = qx_{1} instead of x_{1}, … x_{n}. It allows us to use G_{n+1} := τ G_{n} \overline{τ}. Let us moreover write T_{n} := \overline{τ} T_{1} τ = τ T_{n−1} \overline{τ}; concretely T_{n} operates on x_{n}, x_{n+1} according to (15) and leaves x_{2}, …, x_{n−1} invariant.

Introducing factors \overline{G}_{n+1}G_{n+1} and \overline{τ}τ, one has to show

x_{n}(G_{n+1}\overline{G}_{n+1}) \overline{G}_{n}\overline{τ}\overline{T}_{1} τG_{n} = τG_{n} (\overline{τ}τ)T_{n−1} (\overline{τ}τ) x_{n}(\overline{τ}τ) \overline{G}_{n}\overline{τ} .

Taking into account that \overline{G}_{n+1}\overline{G}_{n} commutes with \overline{τ}T̄_{1}τ = T_{x_{n},x_{n+1}}, we are finally reduced to make sure that

x_{n}G_{n+1}\overline{T}_{n}\overline{G}_{n+1}\overline{G}_{n}G_{n} = G_{n+1}T_{n}x_{n+1}\overline{G}_{n+1} ,

that is

x_{n} \overline{T}_{n} = T_{n}x_{n+1} ,

which is no other than relation (19) for the pair (x_{n}, x_{n+1}).

One similarly checks that T_{1} ⋯ T_{n−1} \overline{τ}τ x_{n} = x_{1}\overline{T}_{1} ⋯ \overline{T}_{n−1} = τx_{0}\overline{τ}\overline{T}_{1} ⋯ \overline{T}_{n−1} = τx_{n}ξ_{n}/q, that is

ξ_{1} τ x_{n} = τ x_{n} ξ_{n}/q .  (37)

There remains the case of Φ ξ_{n} = τG_{n} x_{n}G_{n}\overline{T}_{1} ⋯ \overline{T}_{n−1}. If G_{n} commutes with x_{n}, then

Φ ξ_{n} = τx_{n}\overline{τ} \overline{T}_{1} ⋯ \overline{T}_{n−1} = qT_{1} ⋯ T_{n−1} x_{n} = q (T_{1} ⋯ T_{n−1} x_{n}\overline{G}_{n}\overline{τ}) (τG_{n}) ,

that is

Φ ξ_{n} = qξ_{1} Φ .

QED

The ξ_{i}’s satisfy the same commutation relations with the T_{i}’s as the x_{i}’s do. In fact, one can interpolate between ξ_{i} and x_{i} without losing a commutative algebra.

Theorem 7 Let β ∈ ℂ; for i = 1, …, n − 1, let

ξ_{i}^{β} := ξ_{i} + βx_{i} .

Then the ξ_{i}^{β}’s generate a commutative algebra, and satisfy the relations

ξ_{i+1}^{β}T_{i} = \overline{T}_{i} ξ_{i}^{β} & (ξ_{i}^{β} + ξ_{i+1}^{β})T_{i} = T_{i}(ξ_{i}^{β} + ξ_{i+1}^{β}) , 1 ≤ i ≤ n − 1 , (38)
ξ_{j}^{β}T_{i} = T_{i}ξ_{j}^{β} , j ≠ i, i + 1  (39)
ξ_{i+1}^{β} Φ = Φ ξ_{i}^{β} , i = 1, …, n − 1 .  (40)

Moreover, if G_{n} commutes with multiplication by x_{n}, then

qξ_{1}^{β} Φ = Φ ξ_{n}^{β} .  (41)

Proof. Commutations relations (38,39,40) are a consequence of those for β = 0. Given 1 ≤ i < j ≤ n, the commutator [ξ_{i}^{β}, ξ_{j}^{β}] is equal to β[x_{i}, ξ_{j}^{β}] − β[ξ_{i}^{β}, x_{j}]. Explicitly,

x_{j}ξ_{i}^{β} = x_{j} \underbrace{T_{i} ⋯ T_{j−2}}_{A} T_{j−1} \underbrace{T_{j} ⋯ T_{n−1} x_{n} \overline{G}_{n}}_{B} \overline{τ} \underbrace{\overline{T}_{1} ⋯ \overline{T}_{i−1}}_{C} .

Now, x_{j} commutes with A; x_{j}T_{j−1} = \overline{T}_{j−1}x_{j−1}; x_{j−1} commutes with B; passing through \overline{τ}, it is transformed into x_{j} again and commutes with C.

Therefore

[ξ_{i}^{β}, x_{j}] = A(T_{j−1} − \overline{T}_{j−1})B \overline{τ} Cx_{j} = A(t_{1} + t_{2}) B \overline{τ} Cx_{j} .

Similarly, starting from

x_{i}ξ_{j}^{β} = x_{i} B \overline{τ} C \overline{T}_{i} \underbrace{\overline{T}_{i+1} ⋯ \overline{T}_{j−1}}_{D} ,

one finds that

[ξ_{j}^{β}, x_{i}] = B \overline{τ} Cx_{i+1}D(t_{1} + t_{2}) .

Finally, one remarks that x_{i+1}D = T_{i+1} ⋯ T_{j−1} x_{j} := Ex_{j}, and that A commutes with B, becomes equal to E when passing through \overline{τ} and commutes with C at the end. Therefore the ξ_{i}^{β}’s generate a commutative algebra. QED

One checks, for any 1 ≤ i < n the commutations

T_{i}(ξ_{i}^{β} + ξ_{i+1}^{β}) = (ξ_{i}^{β} + ξ_{i+1}^{β})T_{i} & T_{i}(ξ_{i}^{β}ξ_{i+1}^{β}) = (ξ_{i}^{β}ξ_{i+1}^{β})T_{i} , (42)

and this ensures the commutations of each T_{i} with the elementary symmetric functions in the Cherednik elements. Therefore one has:

Lemma 8 The Hecke algebra generated by the T_{i}’s commutes with the ring of symmetric functions in the ξ_{i}^{β}.

