Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde

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Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde

Hacene Belbachir, Adrien Boussicault, Jean-Gabriel Luque

To cite this version:

Hacene Belbachir, Adrien Boussicault, Jean-Gabriel Luque. Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde. Journal of Algebra, Elsevier, 2008, 320 (11), pp.3911-3925. �hal-00173319�

hal-00173319, version 1 - 19 Sep 2007

Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the

Vandermonde

H. Belbachir

, A. Boussicault

and J.-G. Luque

September 19, 2007

Abstract

We investigate the link between rectangular Jack polynomials and Hankel hyperdeterminants. As an application we give an expression of the even power of the Vandermonde in term of Jack polynomials.

1 Introduction

Few after he introduced the modern notation for determinants [2], Cayley proposed several extensions to higher dimensional arrays under the same name hyperdeterminant [3, 4]. The notion considered here is apparently the simplest one, defined for a kth order tensor M = (Mi1···ik)1≤i1,...,ik≤n on an n-dimensional space by

Det M = 1 n!

X

σ=(σ1,···,σk)∈S^k_n

sign(σ)M^σ,

∗Universit´e des Sciences et de la Technologie Houari Boumediene. BP 32 USTHB 16111 Bab-Ezzouar Alger, Alg´erie. email: hacenebelbachir@gmail.com

†Universit´e de Paris-Est Marne-la-Valle, Institut d’´Electronique et d’Informatique Gaspard-Monge 77454 Marne-la-Valle Cedex 2. email: Adrien.Boussicault@univ-mlv.fr

‡Universit´e de Paris-Est Marne-la-Valle, Institut d’´Electronique et d’Informatique Gaspard-Monge 77454 Marne-la-Valle Cedex 2. email: Jean-Gabriel.Luque@univ-mlv.fr

where sign(σ) = sign(σ₁)· · ·sign(σ_k) is the product of the signs of the per- mutations, M^σ =Mσ1(1)...σk(1)· · ·Mσ1(n)...σk(n) andS_ndenotes the symmetric group. Note that others hyperdeterminants are found in literature. For ex- ample, those considered by Gelfand, Kapranov and Zelevinsky [7] are bigger polynomials with geometric properties. Our hyperdeterminant is a special case of the riciens [26, 21] with only alternant indices. For an hypermatrice with an even number of indices, it generates the space of the polynomial in- variants of lowest degree. Easily, one obtains the nullity of Det whenk is odd and its invariance under the action of the group SL^×k_n . Few references exists on the topics see [26, 29, 30, 11, 1, 8] and before [22] the multidimensional analogues of Hankel determinant do not seem to have been investigated.

In this paper, we discuss about the links between Hankel hyperdeter- minsants and Jack’s symmetric functions indexed by rectangular partitions.

Jack’s symmetric functions are a one parameter (denoted by α in this pa- per) generalization of Schur functions. They were defined by Henry Jack in 1969 in the aim to interpolate between Schur fonctions (α = 1) and zonal polynomials (α = 2) [15, 16]. The story of Jack’s polynomials is closely related to the generalizations of the Selberg integral [1, 12, 14, 19, 27, 22].

The relation between Jack’s polynomials and hyperdeterminants appeared implicitly in this context in [22], when one of the author with J.-Y. Thibon gave an expression of the Kaneko integral [12] in terms of Hankel hyperde- terminant. More recently, Matsumoto computed [25] an hyperdeterminantal Jacobi-Trudi type formula for rectangular Jack polynomials.

The paper is organized as follow. In Section 2, after we recall defini- tions of Hankel and Toeplitz hyperdeterminants, we explain that an Hankel hyperdeterminant can be viewed as the umber of an even power of the Van- dermonde via the substitution R

Y : xⁿ → Λⁿ(Y) where Λⁿ(Y) denotes the nth elementary symmetric functions on the alphabetY. Section 3 is devoted to the generalization of the Matsumoto formula [25] to almost rectangular Jack polynomials. In Section 4, we give an equality involving the substitu- tion R

Y and skew Jack polynomials. As an application, we give in Section 5 expressions of even powers of the Vandermonde determinant in terms of Schur functions and Jack polynomials.

