Macdonald polynomials at t=qk

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Jean-Gabriel Luque

To cite this version:

Jean-Gabriel Luque. Macdonald polynomials at t=qk. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) , Jul 2009, Hagenberg, Austria. �hal-00485888�

Macdonald polynomials at t = q

Jean-Gabriel Luque

Universit´e de Paris Est, Institut d’ ´Electronique et d’Informatique Gaspard-Monge 77454 Marne-la-Vall´ee Cedex 2.

Abstract. We investigate the homogeneous symmetric Macdonald polynomialsPλ(X;q, t) for the specialization t=q^k. We show an identity relying the polynomialsPλ(X;q, q^k)andPλ

“1−q

1−q^kX;q, q^k”

. As a consequence, we describe an operator whose eigenvalues characterize the polynomialsPλ(X;q, q^k).

R´esum´e.Nous nous int´eressons aux propri´et´es des polynˆomes de Macdonald sym´etriquesPλ(X;q, t)pour la sp´ecia- lisationt=q^k. En particulier nous montrons une ´egalit´e reliant les polynˆomesPλ(X;q, q^k)etPλ

“_1−q

1−q^kX;q, q^k” . Nous en d´eduisons la description d’un op´erateur dont les valeurs propres caract´erisent les polynˆomesPλ(X;q, q^k).

Keywords:Symmetric functions, Macdonald polynomials,q-discriminant

1 Introduction

The Macdonald polynomials are(q, t)-deformations of the Schur functions which play an important rˆole in the representation theory of the double affine Hecke algebra [11, 13] since they are the eigenfunctions of the Cherednik elements. More precisely, the non-symmetric Macdonald polynomials are the eigen- functions of the Cherednik elements, but the symmetric Macdonald polynomials are the eigenfunctions of the symmetric functions in the Cherednik elements. The polynomials considered here are the ho- mogeneous symmetric Macdonald polynomialsPλ(X;q, t)and are the eigenfunctions of the Sekiguchi- Debiard-Macdonald operatorM1. For(q, t)generic, the dimension of each eigenspace equals1and each Macdonald polynomial is characterized (up to a multiplicative constant) by the associated eigenvalue of M1. That is no longer true whentis specialized to a rational power ofq(note that the case of the spe- cializationtⁿq^m= 1-nandmbeing integer-has been investigated by Feigin etal.[4] in their study of ideals of symmetric functions defined by vanishing conditions). Hence, it is more convenient to charac- terize the Macdonald (homogeneous symmetric) polynomials by orthogonality (w.r.t.a (q, t)-deformation of the usual scalar product on symmetric functions) and by some conditions on their dominant monomials (seee.g.[12]). In this paper, we consider the specializationt=q^kwherekis a (strictly) positive integer.

One of our motivations is to generalize an identity of [1], which shows that even powers of the discrim- inant are rectangular Jack polynomials. Here, we show that this property follows from deeper relations between the Macdonald polynomialsPλ(X;q, q^k)andPλ

1−q

1−q^kX;q, q^k

(in theλ-ring notation). This result is interesting in the context of the fractional quantum Hall effect [8], since it implies properties of the expansion of the powers of the discriminant in the Schur basis [3, 6, 14]. It implies also that the

1365–8050 c2009 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Macdonald polynomials (att =q^k) are characterized by the eigenvalues of an operatorM(described in terms of isobaric divided differences) whose eigenspaces are of dimension1.

The paper is organized as follows. After recalling notations and background (Section 2) related to Mac- donald polynomials, we give, in Section 3, some properties of the operator which substitutes a complete function to each power of a letter. These properties allow us to show our main result in Section 4 which is an identity involving the polynomialPλ(X;q, q^k)andPλ

1−q

1−q^kX;q, q^k

. As a consequence, we describe (Section 5) an operatorMwhose eigenvalues characterize the Macdonald polynomialsPλ(X;q, q^k). Fi- nally, in Section 6, we give an expression ofMin terms of the Cherednik elements.

2 Notations and background

We recall here the basic definitions and classical properties of the symmetric functions and the Macdonald polynomials.

2.1 Symmetric functions

Consider an alphabetX(potentially infinite). Following [10] we define the symmetric functions onXby the generating functions of the complete homogeneous functionsS^p(X),

σz(X) :=X

i

Sⁱ(X)zⁱ=Y

x∈X

1 1−xz.