For computations, it is convenient to write monomials in the ξ_{i}’s (without repetition) in a canonical form. Write θ for x_{n} \overline{G}_{n} \overline{τ}. Then one checks, from the commutation relations (35), together with T_{i}\overline{T}_{i} = (−t_{1}t_{2}), that:

Lemma 9 Let k be an integer, n ≥ i_{1} > i_{2} > ⋯ > i_{k} ≥ 1 a decreasing sequence. Then the monomial ξ_{i_{1}}ξ_{i_{2}} ⋯ ξ_{i_{k}}(t_{1}t_{2})^{−k(k−1)} is equal to

\begin{aligned}& & i_{1} & ⋯ & n^{−}1 \\ & i_{2} & ⋯ & ⋯ & n^{−}2 \\ & ⋮ & & & ⋮ \\ i_{k} & ⋯ & ⋯ & ⋯ & n^{−}k \end{aligned} θ^{k} \begin{matrix}k & ⋯ & ⋯ & ⋯ & i_{1}^{−}1 \\ k^{−}1 & ⋯ & ⋯ & i_{2}^{−}1 \\ ⋮ & ⋯ & ⋮ \\ 1 & ⋯ & i_{k}^{−}1 \end{matrix} ,

where the planar display on the left must be read (columwise or rowwise) as a product of T_{j}’s, and the display on the right, as a product of \overline{T}_{j}’s.

For example, for n = 8,

ξ_{6}ξ_{3}ξ_{2}(t_{1}t_{2})^{−6} = \begin{matrix}. & . & 6 & 7 \\ 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 5 \end{matrix} θ^{3} \begin{matrix}3 & 4 & 5 \\ 2 & . & . \\ 1 & . & . \end{matrix} = (T_{6}T_{7}T_{3}T_{4}T_{5}T_{6}T_{2}T_{3}T_{4}T_{5}) θ^{3} (\overline{T}_{3}\overline{T}_{4}\overline{T}_{5}\overline{T}_{2}\overline{T}_{1}) .

Exchanging left and right, one gets another commutative algebra generated by the

T_{i−1} ⋯ T_{1} τ \frac{1}{x_{n}} G_{n} \overline{T}_{n−1} ⋯ \overline{T}_{i} + \frac{β}{x_{i}} , 1 ≤ i ≤ n .

In the case of interest for vanishing Macdonald polynomials, one takes G_{n} = x_{n} − (−t_{2})^{n−1}. The following lemma (due to Knop [18]) shows that in that case, one can choose an appropriate β in order to get operators preserving polynomials.

Lemma 10 Let

ξ̂_{i} := T_{i−1} ⋯ T_{1} τ(1 − \frac{(−t_{2})^{n−1}}{x_{n}} )\overline{T}_{n−1} ⋯ \overline{T}_{i} + \frac{(−t_{1}t_{2})^{n−1}}{x_{i}} , 1 ≤ i ≤ n . (43)

Then the ξ̂_{i}’s generate a commutative algebra which preserves polynomials in x_{1}, …, x_{n}.

Proof. Because of the commutation relations (35), we have only to check that ξ̂_{n} preserves polynomials. Recall that x_{i+1}T_{i} = \overline{T}_{i}x_{i}, which implies, because x_{i} + x_{i+1} commute with T_{i}, that

x_{i}T_{i} = T_{i}x_{i+1} + (t_{1} + t_{2})x_{i} .

Therefore, for 1 ≤ i ≤ n − 1, one has

x_{i}T_{n−1} ⋯ T_{1} = T_{n−1} ⋯ T_{1}x_{i+1} + (t_{1} + t_{2})T_{n−1} ⋯ T_{i+1} \overline{T}_{i−1} ⋯ \overline{T}_{1} x_{1} ,

and

x_{i}ξ̂_{n} = T_{n−1} ⋯ T_{1} τ(1 − \frac{(−t_{2})^{n−1}}{x_{n}})x_{i} + \frac{(−t_{1}t_{2})^{n−1}x_{i}}{x_{n}}

+ (t_{1} + t_{2})T_{n−1} ⋯ T_{i+1} \overline{T}_{i−1} ⋯ \overline{T}_{1} τ (x_{0} − \frac{(−t_{2})^{n−1}x_{0}}{x_{n}} )

and this last expression is equal to ξ̂_{n}x_{i} plus a polynomial (because x_{0} = x_{n}/q).

On the other hand,

x_{n}ξ̂_{n} = \overline{T}_{n−1} ⋯ \overline{T}_{1}τ(x_{0} − \frac{(−t_{2})^{n−1}x_{0}}{x_{n}} ) + (−t_{1}t_{2})^{n−1} .

In all, ξ̂_{n} commutes with polynomials, modulo addition of polynomials. Checking that

1 ξ̂_{n} = t_{1}^{n−1}(1 − \frac{(−t_{2})^{n−1}}{x_{n}} ) + \frac{(−t_{1}t_{2})^{n−1}}{x_{n}} = t_{1}^{n−1} ,

one concludes that ξ̂_{n} preserves polynomials. QED

There are also several Cherednik algebras relative to the degenerate affine Hecke algebra. We just state the analogs of the results that we described for the Hecke algebra, keeping the same notations for Cherednik’s elements because they will be used in a different context.

The first Cherednik’s elements are the isobaric version of Dunkl’s elements ([10]).

Let D_{i} be the isobaric derivative f ↦ x_{i} \frac{d}{dx_{i}} (f), and let \overline{π}_{i,j} denote \overline{π}_{x_{i},x_{j}}. Define, following [5],

ξ_{i} := αD_{i} + ∑_{j≠i,j=1}^{n} \overline{π}_{i,j} + 1 − i , 1 ≤ i ≤ n .  (44)

A second variety of elements involves the arbitrary function G_{n} of x_{n} (it can be obtained by degeneration of the ξ_{i}^{β}’s). Define

ξ̂_{i} := x_{i} − σ_{i−1} ⋯ σ_{1} τ G_{n} σ_{n−1} ⋯ σ_{i} 1 ≤ i ≤ n ,  (45)

with σ_{i} = s_{i} + ∂_{i} and τ still such that x_{i}τ = x_{i−1}, x_{0} = x_{n} − α.