2 Hankel and Toeplitz Hyperdeterminants of symmetric functions

2.1 Symmetric functions

Symmetric functions over an alphabet X are functions which are invariant under permutation of the variables. The C-space of the symmetric functions over X is an algebra which will be denoted bySym(X).

Let us consider the complete symmetric functions whose generating series is

σt(X) :=X

i

Sⁱ(X)tⁱ =Y

x∈X

1 1−xt, the elementary symmetric functions

λt(X) :=X

i

Λⁱ(X)tⁱ = Y

x∈X

(1 +xt) =σ−1(X)⁻¹,

and power sum symmetric functions ψt(X) := X

i

Ψⁱ(X)tⁱ

i = log(σt(X)).

When there is no algebraic relation between the letters ofX, Sym(X) is a free (associative, commutative) algebra over complete, elementary or power sum symmetric functions

Sym=C[S¹, S²,· · ·] =C[Λ¹,Λ²,· · ·] =C[Ψ¹,Ψ²,· · ·].

As a consequence, the algebraSym(X) (Xbeing infinite or not) is spanned by the set of the decreasing products of the generators

S^λ =S^λn. . . S^λ1, Λ^λ = Λ^λn. . .Λ^λ1, Ψ^λ = Ψ^λn. . .Ψ^λ1, where λ= (λ1 ≥λ₂ ≥ · · · ≥λ_n) is a (decreasing) partition.

The algebra Sym(X) admits also non multiplicative basis. For example, the monomial functions defined by

m_λ(X) =X

x^λ_i1¹· · ·x^λ_inⁿ,

where the sum is over all the distinct monomialsx^λ_i1¹· · ·x^λ_inⁿ withx_i1, . . . , x_in ∈ X, and the Schur functions defined via the Jacobi-Trudi formula

Sλ(X) = det(S^λi−i+j(X)). (1) Note that the Schur basis admits other alternative definitions. For example, it is the only basis such that

1. It is orthogonal for the scalar product defined on power sums by hΨ_λ,Ψ_µi=δ_λ,µz_λ (2) whereδ_µ,ν is the Kronecker symbol (equal to 1 ifµ=νand 0 otherwise) andzλ =Q

ii^mi(λ)mi(λ)! ifmi(λ) denotes the multiplicity of ias a part of λ.

2. The coefficient of the dominant term in the expansion in the monomial basis is 1,

S_λ =m_λ+X

µ<λ

u_λµm_µ.

When X = {x1, . . . , xn} is finite, a Schur function has another determi- nantal expression

Sλ(X) = det(x^λ_i^j+n−j)

∆(X) , where ∆(X) = Q

i<j(xi−x_j) denotes the Vandermonde determinant.

2.2 Definitions and General properties

A Hankel hyperdeterminant is an hyperdeterminant of a tensor whose entries depend only of the sum of the indices Mi1,...,i_2k =f(i1+· · ·+i2k). One of the authors investigated such polynomials in relation with the Selberg integral [22, 23].

Without lost of generality, we will consider the polynomials H^k_n(X) = Det Λ^{i1+···+i2k}(X)

0≤i1,...,i2k≤n−1, (3) where Λm(X) is themth elementary function on the alphabet X.

Let us consider a shifted version of Hankel hyperdeterminants H^k_v(X) = Det Λ^{i1+···+i2k+vi1}(X)

0≤i1,...,i_2k≤n−1. (4)

where v = (v₀, . . . , v_n−1)∈Zⁿ. Note that (3) implies M_0,...,0 = Λ₀(X) = 1 by convention. But if M0,...,0 6= 0,1, this property can be recovered using a suit- able normalization and, if M_0...0 = 0, by using the shifted version (4) of the Hankel hyperdeterminant. As in [25], one defines To¨eplitz hyperdeterminant by giving directly the shifting version

T^k_v(X) = Det Λ^{i1+···+ik−(ik+1+···+i2k)+vi1}(X)

0≤i1,...,i_2k≤n−1. (5) To¨eplitz hyperdeterminants are related to Hankel hyperdeterminants by the following formulae.

