Zonal polynomials via Stanley's coordinates and free cumulants

TL;DR

This paper explores the combinatorial structure of zonal polynomials using Stanley's coordinates and free cumulants, proving conjectures related to Jack polynomials in this special case.

Contribution

It introduces a combinatorial description of zonal characters via Stanley's coordinates and expresses them as polynomials in free cumulants, confirming recent conjectures.

Findings

Zonal characters admit a combinatorial description using Stanley's multirectangular coordinates.

Zonal characters can be expressed as polynomials in free cumulants with explicit coefficients.

Confirmed two conjectures of Lassalle for Jack polynomials in the zonal case.

Abstract

We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanley's multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of Lassalle for Jack polynomials in the special case of zonal polynomials.