Define moreover

Φ = s_{1} … s_{n−1} x_{n} , Φ̂ = τ G_{n} .

Then one has (cf. [5],[20] for the first part, [18] for the second part when G_{n} = x_{n} + n − 1):

Theorem 11 The ξ_{i}’s (resp. the ξ̂_{i}’s), 1 ≤ i ≤ n, generate a commutative algebra.

They satisfy

ξ_{i+1}s_{i} = s_{i} ξ_{i} − 1 & (ξ_{i} + ξ_{i+1})s_{i} = s_{i}(ξ_{i} + ξ_{i+1}) , 1 ≤ i ≤ n − 1 ,

ξ_{j}s_{i} = s_{i}ξ_{j} , j ≠ i, i + 1  (46)
ξ_{i} Φ = Φ ξ_{i−1} , i = 1, …, n, ξ_{0} = ξ_{n} − α .

ξ̂_{i+1}σ_{i} = σ_{i} ξ̂_{i} − 1 & (ξ̂_{i} + ξ̂_{i+1}) σ_{i} = σ_{i}(ξ̂_{i} + ξ̂_{i+1}) , 1 ≤ i ≤ n − 1 ,

ξ̂_{j} σ_{i} = σ_{i} ξ̂_{j} , j ≠ i, i + 1  (47)
ξ̂_{i} Φ̂ = Φ̂ ξ̂_{i−1} , i = 1, …, n, ξ̂_{0} = ξ̂_{n} − α .

8 Propagation of vanishing properties, and of eigenvectorness

Each edge of a cell in a Yang-Baxter graph is an operator on the ring of polynomials in only two variables, say x, y, as a rank 2 free-module over 𝔖ym(x, y). We shall easily check, in this section, that if a vertex is an eigenvector for Cherednik elements, or vanishes for some values of the x_{i}’s, then its neighbours in a cell have similar properties.

Choosing appropriate functions G_{n} of x_{n}, the same will be true for neighbours connected by an affine edge.

Lemma 12 Let f be a function of x_{1}, …, x_{n}, and i be an integer: 1 ≤ i ≤ n − 1. Then

1) if f is an eigenfunction of ξ_{i}^{β}, ξ_{i+1}^{β}, of respective eigenvalues a, b, then g := f (T_{i} + (t_{1} + t_{2})/(b/a − 1)) is an eigenvector of ξ_{i}^{β}, ξ_{i+1}^{β} with eigenvalues b, a.

2) if f is an eigenfunction of ξ_{1}^{β}, …, ξ_{n}^{β} of respective eigenvalues c_{1}, …, c_{n}, then all elements in the linear span of {f, f T_{i}} are eigenfunctions of ξ_{1}^{β}, …, ξ_{i−1}^{β}, ξ_{i+2}^{β}, …, ξ_{n}^{β} of respective eigenvalues c_{1}, …, c_{i−1}, c_{i+2}, …, c_{n}.

Proof.

fξ_{i+1}^{β}T_{i} = bfT_{i} = bf(T_{i} + \frac{t_{1} + t_{2}}{b/a − 1} − \frac{t_{1} + t_{2}}{b/a − 1}) = bg − b\frac{t_{1} + t_{2}}{b/a − 1}f

Similarly,

f\overline{T}_{i}ξ_{i}^{β} = f(T_{i} + \frac{t_{1} + t_{2}}{b/a − 1} + \frac{b(t_{1} + t_{2})}{a − b})ξ_{i}^{β} = gξ_{i}^{β} + f \frac{ab(t_{1} + t_{2})}{a − b} .

Equality ξ_{i+1}^{β}T_{i} = \overline{T}_{i}ξ_{i}^{β} implies that

gξ_{i}^{β} = g b , gξ_{i+1}^{β} = g a .

This settles the first point, the second being a direct consequence of the commutation relations (38,39,40). QED

Corollary 13 Let f be an eigenfunction of ξ_{1}^{β}, …, ξ_{n}^{β} of respective eigenvalues c_{1}, …, c_{n}. Then all elements of the space f H_{n} (in particular the symmetric function f□_{ω}) are eigenfunctions of ∏_{i=1}^{n}(1 + γξ_{i}) with eigenvalue ∏(1 + γc_{i}), for any γ ∈ ℂ.

Lemma 14 Let f(x, y) be a function of two variables.

1) Suppose that there exist a, b: a ≠ b such that f(a, b) = f(b, a) = 0. Then fT_{x,y}, fs_{x,y}, f∂_{x,y} also vanish in [x, y] = [a, b] and [x, y] = [b, a].

2) Suppose that f(a, b) = 0, with b = (−t_{2}/t_{1})a. Then fT_{x,y} also vanish in [x, y] = [a, b].

Suppose that f(a, b) = 0, with b = a − γ. Then f(s_{x,y} + γ∂_{x,y}) vanish in [x, y] = [a, b].

3) Suppose that f(b, a) = 0, a ≠ b. Then

f(T_{x,y} + \frac{t_{1} + t_{2}}{b/a − 1}) and f(s_{x,y} + γ∂_{x,y} + \frac{γ}{b − a})

vanish in [x, y] = [a, b]

Proof. Write f = f_{1} + yf_{2}, with f_{1}, f_{2} ∈ 𝔖ym(x, y).

If f(a, b) = f(b, a) = 0 and a ≠ b, then both f_{1} and f_{2} vanish in [a, b] and [b, a], and so do fT_{x,y}, fs_{x,y}, f∂_{x,y}. This proves the first point.

The expression fT_{x,y} = t_{1}f_{1} − t_{2}xf_{2} specializes, for [x, y] = [a, b] into t_{1}f_{1}(a, b) − t_{2}af_{2}(a, b) = t_{1}f(a, b), which is null in the second case. Similarly, f(s_{x,y} +γ∂_{x,y}) = f_{1} +xf_{2} −γf_{2} specializes into f_{1}(a, b)+(a−u)f_{2}(a, b), which is null when f(a, b) = 0.