The algebraSymof symmetric functions has aλ-ring structure [10] and many properties of that structure can be understood by manipulatingσ_z. For example, the sum of two alphabetsX+Yis defined by the product

σz(X+Y) :=σz(X)σz(Y) =X

i

Sⁱ(X+Y)zⁱ.

In particular, ifX=Yone hasσz(2X) =σz(X)². This definition is extended to any complex numberα byσ_z(αX) =σ_z(X)^α. For example, the generating series of the elementary functions is

λz(X) := P

Λi(X)zⁱ=Y

x∈X

(1 +xz)

= σ_−z(−X) =P

i(−1)ⁱSⁱ(−X)zⁱ.

The complete functions of the product of two alphabetsXYare given by the Cauchy kernel K(X,Y) :=σ1(XY) =X

i

Sⁱ(XY) =Y

x∈X

Y

y∈Y

1

1−xy =X

λ

Sλ(X)Sλ(Y),

whereS_λdenotes, as in [10], a Schur function. More generally, one has K(X,Y) =X

λ

Aλ(X)Bλ(Y)

for any pair of bases(Aλ)λ and(Bλ)λ in duality for the usual scalar producth,i,i.e. K(X,Y)is the reproducing kernel associated toh,i.

2.2 Macdonald polynomials

The usual scalar product on symmetric functions admits a(q, t)-deformation (seee.g.[12]) defined for a pair of power sum functionsΨ^λandΨ^µ(in the notation of [10]) by

hΨ^λ,Ψ^µiq,t=δλ,µzλ l(λ)

Y

i=1

1−q^λi

1−t^λi, (1)

whereδλ,µ = 1ifλ=µand0otherwise. The family of (symmetric homogeneous) Macdonald polyno- mials(Pλ(X;q, t))λis the unique basis of the symmetric functions orthogonalw.r.t.h, iq,tverifying

Pλ(X;q, t) =mλ(X) +X

µ≤λ

uλµmµ(X), (2)

wherem_λ denotes, as usual, a monomial function [10, 12]. The reproducing kernel associated to this scalar product is

Kq,t(X,Y) :=X

λ

hΨ^λ,Ψ^λi⁻¹_q,tΨλ(X)Ψλ(Y) =σ1

1−t 1−qXY

seee.g.[12, VI.2]. In particular, one has

Kq,t(X,Y) =X

λ

Pλ(X;q, t)Qλ(Y;q, t), (3) whereQ_λ(X;q, t)is the dual basis ofP_λ(Y;q, t)with respect toh,i_q,t,

Qλ(X;q, t) =hPλ, Pλi⁻¹_q,tPλ(X;q, t). (4) The coefficientbλ(q, t) =hPλ, Pλi⁻¹_q,tis known to be

bλ(q, t) = Y

(i,j)∈λ

1−q^λj−i+1t^λ0i−j

1−q^λj−it^λ0i−j+1 (5)

see [12, VI.6]. Writing

Kq,t

1−q 1−t

X,Y

=K(X,Y), (6)

one finds that P_λ

1−q 1−t

X;q, t

λ

is the dual basis of(Q_λ(X;q, t))_λwith respect to the usual scalar producth, i.

Note that there exists an other Kernel type formula which reads λ1(XY) =X

λ

Pλ⁰(X;t, q)Pλ(Y;q, t) =X

λ

Qλ⁰(X;t, q)Qλ(Y;q, t). (7) whereλ⁰denotes the conjugate partition ofλ. This formula can be found in [12, VI.5 p329].

From equalities (6) and (3) , one has σ1(XY) =Kq,t

1−q 1−tX,Y

=X

λ

Qλ

1−q 1−tX;q, t

Pλ(Y;q, t). (8) Applying (7) to

σ₁(XY) =λ₋₁(−XY), one obtains

σ1(XY) =X

λ

(−1)^|λ|Qλ⁰(−X;t, q)Qλ(Y;q, t). (9) Identifying the coefficient ofPλ(Y;t, q)in (8) and (9), one finds the following property.

