Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures

TL;DR

This paper explores symmetric polynomials involving Jucys-Murphy elements, connecting algebraic structures with orthogonal matrix integrals and extending results through Jack polynomials.

Contribution

It introduces new evaluations of symmetric polynomials in Jucys-Murphy elements in terms of zonal spherical functions and extends these results using Jack polynomials.

Findings

Evaluations relate to integrals over orthogonal groups

Extensions involve Jack polynomial measures

Provides algebraic insights into matrix integrals

Abstract

We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give an extension of them, based on Jack polynomials.