In the third case,

f(T_{x,y} + \frac{t_{1} + t_{2}}{b/a − 1}) = (t_{1} + \frac{t_{1} + t_{2}}{b/a − 1})f_{1}(x, y) + (−t_{2}x + \frac{t_{1} + t_{2}}{b/a − 1}y)f_{2}(x, y)

specializes, for [x, y] = [a, b], into

(\frac{t_{1}b/a + t_{2}}{b/a − 1} )f_{1}(a, b) + (\frac{t_{1}b + t_{2}a}{b/a − 1}y)f_{2}(a, b) = af(b, a) ,

that is, vanishes.

Similarly

f(s_{x,y} + γ∂_{x,y} + \frac{γ}{b − a}) = f_{1}(1 + \frac{γ}{b − a}) + f_{2}(x − γ + \frac{γy}{b − a})

specializes, for [x, y] = [a, b], into af(b, a). QED

Propagation of vanishing properties and of eigenvectors along affine edges are insured by the choice of appropriate Φ’s, as shows the following lemma :

Lemma 15 1) Let f be an eigenfunction of the Cherednik elements ξ_{1}, …, ξ_{n}, with eigenvalues c_{1}, …, c_{n}. Let Φ = τG_{n}, with G_{n} commuting with multiplication by x_{n}. Then fΦ is an eigenfunction of ξ_{1}, …, ξ_{n}, with eigenvalues c_{2}, …, c_{n}, qc_{1} (in the case of the Hecke algebra) or c_{2}, …, c_{n}, c_{1} + α (in the case of the degenerate Hecke algebra).

2) Let k be an integer, and let f be a function vanishing in all c(u), |u| ≤ k. Let Φ = τG_{n}, with G_{n}((−t_{2})^{n−1}) = 0 (in the case of the Hecke algebra) and G_{n}(1 − n) = 0 (in the case of the degenerate Hecke algebra). Then f Φ vanishes in all c(u), |u| ≤ k + 1.

Proof. Commutation rules of Cherednik’s elements with Φ insure the first point. If f vanishes in c(u), then fτ vanishes in c(uτ). The vectors u : |u| ≤ k + 1 which are not in the image of τ are such that u_{n} = 0, in which case c_{n}(u) = (−t_{2})^{n−1} or 1 − n, depending on the algebra. In that case, the vanishing of G_{n} implies the vanishing of f Φ. QED

9 Macdonald polynomials

Take Yang-Baxter graph with

σ_{i} := T_{i} , 1 ≤ i ≤ n − 1 , c_{i}(▷) = t_{1}^{n−i}(−t_{2})^{i−1} , 1 ≤ i ≤ n , G_{n} = x_{n} ,

x_{i+n} = qx_{i} , c_{i+n}(▷) = qc_{i}(▷) , ∀i ∈ ℤ  (48)

Definition 16 The Macdonald polynomial M_{v}, v ∈ ℕ^{n}, is the image of the polynomial 1 under the element Y_{v} associated to the Yang-Baxter graph satisfying (48). In other words, one has:

M_{v} := 1 Y_{v} , ∀v ∈ ℕ^{n} .

The easiest characterization of these polynomials (cf. [36], [7]) is that they are eigenfunctions of Cheredenik’s elements ξ_{i} (for the same G), that is for

ξ_{i} := T_{i} … T_{n−1} \overline{τ} \overline{T}_{1} ⋯ \overline{T}_{i−1} .

Theorem 17 Let v ∈ ℕ^{n}. Then the Macdonald polynomial M_{v} is a simultaneous eigenfunction of the ξ_{i}’s, i.e. one has

M_{v} ξ_{j} = M_{v} c_{j}(v) , j : 1 ≤ j ≤ n .

Proof. Relations (35,36) ensure that M_{v} is an eigenfunction, with eigenvalue c(v), if it is true for the origin v = ▷. Indeed, 1 T_{i} = t_{1}, 1 \overline{T}_{i} = −t_{2} implies that 1 ξ_{i} = t_{1}^{n−i}(−t_{2})^{i−1} = c_{i}(▷). QED

Mimachi and Noumi give in [37] a reproducing kernel for the Macdonald polynomials.

The labelling of the affine edges of the Yang-Baxter graph by Φ = τ x_{n} is due to Sahi [40] and Knop [18]. They call Φ a “creation operator”. Baker and Forrester [2] similarly define a lowering operator Φ̂ :

Φ̂ = (T_{n−1} ⋯ T_{1} τ − t_{1}^{n−1})\frac{1}{x_{n}} \overline{T}_{n−1} ⋯ \overline{T}_{1} .  (49)

Lemma 18 Φ̂ preserves polynomials and sends the Macdonald polynomial M_{v} onto a multiple of M_{v\overline{τ}} if v_{n} > 0, and onto 0 otherwise.

Proof. Let \overline{ξ}_{n} := T_{n−1} ⋯ T_{1} τ. Then \overline{ξ}_{n} ξ_{n} = (−t_{1}t_{2})^{n−1} and \overline{ξ}_{n} − t_{1}^{n−1} annihilates all M_{v}, with v_{n} = 0 (because in that case, c_{n}(v) = (−t_{2})^{n−1}). Therefore Φ̂ annihilates those Macdonald polynomials.

Since Φ has zero kernel, we shall evaluate ΦΦ̂ to determine the action of Φ̂ on the other Macdonald polynomials.

