Representation theory and cycle statistics for random walks on the symmetric group

TL;DR

This paper employs representation theory to analyze the mixing times of cycle statistics in random walks on the symmetric group, providing new formulas for character decomposition coefficients and extending results to star transpositions.

Contribution

It introduces a novel approach using the method of moments and a new formula for character coefficients to study cycle statistics in random walks on $S_n$, including star transpositions.

Findings

Derived formulas for character coefficients in $S_n$

Analyzed mixing times for cycle type statistics

Extended results to star transposition walks

Abstract

We use representation theory of $S_n$ to analyze the mixing of permutation cycle type statistics $a_j(\sigma) = ${# of $j$-cycles of $\sigma$} for any fixed $j$ and $\sigma$ resulting from a random $i$-cycle walk on $S_n$. We also derive analogous results for the random star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function $(a_j)^r(\sigma)=\{(\text{# of $j$-cycles of $\sigma$})^r\}$ for any positive integers $r,j$ when $n\geq 2rj$.