Lemma 2.1

Qλ(−X;t, q) = (−1)^|λ|Pλ⁰

1−q 1−tX;q, t

. (10)

Unlike the usual (q=t = 1) scalar product, there is no expression as a constant term for the product h,iq,twhenX={x1, . . . , xn}is finite. But the Macdonald polynomials are orthogonal with respect to an other scalar product defined by

hf, gi⁰_q,t;n= 1

n!C.T.{f(X)g(X^∨)∆_q,t(X)} (11) whereC.T.denotes the constant termw.r.t.the alphabetX,

∆q,t(X) = Y

i6=j

(xix⁻¹_j ;q)_∞ (txix⁻¹_j ;q)∞

,(a;b)_∞ = Y

i≥0

(1−abⁱ)andX^∨ = {x⁻¹₁ , . . . , x⁻¹_n }. The expression of hP_λ, Q_λi⁰_q,t;nis given by ([12, VI.9])

hPλ, Qλi⁰_q,t;n= 1

n!C.T.{∆q,t(X)} Y

(i,j)∈λ

1−qⁱ⁻¹t^n−j+1

1−qⁱt^n−j . (12)

2.3 Skew symmetric functions

Let us define as in [12, VI.7], the skew Macdonald functionsQ_λ/µby

hQ_λ/µ, Pνiq,t:=hQλ, PµPνiq,t. (13) Straightforwardly, one has

Q_λ/µ(X;q, t) =X

ν

hQ_λ, P_νP_µi_q,tQ_ν(X;q, t). (14) And classically, the following property holds (seee.g.[12, VI.7] for a short proof of this identity), Proposition 2.2 LetXandYbe two alphabets, one has

Q_λ(X+Y;q, t) =X

µ

Q_µ(X;q, t)Q_λ/µ(Y;q, t),

or equivalently

Pλ(X+Y;q, t) =X

µ

Pµ(X;q, t)Pλ/µ(Y;q, t).

Equalities (3) and (7) are generalized by identities (15) and (16) as shown in [12, example 6 p.352], X

ρ

Pρ/λ(X;q, t)Qρ/µ(Y;q, t) =Kqt(X,Y)X

ρ

Pµ/ρ(X;q, t)Qλ/ρ(Y;q, t), (15) X

ρ

Qρ⁰/λ⁰(X;t, q)Qρ/µ(Y;q, t) =λ1(XY)X

ρ

Qµ⁰/ρ⁰(X, t, q)Qλ/ρ(Y;q, t). (16)

3 The substitution x^p →S^p(Y) and the Macdonald polynomials

LetX={x1, . . . , xn}be a finite alphabet andYbe an other (potentially infinite) alphabet. For simplicity we will denote byR

Ythe substitution Z

Y

: x^p→S^p(Y), (17)

for eachx∈Xand eachp∈Z. Let us define the symmetric function H^n,k_λ/µ(Y;q, t) := 1

n!

Z

Y

P_λ(X;q, t)Q_µ(X^∨;q, t)∆(X, q, t) (18) whereX^∨={x⁻¹₁ , . . . , x⁻¹_n }.

SetY^tq:=_1−q^1−tYand consider the substitution Z

Y^tq

x^p=S^p Y^tq

=Qp(Y;q, t). (19)

The following result shows thatH^n,k_λ/µis a skew Macdonald polynomial on a suitable alphabet.

Theorem 3.1 LetX= {x1, . . . , xn}be an alphabet,λ= (λ1, . . . , λn)be a partition andµ ⊂λ. The polynomialH^n,k_λ/µ(Y^tq;q, t)is the Macdonald polynomial

H^n,k_λ/µ(Y^tq;q, t) = 1 n!

Y

(i,j)∈λ

1−qⁱ⁻¹t^n−j+1

1−qⁱt^n−j C.T.{∆(X, q, t)}P_λ/µ(Y, q, t) (20) SetY={−y1, . . . ,−ym, . . .}ifY={y1, . . . , ym, . . .}and note that the operationY→Ymakes sense even for virtual alphabet since it sends any homogeneous symmetric polynomialP(Y)of degree pto (−1)^pP(Y). One observes the following phenomenon which is obtained from Theorem 3.1 by applying the operations of theλ-ring structure.

Corollary 3.2 LetX={x1, . . . , xn}be an alphabet,λ= (λ1, . . . , λn)be a partition andµ⊂λ. One has

H^n,k_λ/µ(−Y;q, t) = 1 n!

Y

(i,j)∈λ

1−qⁱ⁻¹t^n−j+1

1−qⁱt^n−j C.T.{∆(X, q, t)}Q_λ⁰_/µ⁰(Y, t, q). (21)

Note that in the case of partitions, one has Corollary 3.3

H^n,k_λ (−Y, q, t) = 1 n!

