Combinatorics of asymptotic representation theory

TL;DR

This paper explores the combinatorial structures and dualities in the asymptotic representation theory of symmetric groups, focusing on Kerov polynomials that relate characters to free cumulants, to better understand large-scale behaviors.

Contribution

It reviews the development of dual combinatorics in symmetric group representations and highlights the role of Kerov polynomials in connecting characters with free cumulants.

Findings

Kerov polynomials express characters in terms of free cumulants.

Dual combinatorics helps analyze asymptotic behaviors of symmetric groups.

The approach simplifies complex combinatorial structures in large n limits.

Abstract

The representation theory of the symmetric groups S_n is intimately related to combinatorics: combinatorial objects such as Young tableaux and combinatorial algorithms such as Murnaghan-Nakayama rule. In the limit as n tends to infinity, the structure of these combinatorial objects and algorithms becomes complicated and it is hard to extract from them some meaningful answers to asymptotic questions. In order to overcome these difficulties, a kind of dual combinatorics of the representation theory of the symmetric groups was initiated in 1990s. We will concentrate on one of its highlights: Kerov polynomials which express characters in terms of, so called, free cumulants.