Proposition 2.1 1. H^k_v(X) = (−1)^kn(n2−1)T^k_v+(k(n−1))n(X) 2. T^k_v(X) = (−1)^kn(n2−1)H^k_v+(k(1−n))n(X).

Proof The equalities (1) and (2) are equivalent and are direct consequences of the definitions (4) and (5),

T^k_v(X) = Det Λ^{i1+···+ik+(n−1−ik+1}+···+(n−1−i_2k−k(n−1))(X)

= (−1)^kn(n−1)2 H_v+(k(1−n))n(X).

.

2.3 The substitution x

→ Λ

( Y )

Let X = {x₁, . . . , x_n} be a finite alphabet and Y be another (potentially infinite) alphabet. For simplicity we will denote by R

Y the substitution Z

Y

x^p = Λ^p(Y), for each x∈Xand each p∈Z.

The main tool of this paper is the following proposition.

Proposition 2.2 For any integer k ∈N− {0}, one has 1

n!

Z

Y

∆(X)^2k =H^k_n(Y) where ∆(X) =Q

i<j(x_i−x_j).

Proof It suffices to develop the power of the Vandermonde determinant

∆(X)^2k = det x^j−1_i ^2k

= X

σ1,···,σ2k∈S_n

sign(σ1· · ·σ2k)Y

i

x^σ_i^{1(i)+···+σ2k(i)−2k}.

Hence, applying the substitution, one obtains 1

n!

Z

Y

∆(X)^2k = 1 n!

X

σ1,···,σ_2k∈S_n

sign(σ1· · ·σ_2k)Y

i

Λ^{σ1(i)+···+σ2k(i)−2k}(Y)

= H^k_n(Y).

More generally, the Jacobi-Trudi formula (1) implies the following result.

Proposition 2.3 One has 1 n!

Z

Y

Sλ(X)∆(X)^2k=H^k_reverse

n(λ)(Y),

where reversen(v) = (vn, . . . , v1) if v = (v1, . . . , vp) is a composition with p≤n and v_p+i = 0for 1≤i≤n−p.

Proof It suffices to remark that

Sλ(X)∆(X)^2k = det(x^λ_i^{n−j+1+j−1}) det x^j−1_i 2k−1

,

and apply the same computation than in the proof of Proposition 2.2.

Example 2.4 If k= 1 then using the second Jacobi-Trudi formula

Sλ = det(Λ^{λ′n−i−i+j}) (6) where λ^′ denotes the conjugate partition ofλ, Proposition 2.3 implies

1 n!

Z

Y

S_λ(X)∆(X)² = (−1)^n(n−1)2 S_(λ+(n−1)ⁿ₎′(Y).

3 Jack Polynomials and Hyperdeterminants

In this section, we will consider the symmetric functions as aλ-ring endowed with the operatorSⁱ, and we will use the definition of addition and multiplica- tion of alphabets in this context (seee.g. [18]). LetXandYbe two alphabets, the symmetric functions over the alphabet X+Y are generated by the com- plete functionsSⁱ(X+Y) defined byσ_t(X+Y) =σ_t(X)σt(Y) =P

iSⁱ(X+Y)tⁱ. If X = Y, one has σt(2X) := σt(X+X) = σt(X)². Similarly one defines σ_t(αX) =σ_t(X)^α. In particular, the equality σ_t(−X) =Q

x(1−xt) =λ_−t(X) gives Sⁱ(−X) = (−1)ⁱΛⁱ(X). The product of two alphabet X and Y is de- fined by σt(XY) = P

Sⁱ(XY)tⁱ = Q

x∈X

Q

y∈Y 1

1−xyt. Note that σ1(XY) = K(X,Y) = P

λS_λ(X)Sλ(Y) is the Cauchy Kernel.