Y

(i,j)∈λ

1−qⁱ⁻¹t^n−j+1

1−qⁱt^n−j C.T.{∆(X, q, t)}Qλ⁰(Y, t, q) (22) Example 3.4 Consider the following equality

H^2,3_41/3(−Y;q, t) = (∗)C.T.{∆(X, q, t)}Q2111/111(Y;t, q).

whereX = {x1, x₂}. The coefficient (∗) is computed as follows. One writes the partition[41] in a rectangle of height2and length4.

×

× × × ×

Each×of coordinates(i, j)is read as the fraction[i, j] := ^1−q_1−qⁱ⁻¹it^2−jt3−j. Hence

(∗) = [1,2][1,1][2,1][3,1][4,1] = (1−t)(1−t²)(1−qt²)(1−q²t²)(1−q³t²) (1−q)(1−qt)(1−q²t)(1−q³t)(1−q⁴t)

4 A formula involving the polynomialsP_λ

1−q

1−q^kX;q, q^k

andP_λ X;q, q^k

Now, we suppose thatt = q^k withk ∈ N. In that case, the constant termC.T.{∆(X, q, t)}admits a closed form and Corollary 3.3 gives

Corollary 4.1

H^n,k_λ (−Y, q, q^k) =β_λ^n,k(q)Q_λ⁰(Y;q^k, q). (23) where

β_λ^n,k(q) =

n−1

Y

i=0

λ_n−i−1 +k(i+ 1) k−1

q

and hn

p

i

q

= ^(1−q(1−q)...(1−q^{n)...(1−qn−p+1r}) ⁾denotes theq-binomial.

Example 4.2 Setk= 2, n= 3and consider the polynomial H^3,2_[320](−Y;q, q²) = 1

n!

Z

−Y

P_[32](x1+x2+x3;q, q²)Y

i6=j

(1−xix⁻¹_j )(1−qxix⁻¹_j ).

One has

H^3,2_[320](−Y;q, q²) = 1−q⁵

1−q⁸

(1−q)² Q_[221](Y;q², q).

Let

Ω_S := 1 n!

Z

X

Y

i6=j

(1−x_ix⁻¹_j ) (24)

and for eachv∈Zⁿ,

S˜_v(X) = det

x^v_i^j+n−j Y

i<j

(x_i−x_j)⁻¹.

Lemma 4.3 Ifvis any vector inZⁿ, one has

ΩSS˜v(X) =Sv(X) := det(S^vi−i+j(X)) (25) In particular,ΩSleaves invariant any symmetric polynomial. The operator

Am:= ΩSΛⁿ(X)^−m (26) acts on symmetric polynomials by substracting m from each part of the partitions appearing in their expansion in the Schur basis.

Example 4.4 IfX={x1, x2, x3}andλ= [320], one has P₃₂(X;q, t) =S₃₂(X) +(−q+t)S311(X)

qt−1 +(q+ 1) qt²−1

(−q+t)S221(X) (qt−1)²(qt+ 1) . Hence,

A1P32(X;q, t) = ^(−q+t)S_qt−1^2(X)+^(q+1)(^qt2−1)^(−q+t)S11(X) (qt−1)²(qt+1)

= (−q+t)(t+1)(^q2t−1)^P11(X;q,t)

(qt−1)²(qt+1) +^(−q+t)P_qt−1^2(X;q,t). Theorem 4.5 Ifλdenotes a partition of length at mostn, one has

A_{(k−1)(n−1)}Pλ(X;q, q^k)

k−1

Y

l=1

Y

i6=j

(xi−q^lxj) =β_λ^n,k(q)Pλ

1−q 1−q^kX;q, q^k

(27)

Example 4.6 Setk= 2,n= 3andλ= [2]. One has P_[2](x1+x2+x3;q, q²)Y

i6=j

(xi−qxj) =−q³S_[6,2]+q²q³−1 q−1 S_[6,1,1]

+q²(q⁵−1)

q³−1 S_[5,3]−q(q²+ 1)(q⁵−1)

q³−1 S_[5,2,1]−q(q⁷−1)

q³−1 S_[4,3,1]+q⁷−1 q−1S_[4,2,2]. And,

A2P[2](x1+x2+x3;q, q²)Y

i6=j

(xi−qxj) =q⁷−1 q−1 S[2]. Since,

P_[2]

x1+x2+x3

1 +q ;q, q²

= q−1 q³−1S_[2]

one obtains

A2P_[2](x1+x2+x3;q, q²)Y

i6=j

(xi−qxj) = 1

1

q

3 1

q

7 1

q

P_[2]

x1+x2+x3

1 +q ;q, q²

.

