Deformation of Cayley's Hyperdeterminants

Tommy Wuxing Cai,Naihuan Jing

doi:10.37236/8091

Abstract

We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we obtain a generalization of Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions.

Highlights

For a 2m dimensional hypermatrix A = (A(i1, · · ·, i2m))1 i1,...,i2m n Cayley’s second hyperdeterminant [5] of A is a generalization of the usual determinant: det[2m](A) = 1 n!2m n sgn(σi) A(σ1(i), . . . , σ2m(i)). (1)σ1,...,σ2m∈Sn i=1 i=1In [7] properties of hyperdeterminants and discriminants are studied in general
A q-analog Hankel determinant has been studied in [8] and q-analog of the hyperdeterminant for non-commuting matrices has been introduced in the context of quantum groups [9]
Recall that the classical Jacobi-Trudi formula expresses the Schur function associated to partition μ as a determinant of simpler Schur functions of row shapes: sμ(x) = det(sμi−i+j(x))

Summary

Introduction

For a 2m dimensional hypermatrix A = (A(i1, · · · , i2m))1 i1,...,i2m n Cayley’s second hyperdeterminant [5] of A is a generalization of the usual determinant: det[2m](A) = 1 n! We introduce a λ-hyperdeterminant (of commuting entries) by replacing permutations with alternating sign matrices in a natural and nontrivial manner. To introduce the λ-hyperdeterminant, we first consider generalization of the permutation sign in the usual determinant, and generalize Cayley’s hyperdeterminants. Each row or column of a fixed permutation matrix has only one nonzero entry along each row and each column.

Results

Conclusion