ΦΦ̂ = τ x_{n} T_{n−1} ⋯ T_{1}τ \frac{1}{x_{n}} \overline{T}_{n−1} ⋯ \overline{T}_{1} − t_{1}^{n−1}τ x_{n} \frac{1}{x_{n}} \overline{T}_{n−1} ⋯ \overline{T}_{1}
= τ \overline{T}_{n−1} ⋯ \overline{T}_{1} (x_{1}τ \frac{1}{x_{n}}) \overline{T}_{n−1} ⋯ \overline{T}_{1} − t_{1}^{n−1}τ \overline{T}_{n−1} ⋯ \overline{T}_{1}
= \frac{1}{q} \overline{ξ}_{1}^{2} − t_{1}^{n−1}\overline{ξ}_{1} ,

with \overline{ξ}_{1} := τ \overline{T}_{n−1} ⋯ \overline{T}_{1}. Therefore Φ̂ preserves polynomials, and sends M_{v} on a non-zero multiple of M_{v\overline{τ}} if v_{n} ≠ 0. QED

A simpler lowering operator is

ˆΦ̂ := (T_{n−1} ⋯ T_{1} τ − t_{1}^{n−1})\frac{1}{x_{n}} \overline{τ} .  (50)

Since the eigenvalues of M_{v} under the ξ_{j}’s are permuted of those of M_{vs_{i}}, any linear combination of M_{v}, where the v’s are in the orbit under 𝔖_{n}, is an eigenfunction of any symmetric function in the ξ_{j}. In particular, let Ξ := ξ_{1} + ⋯ + ξ_{n}. One has the following theorem ([36], [7]).

Theorem 19 For each partition λ ∈ ℕ^{n}, there exists a unique polynomial M_{λ}^{𝔖}, symmetrical in x_{1}, …, x_{n}, whose expansion in the basis of Schur functions is of the type M_{λ}^{𝔖} = S_{λ} + ∑_{μ<λ} c_{λ}^{μ}S_{μ} and which is an eigenfunction of Ξ (the sum being over all partitions smaller than λ with respect to the dominance order).

Given any v in the 𝔖_{n}-orbit of λ, then up to multiplication by a scalar (we write ˙=), one has

M_{λ}^{𝔖} ˙=M_{v} ∑_{w∈𝔖_{n}}(−t_{2})^{\binom{n}{2}−ℓ(w)}T_{w} = M_{v} ∏_{1≤i<j<≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω} = M_{v} □_{ω} .

Proof. Since Ξ commutes with □_{ω}, M_{v} □_{ω} is a symmetric eigenfunction of Ξ with eigenvalue q^{λ_{1}}t_{1}^{n−1} + ⋯ + q^{λ_{n}}(−t_{2})^{n−1} which characterizes the partition λ. One can check that the ξ_{i}’s have a triangular action on polynomials by putting an appropriate order on monomials, related to the Bruhat order on the affine symmetric group. We refer for that point to [20] and [17]. Thus M_{λ}^{𝔖} is not only characterized by an eigenvalue, but also by a leading term. QED

The first polynomials, in the basis of Schur functions are, for two variables

M_{0,0}^{𝔖} = 1, M_{1,0}^{𝔖} = S_{1}, M_{2,0}^{𝔖} ˙=(qt_{1} + t_{2})S_{2} + (qt_{2} + t_{1})S_{1,1}, M_{1,1}^{𝔖} = S_{1,1}

M_{3,0}^{S} ˙=(t_{2} + q^{2}t_{1})S_{3} − (q + 1)(t_{1} + qt_{2})S_{2,1},

M_{4,0}^{𝔖} ˙=(t_{2} + q^{3}t_{1})(t_{2} + q^{2}t_{1})S_{4} + (1 + q + q^{2})(t_{2} + q^{2}t_{1})(t_{1} + qt_{2})S_{3,1} + (q + q^{3})(t_{1} + t_{2})(t_{1} + qt_{2})S_{2,2}

From the expression of the ξ_{i}’s one sees that all □_{ω} ξ_{i} □_{ω}, and thus also □_{ω} Ξ □_{ω}, □_{ω} Ξ, are proportional to □_{ω} \overline{τ} □_{ω} . But now, □_{ω} acts as a scalar on symmetric function, and therefore the action of Ξ(1 □_{ω}) on symmetric functions is just

f ↦ f τ □_{ω} = f τ ∏_{1≤i<j<≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω}

= f(x_{2}, …, x_{n}, qx_{1}) ∏_{1≤i<j<≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω} , (51)

which is the usual expression of the Debiard-Sekiguchi-Macdonald operator.

Creation (or raising) operators on symmetric Macdonald polynomials are similar operators ([23])

B_{k} := (t_{1}τ − (−t_{2})) ⋯ (t_{1}^{k}τ − (−t_{2})^{k}) x_{1} ⋯ x_{k} □_{ω} .

The image of M_{λ}^{𝔖}, when ℓ(λ) ≤ k, is proportional to M_{λ+1^{k}}^{𝔖} . Other creation operators are given in [26], [14], [15], [16].

10 Vanishing Macdonald polynomials

Take Yang-Baxter graph with

σ_{i} := T_{i} , 1 ≤ i ≤ n − 1 , c_{i}(▷) = t_{1}^{n−i}(−t_{2})^{i−1} , 1 ≤ i ≤ n , G_{n} = x_{n} − (−t_{2})^{n−1} ,

x_{i+n} = qx_{i} , c_{i+n}(▷) = qc_{i}(▷) , ∀i ∈ ℤ  (52)

Definition 20 The vanishing Macdonald polynomial M̂_{v}, v ∈ ℕ^{n}, is the image of 1 under the element Y_{v} associated to the Yang-Baxter graph satisfying (52):

M̂_{v} := 1 Y_{v} .

The appropriate Cherednik’s elements are [18]

ξ̂_{i} := T_{i−1} ⋯ T_{1} τ(1 − \frac{(−t_{2})^{n−1}}{x_{n}} )\overline{T}_{n−1} ⋯ \overline{T}_{i} + \frac{(−t_{1}t_{2})^{n−1}}{x_{i}} , 1 ≤ i ≤ n

Theorem 21 Let v ∈ N^{n}, and k = |v|. Let Pol(v) be the subspace of polynomials f of degree ≤ |v| such that

∀u ∈ ℕ^{n}, |u| ≤ |v|, u ≠ v ⇒ f(c(u)) = 0 .

Then Pol(v) is 1-dimensional and contains M̂_{v}.

The polynomial M̂_{v} is an eigenfunction of the ξ̂_{i}’s with eigenvalues

(−t_{1}t_{2})^{n−1}/c_{1}(v), …, (−t_{1}t_{2})^{n−1}/c_{n}(v) .