3.1 Jack polynomials

One considers a one parameter generalization of the scalar product (2) defined byhΨλ,Ψµiα =δλ,µzλα^l(λ),wherel(λ) =ndenotes the length of the partition λ = (λ1 ≥ · · · ≥ λ_n) with λ_n > 0. The Jack polynomials P_λ^(α) are the unique symmetric functions orthogonal for h,iα and such thatP_λ^(α) =mλ+ P

µ<λu^(α)_λµm_µ. Note that in the case when α = 1, one recovers the definition of Schur functions,i.e. P_λ⁽¹⁾ =Sλ. Let (Q^(α)_λ ) be the dual basis of (P_λ^(α)). The polynomials P_λ^(α) and Q^(α)_λ are equal up to a scalar factor and the coefficient of proportionality is computed explicitly in [24] VI. 10:

b^(α)_λ := P_λ^(α)

Q^(α)_λ =hP_λ^(α), P_λ^(α)i⁻¹ = Y

(i,j)∈λ

α(λi−j) +λ^′_j −i+ 1

α(λi−j + 1) +λ^′_j−i. (7) LetX={x1,· · · , x_n} be a finite alphabet and denote by X^∨ the alphabet of the inverse {x⁻¹₁ , . . . , x⁻¹_n }. Let us introduce the second scalar product by

hf, gi^′_n,α = 1

n!C.T.{f(X)g(X^∨)Y

i6=j

(1−xix⁻¹_j )^1α},

see [24] VI. 10. The polynomials P_λ^(α) and Q^(α)_λ are also orthogonal for this scalar product.

For simplicity, we will consider also another normalisation defined by R^(α),n_λ (Y) :=hP_λ^(1/α)′ , Q^(1/α)_λ′ i^′_n,1

αQ^(α)_λ (Y).

Note that the polynomial R^(α),n_λ is not zero only when l(λ)≤ n and in this case the value of the coefficient hP_λ^(1/α)′ , Q^(1/α)_λ′ i^′_n,1

α

is known to be

hP_λ^(1/α)′ , Q^(1/α)_λ′ i^′_n,1 α

= 1 n!

Y

(i,j)∈λ^′

n+ _α¹(j −1)−i+ 1 n+_α^j −i C.T.

( Y

i6=j

1−x_ix⁻¹_j α

)

(8) see [24] VI 10.

3.2 The operator Z

Y

and almost rectangle Jack poly- nomials

Suppose that X = {x1, . . . , xn} is a finite alphabet. Let Y = {y1,· · · } be another (potentially infinite) alphabet and consider the integral

I_n,k(Y) = 1

n!C.T.{Λⁿ(X^∨)^p+k(n−1)Λ^l(X^∨)Y

(1 +x_iy_j)∆(X)^2k}. (9) Note that,

In,k(Y) = (−1)^kn(n−1)2

n! C.T.{Λⁿ(X^∨)^pΛ^l(X^∨)Y

(1 +xiyj)Y

i6=j

(1− xi

x_j)^k}.

(10) But

Λⁿ(X^∨)^pΛ^l(X^∨) =P_(p+1)^(1/k)lp^n−l(X^∨), (11)

and Y

(1 +x_iy_j) =X

λ

Q^(1/k)_λ (X)Q^(k)_λ′ (Y). (12) Hence, from the orthogonality ofP_λ^(α) andQ^(α)_λ , equalities (10), (11) and (12) imply

I_n,k(Y) = (−1)^kn(n2−1)R^(k),n_npl (Y). (13) On the other hand, one has the equality

Z

Y

x^m = Λ_m(Y) = C.T.{x^−mY

i

(1 +xy_i)}.

Remarking that ∆(X^∨) = (−1)^n(n2−1)_Λn^∆(X)(X)ⁿ⁻¹,and Λⁿ(X)^mΛ^l(X) = S_(mⁿ₎₊₍₁^l₎(X) for each m∈Z, Equality (9) can be written as

I_n,k(Y) = 1 n!

Z

Y

S((p−k(n−1))ⁿ)+(1^l)(X)∆(X)^2k = (−1)^kn(n−1)2 T^k

p^n−l(p+1)^l(X).

