Hyperdeterminantal calculations of Selberg's and Aomoto's integrals

Abstract

The hyperdeteminants considered here are the simplest analogues of determinants for higher rank tensors which have been defined by Cayley, and apply only to tensors with an even number of indices. We have shown in a previous article that the calculation of certain multidimensional integrals could be reduced to the evaluation of hyperdeterminants of Hankel type. Here, we carry out this computation by purely algebraic means in the cases of Selberg's and Aomoto's integrals.

Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

“…Note that J.-Y. Thibon with one of the authors have already investigated the links between combinatorics and Selberg integrals [30][31][32] .…”

Staircase Macdonald polynomials and the $q$-Discriminant

“…This kind of hyperdeterminant have been already considered by the authors in collaboration with Thibon and Belbachir [19,20,2]. In particular, it is shown that the coefficients C λ (n, l) arising in the expression…”

“…The polynomial considered here is the simplest one in the sense that it generalizes the expansion of determinant as an alternated sum. Reader can refer to [19,20,21,23,26] for more informations on the subject.…”

“…The powers of the discriminant (q = 1) are encountered also in the context of generalizations of the Selberg integral [10,12,25]. These integrals are closely related to the notion of Hankel hyperdeterminant [19,20] and Jack polynomials [8,9]. The Selberg integral admits q-analogue involving the q-discriminant (see e.g.…”

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

“…One of the authors investigated such polynomials in relation with the Selberg integral [22,23]. Without lost of generality, we will consider the polynomials…”