As a consequence, one has

Corollary 4.7 Ifλ=µ+ [((k−1)(n−1))ⁿ],

Pµ(X;q, q^k)

k−1

Y

l=1

Y

i6=j

(xi−q^lxj) =β_λ^n,k(q)Pλ

1−q

1−q^kX;q, q^k

.

Example 4.8 Setk= 3,n= 2andλ= [5,2]. One has

P_[5,2](x₁+x₂;q, q³)(x₁−qx₂)(x₁−q²x₂)(x₂−qx₁)(x₂−q²x₁) = q³S_[9,2]+(1−q⁷)(1 +q⁴)

1−q⁵ S_[7,4]−(1−q²)(1 +q)(1 +q²)(1 +q⁴)

1−q⁵ S_[8,3].

This implies

A2P_[5,2](x1+x2;q, q³)(x1−qx2)(x1−q²x2)(x2−qx1)(x2−q²x1) = (x₁x₂)⁻²P_[5,2](x₁+x₂;q, q³)(x₁−qx₂)(x₁−q²x₂)(x₂−qx₁)(x₂−q²x₁) = P_[3](x1+x2;q, q³)(x1−qx2)(x1−q²x2)(x2−qx1)(x2−q²x1).

One verifies that

P_[3](x₁+x₂;q, q³)(x₁−qx₂)(x₁−q²x₂)(x₂−qx₁)(x₂−q²x₁) = 4

2

q

10 2

q

P_[5,2]( x1+x2

1 +q+q²;q, q³).

Remark 4.9 Ifµis the empty partition, Corollary 4.7 gives

k−1

Y

l=1

Y

i6=j

(xi−q^lxj) =β_λ^n,k(q)P[((k−1)(n−1))ⁿ]

1−q 1−q^kX;q, q^k

. (28)

This equality generalizes an identity given in [1]:

Y

i<j

(xi−xj)^2(k−1)=(−1)((k−1)n(n−1) 2

n!

kn k, . . . , k

P_n^(k)_{(n−1)(k−1)}(−X),

whereP_λ^(k)(X) = lim

q→1P_λ^(α)(X;q, q^k)denotes a Jack polynomial (seee.g.[12]).

The expansion of the powers of the discriminant and theirq-deformations in different basis of symmetric functions is a difficult problem having many applications, for example, in the study of Hua-type integrals (seee.g.[5, 7]) or in the context of the fractional quantum Hall effect (e.g.[3, 6, 8, 14]).

Note that in [2], we gave an expression of an otherq-deformation of the powers of the discriminant as staircase Macdonald polynomials. This deformation is also relevant in the study of the expansion of Q

i<j(xi−xj)^2kin the Schur basis (for example, we generalized in [2] a result of [6]).

5 Macdonald polynomials at t = q^k as eigenfunctions

LetY={y1, . . . , ykn}be an alphabet of cardinalityknwithy1=x1, . . . , yn=xn. One considers the symmetrizerπωdefined by

πωf(y1, . . . , ykn) =Y

i<j

(xi−xj)⁻¹ X

σ∈S_kn

sign(σ)f(y_σ(1), . . . , y_σ(kn))y^kn−1_σ(1) . . . yσ(kn−1).

Note thatπωis the isobaric divided difference associated to the maximal permutationωinSkn.

This operator applied to a symmetric function of the alphabetXincreases the alphabet fromXtoYin its expansion in the Schur basis, since

π_ωS_λ(X) =S_λ(Y). (29)

Indeed, the image of the monomialy₁ⁱ¹. . . y_kn^ikn is the Schur functionSI(Y). Since πωSλ(X) =πωx^λ₁¹. . . x^λ_nⁿ=πωy^λ₁¹. . . y_n^λny⁰_n+1. . . y⁰_kn, one recovers equality (29).

One defines the operatorπ^tqwhich consists in applyingπ_ωand specializing the result to the alphabet X^tq :={x1, . . . , xn, qx1, . . . , qxn, . . . , q^k−1x1, . . . , q^k−1xn}.