The component of top degree of M̂_{v} is M_{v}.

The proof is the same as for Macdonald polynomials for what concerns eigenfunctions. One checks that 1 is an eigenfunction of the ξ̂_{i}’s with eigenvalues (−t_{2})^{n−1}, (−t_{2})^{n−2}t_{1}, …, t_{1}^{n−1}, because 1T_{i} = t_{1}, 1\overline{T}_{i} = −t_{2}. On the other hand, it is not difficult to see that Pol(v) is 1-dimensional. The choice of G_{n} = x_{n}−(−t_{2})^{n−1} ensures that vanishing propagates along an affine edge.

Restricting G_{n} to its top-degree component x_{n} gives M_{v} instead of M̂_{v}. In the case n = 2, the first polynomials, and the vectors c(v) are

\begin{matrix}[0, 0] \\ [t_{1}, −t_{2}] \\ 1 \end{matrix}⟶^{Φ}\begin{matrix}[0, 1] \\ [−t_{2}, qt_{1}] \\ x_{2} + t_{2} \end{matrix}⟶\begin{matrix}[1, 0] \\ [qt_{1}, −t_{2}] \\ ^{−} \frac{(t_{1}+t_{2})t_{2}(x_{2}+t_{2})}{qt_{1}+t_{2}}^{−}t_{2}(x_{1}^{−}t_{1}) \end{matrix}⟶^{Φ}\begin{matrix}[0, 2] \\ [−t_{2}, q^{2}t_{1}] \\ t_{2}(x_{2}^{+}t_{2})(t_{1}^{−} \frac{x_{2}}{q}^{−} \frac{t_{1}+t_{2}}{qt_{1}+t_{2}} (x_{1}^{+}t_{2})) \end{matrix}

Given any v ∈ ℕ^{n}, there is a unique symmetric polynomial (up to normalization) in the linear span of M̂_{v w}, w ∈ 𝔖_{n}. This polynomial is obtained as the image of M̂_{v} under

□_{ω} = ∑_{w∈𝔖_{n}}(−t_{2})^{\binom{n}{2}−ℓ(w)}T_{w} = ∏_{1≤i<j<≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω}

and clearly inherits from M̂_{v} vanishing and eigenfunction properties, which are stated in the next theorem.

Theorem 22 Let n be an integer, λ ∈ ℕ^{n} be a partition, v any vector in its orbit under S_{n}. Then

M̂_{v}^{𝔖} := M̂_{v} ∏_{1≤i<j<≤n}(t_{1}x_{i} + t_{2}x_{j}) ∂_{ω}

is, up to a scalar, the unique symmetric polynomial vanishing in all c(μ), μ partition ∈ ℕ^{n}, |μ| ≤ |λ|, μ ≠ λ.

It is also the unique eigenfunction of ξ̂_{1} + ⋯ + ξ̂_{n} with eigenvalue

(−t_{2})^{n−1}q^{−λ_{1}} + (−t_{2})^{n−2}t_{1}q^{−λ_{2}} + ⋯ + t_{1}^{n−1}q^{−λ_{n}} .

For n = 2, one has the following polynomials, in terms of Schur functions:

M̂_{0,0}^{𝔖} = 1 , M̂_{1,0}^{𝔖} = S_{1} + t_{2} − t_{1} , M̂_{1,1}^{𝔖} = S_{1,1} + t_{2}S_{1} + t_{2}^{2} ,

M̂_{2,0}^{𝔖} ˙=(qt_{1} + t_{2})S_{2} + (qt_{2} + t_{1})S_{1,1} + (q + 1)(t_{2}^{2} − qt_{1}^{2})S_{1} + q(t_{1} − t_{2})(qt_{1}^{2} − t_{2}^{2}) .

11 Jack polynomials

Take Yang-Baxter graph with

σ_{i} := s_{i} , 1 ≤ i ≤ n − 1 , c_{i}(▷) = 1 − i , 1 ≤ i ≤ n , G_{n} = x_{n} ,
x_{i+n} = x_{i} , c_{i+n}(▷) = c_{i}(▷) + α , ∀i ∈ ℤ  (53)

Definition 23 The non symmetric Jack polynomial J_{v}, v ∈ ℕ^{n}, is the image of 1 under the Yang-Baxter element Y_{v} given by the graph satisfying (53):

J_{v} := 1 Y_{v} .

For example, for n = 2, one has the following sequence of polynomials, writing v, c(v), J_{v} on top of each other as before, and writing σ for σ_{1} = s_{1}:

\begin{matrix}[0, 0] \\ [0,^{−}1] \\ 1 \end{matrix}⟶^{Φ}\begin{matrix}[0, 1] \\ [^{−}1, α] \\ x_{2} \end{matrix}⟶^{σ}\begin{matrix}[1, 0] \\ [α,^{−}1] \\ x_{1} + \frac{x_{2}}{1+α} \end{matrix} ⟶^{Φ}\begin{matrix}[0, 2] \\ [^{−}1, 2α] \\ x_{2}(x_{2} + \frac{x_{1}}{1+α}) \end{matrix}⟶^{σ}\begin{matrix}[2, 0] \\ [2α,^{−}1] \\ x_{1}^{2} + \frac{2x_{1}x_{2}+x_{2}^{2}}{1+2α} \end{matrix} ⟶^{Φ} ⋯

The symmetric polynomials are obtained, up to normalization, as images of non-symmetrical ones under Δ(X)∂_{ω} = ∑_{w∈𝔖_{n}} w, i.e. for any partition λ ∈ ℕ, any v in the orbit of λ under 𝔖_{n}, one has

J_{λ}^{𝔖} = J_{v} Δ(X) ∂_{ω} = ∑_{w∈𝔖_{n}}(J_{v})^{w} .

One checks that 1 is an eigenvector of the Cherednik elements (defined in (44))

ξ_{i} := αD_{i} + ∑_{j≠i,j=1}^{n} \overline{π}_{i,j} + 1 − i , 1 ≤ i ≤ n ,

with eigenvalues 0,^{−}1, …,^{−}n^{+}1. These values are our initial spectral parameters, and therefore the commutation relations (46) imply the following theorem due to Opdam ( [39], [20]).

