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 Id: hal-00173319 https://hal.archives-ouvertes.fr/hal-00173319 Submitted on 19 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 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 1 Det M = sign(σ)M σ, n! Sk σ=(σ1,X··· ,σk)∈ n ∗Universit´edes Sciences et de la Technologie Houari Boumediene. BP 32 USTHB 16111 Bab-Ezzouar Alger, Alg´erie. email: [email protected] †Universit´e de Paris-Est Marne-la-Valle, Institut d’Electronique´ et d’Informatique Gaspard-Monge 77454 Marne-la-Valle Cedex 2. email: [email protected] ‡Universit´e de Paris-Est Marne-la-Valle, Institut d’Electronique´ et d’Informatique Gaspard-Monge 77454 Marne-la-Valle Cedex 2. email: [email protected] hal-00173319, version 1 - 19 Sep 2007 1 where sign(σ) = sign(σ1) ··· sign(σk) is the product of the signs of the per- σ S mutations, M = Mσ1(1)...σk(1) ··· Mσ1(n)...σk(n) and n denotes 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 when k is odd ×k and its invariance under the action of the group SLn . 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- n n n dermonde via the substitution Y : x → Λ (Y) where Λ (Y) denotes the nth elementary symmetric functions on the alphabet Y. Section 3 is devoted R to the generalization of the Matsumoto formula [25] to almost rectangular Jack polynomials. In Section 4, we give an equality involving the substitu- tion Y and skew Jack polynomials. As an application, we give in Section 5 expressions of even powers of the Vandermonde determinant in terms of R Schur functions and Jack polynomials. 2 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 by Sym(X). Let us consider the complete symmetric functions whose generating series is 1 σ (X) := Si(X)ti = , t 1 − xt i X X xY∈ the elementary symmetric functions i i −1 λt(X) := Λ (X)t = (1 + xt)= σ−1(X) , i X X xY∈ and power sum symmetric functions ti ψ (X) := Ψi(X) = log(σ (X)). t i t i X When there is no algebraic relation between the letters of X, Sym(X)isa 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 algebra Sym(X)(X being infinite or not) is spanned by the set of the decreasing products of the generators Sλ = Sλn ...Sλ1 , Λλ =Λλn . Λλ1 , Ψλ =Ψλn . Ψλ1 , where λ =(λ1 ≥ λ2 ≥···≥ λn) is a (decreasing) partition. The algebra Sym(X) admits also non multiplicative basis. For example, the monomial functions defined by m X xλ1 xλn , λ( )= i1 ··· in X 3 xλ1 xλn x ,...,x where the sum is over all the distinct monomials i1 ··· in with i1 in ∈ X, and the Schur functions defined via the Jacobi-Trudi formula λi−i+j Sλ(X) = det(S (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 hΨλ, Ψµi = δλ,µzλ (2) where δµ,ν is the Kronecker symbol (equal to 1 if µ = ν and 0 otherwise) mi(λ) and zλ = i i mi(λ)! if mi(λ) denotes the multiplicity of i as a part of λ. Q 2. The coefficient of the dominant term in the expansion in the monomial basis is 1, Sλ = mλ + uλµmµ. Xµ<λ When X = {x1,...,xn} is finite, a Schur function has another determi- nantal expression det(xλj +n−j) S (X)= i , λ ∆(X) X where ∆( )= i<j(xi − xj) denotes the Vandermonde determinant. Q 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,...,i2k = 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 Hk (X) = Det Λi1+···+i2k (X) , (3) n 0≤i1,...,i2k≤n−1 where Λm(X) is the mth elementary function on the alphabet X. Let us consider a shifted version of Hankel hyperdeterminants k i +···+i +v H (X) = Det Λ 1 2k i1 (X) . (4) v 0≤i1,...,i2k≤n−1 4 n where v =(v0,...,vn−1) ∈ Z . Note that (3) implies M0,...,0 =Λ0(X) = 1 by convention. But if M0,...,0 =06 , 1, this property can be recovered using a suit- able normalization and, if M0...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 k i +···+i −(i +···+i )+v T (X) = Det Λ 1 k k+1 2k i1 (X) . (5) v 0≤i1,...,i2k≤n−1 To¨eplitz hyperdeterminants are related to Hankel hyperde terminants by the following formulae. − k kn(n 1) k H X 2 T X Proposition 2.1 1. v ( )=(−1) v+(k(n−1))n ( ) − k kn(n 1) k T X 2 H X 2. v( )=(−1) v+(k(1−n))n ( ). Proof The equalities (1) and (2) are equivalent and are direct consequences of the definitions (4) and (5), k X i1+···+ik+(n−1−ik+1+···+(n−1−i2k−k(n−1)) X Tv( ) = Det Λ ( ) kn(n−1) 2 H n X = (−1) v+(k(1−n)) ( ). . 2.3 The substitution xn → Λn(Y) Let X = {x1,...,xn} be a finite alphabet and Y be another (potentially infinite) alphabet. For simplicity we will denote by Y the substitution R xp =Λp(Y), Y Z for each x ∈ X and 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 X 2k k Y ∆( ) = Hn( ) n! Y Z X where ∆( )= i<j(xi − xj). Q 5 Proof It suffices to develop the power of the Vandermonde determinant 2k X 2k j−1 σ1(i)+···+σ2k(i)−2k ∆( ) = det xi = sign(σ1 ··· σ2k) xi . σ1,··· ,σ ∈Sn i X2k Y Hence, applying the substitution, one obtains 1 1 2k σ1(i)+···+σ2k(i)−2k ∆(X) = sign(σ1 ··· σ2k) Λ (Y) n! Y n! S Z σ1,··· ,σ2k∈ n i k Y X Y = Hn( ). More generally, the Jacobi-Trudi formula (1) implies the following result.