From equality (29), one has

π^tq_ωSλ(X) =Sλ (1 +q+. . .+q^k−1)X

, (30)

forl(λ)≤n. Furthermore, the expansion ofSλ (1 +q+. . .+q^k−1)X

in the Schur basis is triangular, so the operator π^tq defines an automorphism of the space Sym≤n generated by the Schur functions indexed by partitions whose length are less or equal ton,i.e.for each functionf ∈Sym_≤n, one has

π^tqf(X) =f(X^tq). (31)

In particular,

Lemma 5.1 Letλbe a partition such thatl(λ)≤nthen π_ω^tqPλ

1−q

1−q^kX;q, t=q^k

=Pλ(X, q, q^k). (32)

Consider the operatorM:f →Mf defined by

M:= (x₁. . . x_n)^{(k−1)(1−n)}π_ω^tq

k−1

Y

l=1

Y

i6=j

(x_i−q^lx_j).

The following theorem shows that the Macdonald polynomials are the eigenfunctions of the operatorM.

Theorem 5.2 The Macdonald polynomialsPλ(X;q, q^k)are eigenfunctions ofM. The eigenvalue asso- ciated toP_µ(X;q, q^k)isβµ+((k−1)(n−1))^{n,k n}(q). Furthermore, ifk >1, the dimension of each eigenspace is1.

Example 5.3 Ifn= 5, the eigenvalues associated to the partitions of4are

β4,k

[4k,4k−4,4k−4,4k−4,4k−4] = h5k−5 k−1 i

q h6k−5

k−1 i

q h7k−5

k−1 i

q h8k−5

k−1 i

q h9k−1

k−1 i

q(λ= [4,0,0,0,0]), β4,k

[4k−1,4k−3,4k−4,4k−4,4k−4] = h5k−5

k−1 i

q h6k−5

k−1 i

q h7k−5

k−1 i

q h8k−4

k−1 i

q h9k−2

k−1 i

q(λ= [3,1,0,0,0]), β4,k

[4k−2,4k−2,4k−4,4k−4,4k−4] = h5k−5 k−1 i

q h6k−5

k−1 i

q h7k−5

k−1 i

q h8k−3

k−1 i

q h9k−3

k−1 i

q(λ= [2,2,0,0,0]), β4,k

[4k−2,4k−3,4k−3,4k−4,4k−4] = h5k−5

k−1 i

q h6k−5

k−1 i

q h7k−4

k−1 i

q h8k−4

k−1 i

q h9k−3

k−1 i

q(λ= [2,1,1,0,0]), β4,k

[4k−3,4k−3,4k−3,4k−3,4k−4] = h5k−5

k−1 i

q h6k−4

k−1 i

q h7k−4

k−1 i

q h8k−4

k−1 i

q h9k−4

k−1 i

q(λ= [1,1,1,1,0]).

6 Expression of Min terms of the Cherednik elements

In this paragraph, we restate Proposition 5.2 in terms of Cherednik operators. Cherednik’s operators {ξi;i∈ {1, . . . , n}}=: Ξare commutative elements of the double affine Hecke algebra. The Macdonald polynomialsP_λ(X;q, t)are eigenfunctions of symmetric polynomialsf(Ξ)and the eigenvalues are ob- tained substituting each occurrence ofξ_iinf(Ξ)byq^λitⁿ⁻ⁱ(see [11] for more details).

Suppose thatk >1and consider the operatorM˜ :f →Mf˜ defined by M˜ :=

k−1

Y

i=1

(1−qⁱ)ⁿM. (33)

From Proposition 5.2, one has

MP˜ λ(X;q, q^k) =

n−1

Y

i=0 k−1

Y

j=1

(1−q^{λn−i+k(i+1)−j})Pλ(X;q, q^k). (34)

The following proposition shows thatM˜ admits a closed expression in terms of the Cherednick elements.

Proposition 6.1 One supposes thatk >1. For any symmetric functionf, one has

Mf˜ (X) =

k−1

Y

l=1 n

Y

i=1

(1−q^l+kξi)f(X). (35)

AcknowledgmentsThe author is grateful to Alain Lascoux and Jean-Yves Thibon for fruitful discussions.

I also thank Christophe Tollu for its corrections.

References

[1] H. Belbachir, A. Boussicault, J.-G. Luque, Hankel hyperdeterminants, rectangular Jack polynomials and even power of the Vandermonde, Journal of Algebra320(2008) 3911-3925.

[2] A. Boussicault, J.G. Luque, Staircase Macdonald polynomials and the q-Discriminant Formal Power Series and Algebraic Combinatorics June 23-27, 2008 Valparaiso Vi˜na del Mar - Chile, preprint arXiv:0801.2443.

[3] P. Di Francesco, M. Gaudin, C. Itzykson, F. Lesage, Laughlin’s wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A9(1994) 4257-4351.