Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

TL;DR

This paper establishes precise bounds on virtual degrees and characters of symmetric groups, improving understanding of their Witten zeta function and applications to random walk mixing times.

Contribution

It provides new sharp bounds on virtual degrees and characters, and refines bounds on the Witten zeta function, with applications to random walk mixing times.

Findings

Sharp bounds on virtual degrees of symmetric groups

Improved bounds on the Witten zeta function

Characterization of fixed-point free conjugacy classes for rapid mixing

Abstract

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.