Theorem 24 For any v ∈ ℕ^{n}, the polynomial J_{v} is a simultaneous eigenfunction of ξ_{1}, …, ξ_{n}, with eigenvalues c_{1}(v), …, c_{n}(v).

For any partition λ ∈ ℕ^{n}, and any symmetric function f ∈ 𝔖ym(n), the Jack polynomial J_{λ}^{𝔖} is an eigenvector of f(ξ_{1}, …, ξ_{n}) with eigenvalue f(c_{1}(λ), …, c_{n}(λ)).

Baker and Forrester [1] give another symmetrization, using Sahi’s scalar product. Introduce another set of variables Y = {y_{1}, …, y_{n}}. Then, according to Sahi [41], the kernel

Ω(X, Y) := ∏_{i=1}^{n}(1 − x_{i}y_{i})^{−1} ∏_{i=1}^{n} ∏_{j=1}^{n}(1 − x_{i}y_{j})^{−1/α}

diagonalizes in the basis of Jack polynomials: there exist non-zero constants m_{v} such that

Ω(X, Y) = ∑_{v∈ℕ^{n}} m_{v}J_{v}(X)J_{v}(Y) .

Recall that, thanks to the Cauchy determinant,

∏_{i=1}^{n}(1 − x_{i}y_{i})^{−1}∂_{ω}^{X} = Δ(Y) ∏_{i,j}(1 − x_{i}y_{j})^{−1} ;

this last function is sent, under ∂_{ω}^{Y} onto n! ∏_{i,j}(1 − x_{i}y_{j})^{−1}.

Therefore, the image of Ω(X, Y) under ∂_{ω}^{X}∂_{ω}^{Y} is n! ∏_{i,j}(1 − x_{i}y_{j})^{−1−1/α}. This last function diagonalizes in the basis of symmetric Jack polynomials, but this time for the parameter 1/(1 + 1/α) = α/(1 + α).

Therefore, all J_{vw}∂_{ω}, for a fixed v ∈ ℕ^{n}, coincide, up to non zero-scalars, and are proportional to the same symmetric Jack polynomial ( one has to check that the polynomial of index a permutation of [λ_{1}^{+}n^{−}1, …, λ_{n}^{+}0] is sent to the polynomial of index the (decreasing) partition λ. As for normalization, one can check that, for a weakly increasing u ∈ ℕ^{n},

J_{u_{1},u_{2}+1,...,u_{n}+n−1}∂_{ω} = ∑w^{′}(J_{u})^{w} ,

where the sum on the right is over cosets representatives of 𝔖_{n} modulo the stabilizer of u, and the polynomial J_{u} is for the parameter α/(1 + α). For example,

J_{014; α}∂_{321} = J_{002; α/(1+α)}(1 + s_{2} + s_{2}s_{1}) .

Knop and Sahi [20] show that symmetric Jack polynomials can be obtained through still another symmetrization :

Lemma 25 Let λ be a partition of length k ≤ n, and v be any permutation of [λ_{1} − 1, …, λ_{k} − 1, 0^{n−k}]. Then

J_{λ}^{𝔖} = ∑_{w∈𝔖_{n}}(J_{v} x_{1} ⋯ x_{k})^{w} .

Proof.

J_{0^{n−k},λ_{1},...,λ_{k}} = J_{λ_{1}−1,...,λ_{k}−1,0^{n−k}} τ^{k} x_{n} ⋯ x_{n−k+1} = J_{λ_{1}−1,...,λ_{k}−1,0^{n−k}} x_{k} ⋯ x_{1} τ^{k}.

The image of the left hand-side under ∑ w is J_{λ}^{𝔖}. However,

∑_{w∈𝔖_{k}×𝔖_{n−k}}(J_{λ_{1}−1,...,λ_{k}−1,0^{n−k}})^{w} = ∑_{w∈𝔖_{k}×𝔖_{n−k}}(J_{v})^{w}

and this allows to replace J_{λ_{1}−1,...,λ_{k}−1,0^{n−k}} by J_{v} in the summation on 𝔖_{n}. QED

There is an abundant literature about symmetric Jack polynomials (cf. the references given in [35]). They can be obtained through “creation operators” [25]. Their simplest explicit expression seems to be a determinant given in [24]. They are eigenfunctions of the Sekiguchi operator

(D_{1} + (n^{−}1)β + t)(D_{2} + (n^{−}2)β + t) ⋯ (D_{n} + 0β + t) x_{1}^{n−1} ⋯ x_{n}^{0} ∂_{ω} ,

D_{i} still being f ↦ x_{i} \frac{d}{dx_{i}} (f), and β = 1/α.

12 Vanishing Jack polynomials

Take Yang-Baxter graph with

σ_{i} := s_{i} + ∂_{i} , 1 ≤ i ≤ n − 1 , c_{i}(▷) = 1 − i , 1 ≤ i ≤ n , G_{n} = x_{n} + n − 1 ,

x_{i+n} = x_{i} + α , c_{i+n}(▷) = c_{i}(▷) + α , ∀i ∈ ℤ  (54)

and let the Cherednik algebra be generated by the

ξ̂_{i} := x_{i} − σ_{i−1} ⋯ σ_{1} τ (x_{n}^{+}n^{−}1) σ_{n−1} ⋯ σ_{i} , 1 ≤ i ≤ n .  (55)

Definition 26 The vanishing Jack polynomial Ĵ_{v}, v ∈ ℕ^{n}, is the image of 1 under the Yang-Baxter element Y_{v} associated to graph satisfying (54):

Ĵ_{v} := 1 Y_{v} .

