A Spin Analogue of Kerov Polynomials

TL;DR

This paper introduces spin analogues of Kerov polynomials, expressing spin irreducible characters as polynomials in even free cumulants, and conjectures positivity of their coefficients.

Contribution

It develops a new framework for spin representations by defining polynomial relations similar to Kerov polynomials for symmetric groups.

Findings

Spin irreducible characters are polynomials in even free cumulants.

Proposes a conjecture on positivity of polynomial coefficients.

Extends Kerov polynomial concepts to spin (projective) representations.

Abstract

Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.