(14) One deduces an hyperdeterminantal expression for a Jack polynomial indexed by the partition n^pl.

Proposition 3.1 For any positive integers n, p, l and k, one has.

R^(k),n_npl =T^k_pn−l(p+1)^l.

The constant term appearing in (8) is a special case of the the Dyson Con- jecture [6]. The conjecture of Dyson has been proved the same year indepen- dently by Gunson [10] and Wilson [31] ( in 1970 I. J. Good [9] have shown an elegant elementary proof involving Lagrange interpolation),

C.T.Y

i6=j

1−x_ix⁻¹_j ai

=

a1+· · ·+an

a₁, . . . , a_n

,

for a₁, . . . , a_n∈N. Hence, one has Q^(k)_npl(Y) = n!

kn k,· · · , k

−1

κ(n, p, l;k)T^k_pn−l(p+1)^l(Y) where

κ(n, p, l;k) =

n

Y

i=1 p

Y

j=1

j +k(i−1) j −1 +ki

l

Y

i=1

p+ 1 +k(n−i) p+k(n−i+ 1). In particular, whenl = 0, one recovers a theorem by Matsumoto.

Corollary 3.2 (Matsumoto [25]) P_n^(k)p (Y) =n!

kn k, . . . , k

−1

T^k_pn(Y).

Proof From equalities (8) and (7), one has hP_p^(1/k)n , Q^(1/k)_pⁿ i^′_1/k,n= 1

n!

kn k, . . . , k

hP_n^(k)p_l, P_n^(k)plik. Applying Proposition 3.1, one finds the result.

Setting p = k(n−1), one obtains the expression of an Hankel hyperde- terminant as a Jack polynomials.

Corollary 3.3

H^k_n(Y) = (−1)^kn(n−1)2 n!

kn k, . . . , k

P_n^(k)_k(n−1)(Y).

3.3 Jack polynomials with parameter α =

Let Y = {y1, y₂,· · · } be a (potentially infinite alphabet). Consider the endomorphism defined on the power sums symmetric functions Ψ_p(Y) by ωα(Ψp(Y)) := Ψp(−αY) = (−1)^p−1αΨp(Y) (see [24] VI 10), where Y = {−y1,−y2,· · · } . This map is known to satisfy the identities

ωαP_λ^(α)(Y) =Q⁽

1 α) λ^′ (Y) and

ω_αΛⁿ(Y) = g₍ⁿ1

α)(Y) := Λⁿ(−αY).

Applyingωk on Proposition 3.1, one obtains the expression of a Jack polyno- mial with parameter α = ¹_k for an almost rectangular shape λ= (p+ 1)^lp^n−l as a shifted Toeplitz hyperdeterminant whose entries are

Mi1...i_2k =gⁱ1^{1+···+ik−ik+1−···−i2k+λn−i1+1} k

.

Proposition 3.4 One has P⁽

1 k)

(p+1)^lp^n−l(Y) =n!

kn k, . . . , k

−1

κ(n, p, l;k)T^(k)

p^n−l(p+1)^l(−kY).

Proof It suffices to apply Proposition 3.1 with the alphabet −αY to find Q^(k)_npl(−kY) =n!

kn k, . . . , k

−1

κ(n, p, l;k)T^(k)

p^n−l(p+1)^l(−kY).

The result follows from

Q^(k)_npl(−kY) =ω_kQ^(k)_npl(Y) =P_(p+1)^(1k) lp^n−l(Y).

4 Skew Jack polynomials and Hankel hyper- determinants

4.1 Skew Jack polynomials

Let us define as in [24] VI 10, the skew Q functions by hQ^(α)_λ/µ, P_ν^(α)i:=hQ^(α)_λ , P_µ^(α)P_ν^(α)i.

Straightforwardly, one has

Q^(α)_λ/µ =X

ν

hQ^(α)_λ , P_ν^(α)P_µ^(α)iQ^(α)_ν . (15) Classically, the skew Jack polynomials appear when one expands a Jack polynomial on a sum of alphabet.

