Jack polynomials and free cumulants

TL;DR

This paper explores the relationship between Jack polynomials and free cumulants, expressing expansion coefficients as polynomials with a conjecture about their nonnegativity, extending symmetric group character results.

Contribution

It introduces a novel polynomial expression for Jack polynomial coefficients in terms of free cumulants and conjectures their nonnegativity, extending prior symmetric group character findings.

Findings

Coefficients expressed as polynomials in free cumulants

Conjecture of nonnegative integer coefficients

Extension of symmetric group character results

Abstract

We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group.