For example, for n = 2, one has the following sequence of polynomials :

\begin{matrix}[0, 0] \\ [0,^{−}1] \\ 1 \end{matrix}⟶^{Φ}\begin{matrix}[0, 1] \\ [^{−}1, α] \\ x_{2}^{+}1 \end{matrix}⟶^{σ}\begin{matrix}[1, 0] \\ [α,^{−}1] \\ x_{1}^{+} \frac{x_{2}+1}{1+α} \end{matrix} ⟶^{Φ}\begin{matrix}[0, 2] \\ [−1, 2α] \\ (x_{2}^{+}1)(x_{2}^{−}α^{+} \frac{x_{1}+1}{1+α}) \end{matrix}⟶^{σ}\begin{matrix}[2, 0] \\ [2α,^{−}1] \\ x_{1}(x_{1}^{−}α)^{+} \frac{(x_{2}+1)(2x_{1}+x_{2}+1−3α)}{1+2α} \end{matrix}

Knop and Sahi [19] have shown that these polynomials can be characterized by vanishing conditions. Let Pol_{k} be the space of polynomials in x_{1}, …, x_{n} of degree ≤ k.

Theorem 27 Let v ∈ N^{n}, and k = |v|. Let Pol(v) be the subspace of polynomials f of degree ≤ |v| such that

∀u ∈ ℕ^{n}, |u| ≤ |v|, u ≠ v ⇒ f(c(u)) = 0 .

Then Pol(v) is 1-dimensional and contains Ĵ_{v}.

The polynomial Ĵ_{v} is an eigenfunction of the ξ̂_{i}’s with eigenvalues c(v). The restriction of Ĵ_{v} to its component of top degree ( = |v|) is equal to

J_{v}.

Proof. The dimension of Pol_{k} is equal to the number of equations +1, this accounts for the existence of a vanishing polynomial, unicity follows by putting an appropriate order on monomials. Moreover, restricting s_{i} + ∂_{i} and G_{n} = x_{n} + n − 1 to their terms of highest degree shows that the leading term of Ĵ_{v} is J_{v}.

The labels on the edges of cells have been chosen to propagate vanishing conditions, at the same time as eigenvector properties. The choice of G = x_{n} + n − 1 insures that for any f, fΦ vanishes in c(v) for any v such that v_{n} = 0 (in which case c_{n}(v) = 1 − n).

Because of the commutation relations (47), it is sufficient to test that 1 is an eigenfunction for ξ̂_{n}, with the right eigenvalue, to get that all ξ̂_{i} diagonalize. Indeed

1 ξ̂_{n} = x_{n} − 1σ_{n−1} ⋯ σ_{1} τ (x_{n} + n − 1) = −n + 1 .

QED

Given any v ∈ ℕ^{n}, the linear span of Ĵ_{vw}, w ∈ 𝔖_{n} contains, up to normalization, a unique symmetric polynomial (27)

Ĵ_{v}^{𝔖} := ∑_{w∈𝔖_{n}} Ĵ_{v} σ_{w} = Ĵ_{v} ∏_{1≤i,j≤n}(1 + x_{i} − x_{j}) ∂_{ω}

(Sahi and Knop give another symmetrization), and this symmetric polynomial inherits from Ĵ_{v} the properties stated by the following theorem :

Theorem 28 Let n be an integer, λ ∈ ℕ^{n} be a partition, v any vector in its orbit under 𝔖_{n} and let γ be an indeterminate. Then

Ĵ_{v}^{𝔖} := Ĵ_{v} ∏_{1≤i<j<≤n}(1 + x_{i} − x_{j}) ∂_{ω}

is, up to a scalar, the unique symmetric polynomial vanishing in all c(u), u ∈ ℕ^{n}, |u| ≤ |v|, u not a permutation of v.

It is also the unique eigenfunction of (γ + ξ̂_{1}) ⋯ (γ + ξ̂_{n}) with eigenvalue

(γ + αλ_{1} − 0)(γ + αλ_{2} − 1) ⋯ (γ + αλ_{n} − n^{+}1) .

For n = 2 the first polynomials, in the basis of Schur functions, are

Ĵ_{0,0}^{𝔖} = 1 , Ĵ_{1,0}^{𝔖} = S_{1} + 1 , Ĵ_{1,1}^{𝔖} = S_{1,1} + S_{1} + 1 ,

Ĵ_{2,0}^{𝔖} = (α + 1)S_{2} + (1 − α)S_{1,1} + ((1 − α)(2 + α)S_{1} − (α^{2} + 2α − 1)

Ĵ_{3,0}^{𝔖} = (2α + 1)S_{3} − 2(α − 1)S_{2,1} − 3(α + 1)(2α − 1)S_{2} + 3(2α − 1)(α − 1)S_{1,1} + (−12α + 4α^{3} + 2α^{2} + 3)S_{1} + (2α − 1)(2α^{2} + 5α − 1) .

The symmetric Jack polynomials specialize, for α = 1, into Schur functions. Similarly, vanishing symmetric Jack polynomials specialize into some Schubert polynomials. Indeed, Schubert polynomials are polynomials in two sets of variables, indexed by permutations, or by integral vectors (which are the codes of the permutations), which have vanishing properties related to the Bruhat order [32]. In the special case that we need now, let λ ∈ ℕ^{n} be a partition and X_{λ} be the Schubert polynomial in x_{1}, …, x_{n}, y_{1}, …, y_{∞}, of index the increasing reordering of λ (it is a Grassmannian Schubert polynomial [34]). Then X_{λ} vanishes in every partition μ ∈ ℕ: |μ| ≤ |λ|, μ ≠ λ, i.e X_{λ}, evaluated in

x_{1} = μ_{1} + n − 1, …, x_{n} = μ_{n} + 0 , y_{1} = 0, y_{2} = 1, y_{3} = 2, …, y_{i} = i^{−}1, …

is null. Therefore, one has the following lemma (stated by Knop and Sahi [19] in terms of factorial Schur functions; in the literature, X_{λ} is rather denoted 𝕐_{λ_{n},...,λ_{1}}).

Lemma 29 For any partition λ ∈ ℕ^{n}, the specialization α=1 of Ĵ_{λ}^{𝔖} is equal to the Schubert polynomial X_{λ} evaluated in {x_{1} + n − 1, …, x_{n} + n − 1}, {0, 1, 2, …}.

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