Proposition 4.1 Let X and Y be two alphabets, one has Q^(α)_λ (X+Y) =X

µ

Q^(α)_µ (X)Q^(α)_λ/µ(Y), or equivalently

P_λ^(α)(X+Y) =X

µ

P_µ^(α)(X)P_λ/µ^(α)(Y).

Proof See [24] VI.7 for a short proof of this identity.

An other important normalisation is given by

J_λ^(α)=c_λ(α)P_λ^(α)=c^′_λ(α)Q^(α)_λ , where

cλ(α) = Y

(i,j)∈λ

(α(λi−j) +λ^′_j−i+ 1), and

c^′_λ(α) = Y

(i,j)∈λ

(α(λi−j+ 1) +λ^′_j −i), if λ^′ denotes the conjugate partition ofλ.

If one defines skew J function by J_λ/µ^(α) :=X

ν

hJ_λ^(α), Jµ^(α)Jν^(α)i_α hJν^(α), Jν^(α)iα

J_ν^(α)

then J_λ/µ^(α) is again proportional to P_λ/µ^(α) and Q^(α)_λ/µ :

J_λ/µ^(α) =c_λ(α)c^′_µ(α)P_λ/µ^(α) =c^′_λ(α)c_µ(α)Q^(α)_λ/µ. (16)

4.2 The operator R

Y

and the skew Jack symmetric functions

LetX,Yand Zbe three alphabets such that♯X=n <∞and ♯Z=m <∞.

Consider the polynomial In,k(Y,Z) = 1

n!

Z

Y

Y

i

x^−m_i Y

i,j

(xi+zj)∆^2k(X).

Remarking that Z

Y

x^p−mY

i

(x+zi) =

m

X

i=0

Λ^p+i−m(Y)Λ^m−i(Z)

= Λ^p(Y+Z)

= Z

Y+Z

x^p, one obtains

In,k(Y,Z) = H^k_n(Y+Z)

= ⁽⁻¹⁾

kn(n−1) 2

n!

nk k,...,k

P_n^(k)_k(n−1)(Y+Z). (17) Hence, the image of Q(k¹)

λ (X^∨)∆(X)^2k by Z

Y

is a Jack polynomial.

Corollary 4.2 One has, Z

Y

Q(¹k)

λ (X^∨)∆(X)^2k= (−1)^kn(n−1)2

nk k, . . . , k

(b^(k)_nk(n−1))⁻¹Q^(k)_{nk(n−1)/λ′}(Y), where b^(k)_nk(n−1) = (2(n−1))!(nk)!((n−1)k)!

kn!(n−1)!((2n−1)k−1)!.

Proof The equality follows from Y

i

x^−m_i Y

i,j

(xi+zj) =Y

i,j

(1 + zj

x_i) =X

λ

Q^(k)_λ′ (Z)Q(¹k)

λ (X^∨).

Indeed, one has

I_n,k(Y,Z) = 1 n!

X

λ

b^(k)_λ′ P_λ^(k)′ (Z) Z

Y

Q(¹k)

λ (X^∨)∆(X)^2k. And in the other hand, by (17) one obtains

I_n,k(Y,Z) = (−1)^kn(n−1)2 n!

nk k, . . . , k

X

λ

P_λ^(k)(Z)P_n^(k)k(n−1)/λ(Y).

Identifying the coefficient of P_λ^(k)′ (Z) in the two expressions, one finds, Z

Y

Q(¹k)

λ (X^∨)∆(X)^2k = (−1)^kn(n−1)2

nk k, . . . , k

(b^(k)_λ )⁻¹P_n^(k)_{k(n−1)/λ′}(Y), where the value ofb^(k)_λ := ^P

(k) λ

Q^(k)_λ is given by equality (7). But, from 16,P_λ/µ^(α) =

b^(α)_λ b^(α)µ

Q^(α)_λ/µ. Hence, Z

Y

Q(¹k)

λ (X^∨)∆(X)^2k= (−1)^kn(n−1)2

nk k, . . . , k

(b^(k)_nk(n−1))⁻¹Q^(k)_{nk(n−1)/λ′}(Y).

