Stanley character polynomials

TL;DR

This paper discusses Stanley's polynomial approach to symmetric group characters on multirectangular Young diagrams, highlighting its significance in combinatorics and asymptotic representation theory.

Contribution

It reviews Stanley's conjecture on explicit polynomial forms of normalized characters and their applications in dual combinatorics and asymptotic analysis.

Findings

Stanley's polynomials are polynomial in rectangle side lengths.

They provide a powerful parametrization for Young diagrams.

Applications include insights into Kerov polynomials.

Abstract

Stanley considered suitably normalized characters of the symmetric groups on Young diagrams having a special geometric form, namely multirectangular Young diagrams. He proved that the character is a polynomial in the lengths of the sides of the rectangles forming the Young diagram and he conjectured an explicit form of this polynomial. This Stanley character polynomial and this way of parametrizing the set of Young diagrams turned out to be a powerful tool for several problems of the dual combinatorics of the characters of the symmetric groups and asymptotic representation theory, in particular to Kerov polynomials.