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

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 terms of Jack polynomials.

Hyperdeterminantal computation for the Laughlin wavefunction

“…The Hankel hyperdeterminants appear in the literature in the work of Lecat [13] (see also [12,13]), but few properties have been considered. More recently, one of the authors with Jean-Yves Thibon [16,17] and two of the authors with Hacene Belbachir [2] investigated the links between these polynomials and the Selberg integral and the Jack polynomials. More generally, one defines a shifted Hankel hyperdeterminant depending on 2k decreasing vectors λ (1) , .…”

Macdonald symmetric functions of rectangular shapes

“…Let us study the special case of Jack functions. We will show that our theorem on Macdonald functions implies the hyperdeterminant formula or Jacobi-Trudi formula for Jack functions of almost rectangular shapes [2]. Recall that the Macdonald function Q λ (q, q β ) goes to Jack function Q λ (β −1 ) when q goes to 1.…”

“…The q = 1 case of (1.4) is equivalent to Matsumoto's Hyperdeterminant formula for rectangular Jack functions [14]. Matsumoto's formula was generalized to almost rectangular shapes (Q λ with λ = ((k + 1) t , k s )) in [2], and this generalized formula is equivalent to the q = 1 case of our formula (5.2). Formula (1.4) can also be specified to the case of Hall-Littlewood functions.…”

Deformation of Cayley's Hyperdeterminants

“…A q-analog Hankel determinant has been studied in [8] (see also [1]) and q-analog of the hyperdeterminant for non-commuting matrices has been introduced in the context of quantum groups [9].…”