The value of b^(k)_nk(n−1) is obtained from Equality (7) after simplification.

5 Even powers of the Vandermonde determi- nant

5.1 Expansion of the even power of the Vandermonde on the Schur basis

The expansion of even power of the Vandermonde polynomial on the Schur functions is an open problem related the fractional quantum Hall effect as

described by Laughlin’s wave function [20]. In particular is of considerable in- terest to determine for what partitions the coefficients of the Schur functions in the expansion of the square of Vandermonde vanishe [5, 32, 33, 28, 17].

The aim of this subsection is to give an hyperdeterminantal expression for the coefficient of Sλ(X) in ∆(X)^2k.

Let us denote by A₀ the alphabet verifying Λⁿ(A₀) = 0 for each n 6= 0 (and by convention Λ⁰(A₀) = 1). The second orthogonality of Jack polyno- mials can be written as

hf, gi^′_n,α = 1 n!

Z

A⁰

f(X)g(X^∨)Y

i6=j

(1−x⁻¹_i x_j)^α1.

In the case when α = 1, it coincides with the first scalar product. In partic- ular,

hSλ(X), Sµ(X)i^′_n,1 =δ_λµ.

Hence, the coefficient of S_λ(X) in the expansion of ∆(X)^2k is hS_λ(X),∆(X)^2ki^′_n,1 = 1

n!

Z

A⁰

S_λ(X)∆(X^∨)^2kY

i6=j

(1−x⁻¹_i x_j).

One has

hSλ(X),∆(X)^2ki^′_n,1 = (−1)^n(n2−1) 1 n!

Z

A⁰

S_λ(X)Λⁿ(X)(2k+1)(1−n)∆(X)^2(k+1)

= (−1)^n(n−1)2 1 n!

Z

A⁰

Sλ+((2k+1)(1−n))ⁿ(X)∆(X)^2(k+1). By Proposition 2.3, one obtains an hyperdeterminantal expression for the coefficients of the Schur functions in the expansion of the even power of the Vandermonde determinant.

Corollary 5.1 The coefficient of Sλ(X) in the expansion of ∆(X)^2k is the hyperdeterminant

hSλ(X),∆(X)^2ki^′_n,1 = (−1)^n(n−1)2 H^k+1

reversen(λ)−((2k+1)(n−1))ⁿ(A₀).

It should be interesting to study the link between the notion of admissible partitions introduced by Di Francesco and al [5] and such an hyperdetermi- nantal expression.

5.2 Jack polynomials over the alphabet − X

In this paragraph, we work with Laurent polynomials in X ={x1,· · · , x_n}.

The space of symmetric Laurent polynomials is spanned by the family in- dexed by decreasing vectors ( ˜S_λ(X))(λ1≥···≥λn)∈Zⁿ and defined by

S˜_λ(X) = det(x^λ_i^j+n−j)

∆(X) .

Indeed, each symmetric Laurent polynomial f can be written as f(X) = Λⁿ(X)^−mg(X)

whereg(X) is a symmetric polynomial in X. As,g(x) is a linear combination of Schur functions, it follows that f(X) is a linear combination of ˜S_λ’s. Let X = {−x1, . . . ,−xn} be the alphabet of the inverse of the letters of X. We consider the operation R

−X (i.e. the substitution sending each x^p for x ∈X and p∈Z to the complete symmetric function Sⁿ(X)).

Consider the alternant

a_λ(X) := X

σ

ǫ(σ)x^σλ

= det(x^λ_i^j)

= S˜λ−δ(X)∆(X),

where δ = (n −1, n −2, . . . ,1,0). From this definition, one obtains that the operator _n!¹R

−X sends the product of 2k alternants aλ, aµ, . . . , aρ is an hyperdeterminant

1 n!

Z

−X

aλ(X)aµ(X)· · ·aρ(X) =

Det(S^{λi1+µi2+···+ρi2ki}(X))1≤i1,...,i2k≤n.

