Combinatorial interpretation and positivity of Kerov's character polynomials

TL;DR

This paper proves a positivity property of Kerov's polynomials' coefficients, extending a conjecture, and provides combinatorial interpretations and explicit formulas for these coefficients, enhancing understanding of character values in representation theory.

Contribution

It establishes the positivity of Kerov's polynomial coefficients and offers new combinatorial descriptions and explicit formulas for these coefficients.

Findings

Proved positivity of Kerov's polynomial coefficients.

Provided combinatorial interpretations for coefficients.

Computed subdominant terms for character values.

Abstract

Kerov's polynomials give irreducible character values in term of the free cumulants of the associated Young diagram. We prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Our method, through decomposition of maps, gives a description of the coefficients of the k-th Kerov's polynomials using permutations in S(k). We also obtain explicit formulas or combinatorial interpretations for some coefficients. In particular, we are able to compute the subdominant term for character values on any fixed permutation (it was known for cycles).