Asymptotics of Jack characters

TL;DR

This paper investigates the asymptotic behavior of Jack characters, providing explicit formulas for their leading terms, analyzing their algebraic structure, and demonstrating properties that support Gaussian fluctuations in random Young diagrams.

Contribution

It introduces explicit formulas for the top-degree asymptotic part of Jack characters and explores their multiplicative structure and factorization properties.

Findings

Explicit formulas for asymptotically top-degree Jack characters.

Proof of approximate factorization property for Jack characters.

Support for Gaussian fluctuations in random Young diagrams.

Abstract

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack characters from the viewpoint of the asymptotic representation theory. In particular, we give explicit formulas for their asymptotically top-degree part, in terms of bicolored oriented maps with an arbitrary face structure. We also study their multiplicative structure and their structure constants and we prove that they fulfill approximate factorization property, a convenient tool for proving Gaussianity of fluctuations of random Young diagrams.