(18) Consider the linear operator Ω⁺ defined by

Ω⁺S˜_λ(X) :=S_λ(X) := det(S^λi+i−j(X)). (19) In particular, the operator Ω⁺ lets invariant the symmetric polynomials.

Furthermore, it admits an expression involving R

−X. Lemma 5.2 One has

Ω⁺ = 1 n!

Z

−X

a_δ(X)a−δ(X).

Proof It suffices to show that 1

n!

Z

−X

a_δ(X)a−δ(X) ˜S_λ(X) =S_λ(X).

But

aδ(X)a−δ(X) ˜Sλ(X) =aλ+δ(X)a−δ(X), and by (18), one obtains the result. .

Proposition 5.3

Let X={x1,· · · , xn} be a finite alphabet and0≤l ≤p∈N. R^(k),n_{np+(k−1)(n−1)l}(−X) = (−1)(k−1)n(n−1)

2 +np+lS_(p+1)^l_p^n−l(X)∆(X)^2(k−1). (20) Proof By Lemma 5.2, one has

Sλ(X)∆(X)^2(k−1) = Ω⁺Sλ(X)∆(X)^(2k−1)

= 1

n!

Z

−X

aλ+δ(X)aδ(X)^2(k−1)(X)a−δ(X).

By Equality (18), one obtains

Sλ(X)∆(X)^2(k−1) = Det S^{λi1+n−i1+···+n−i2k−1+i2k−n}(X)

1≤i1,...,i2k≤n

= Det S^{λn−i1+i1+···+i2k−1−i2k}(X)

0≤i1,...,i_2k≤n−1

= (−1)^n(n−1)2 Det S^{λn−i1+1−n+i1+···+i2k}(X)

0≤i1,...,i2k≤n−1

= (−1)^n(n−1)2 H_{reversen(λ)−[(n−1)}n](−X)

= (−1)^{n(n−1)(k2 −1)}T_reversen(λ)−[((k−1)(n−1))ⁿ](−X).

In particular, from Proposition 3.1 R^(k),n_[np+(k−1)(n−1)l](−X) = T^(k)

[((k−1)(n−1))ⁿ]+[p^n−l(p+1)^l](−X)

= (−1)n(n−1)(k−1)

2 S_[(p+1)^l_pⁿ−l](X)∆(X)^2(k−1). But, the Jack polynomial R^(α),n_λ being homogeneous, one has

R^(α),n_λ (−X) = (−1)^|λ|R^(α),n_λ (−X).

The result follows.

Remark 5.4 1. Note that a special case of Proposition 5.3 appeared in [22].

2. Proposition 5.3 can be reformulated as

The polynomialsP_n^(k)p+(k−1)(n−1)l(−X) andP_(p^(k)n)+(1^l)(X)∆(X)^2(k−1)are pro- portional.

This kind of identities relying Jack polynomials in X and in −X can be deduced from more general ones involving Macdonald polynomials when t is specialized to a power of q. This will be investigated in a forthcoming paper.

As a special case of Proposition 5.3, the even powers of the Vandermonde determinants ∆(X)^2k are Jack polynomials on the alphabet −X.

Corollary 5.5 Setting l=p= 0in Equality (20), one obtains

∆(X)^2k = (−1)^(kn(n−1)2 n!

(k+ 1)n k+ 1, . . . , k+ 1

P_n^(k+1)_(n−1)k(−X).

In the same way, using Corollary 4.2, one finds a surprising identity relying Jack polynomials in the alphabets −X and X^∨.

Proposition 5.6 One has

Ω⁺Q^(1/k)_λ (X^∨)∆(X)^2(k−1)Λⁿ(X)ⁿ⁻¹ = (−1)n(n−1)(k−1)

2 +|λ|

n!

nk k, . . . , k

(b^(k)_nk(n−1))⁻¹Q^(k)_nk(n−1)/λ^′(−X).

Proof The result is a straightforward consequence of Lemma 5.2 and Corol- lary 4.2.

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