Asymptotics of characters of symmetric groups: structure of Kerov character polynomials

TL;DR

This paper investigates the asymptotic behavior of symmetric group characters, revealing universal symmetric polynomials that describe Kerov character polynomial coefficients across different genera, confirming a conjecture by Lassalle.

Contribution

It proves the existence of universal symmetric polynomials for Kerov character coefficients at each genus, advancing understanding of symmetric group character asymptotics.

Findings

Existence of universal symmetric polynomials for each genus

Confirmation of Lassalle's conjecture

Explicit structure of Kerov character polynomial coefficients

Abstract

We study asymptotics of characters of the symmetric groups on a fixed conjugacy class. It was proved by Kerov that such a character can be expressed as a polynomial in free cumulants of the Young diagram (certain functionals describing the shape of the Young diagram). We show that for each genus there exists a universal symmetric polynomial which gives the coefficients of the part of Kerov character polynomials with the prescribed homogeneous degree. The existence of such symmetric polynomials was conjectured by Lassalle.