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

2025/12/16 by Arcona, Dominic
Biochemistry, Genetics and Molecular Biology · Mathematics · #Genome Rearrangement Algorithms #Advanced Combinatorial Mathematics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2512.13969

Abstract

We use representation theory of S_n to analyze the mixing of permutation cycle type statistics a_j(σ) = # of j-cycles of σ for any fixed j and σ 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(σ)=\(# of j-cycles of σ)^r\ for any positive integers r,j when n≥ 2rj.

Citations

• Fixed points of non-uniform permutations and representation theory of the symmetric group
• The Limit Profile of Star Transpositions
• Limit profile for random transpositions
• Partition Algebras and the Invariant Theory of the Symmetric Group
• Dimensions of irreducible modules for partition algebras and tensor\n power multiplicities for symmetric and alternating groups

Related

• The random (n-k)-cycle to transpositions walk on the symmetric group
• Hitting time mixing for random k-cycles
• Likelihood Orders for some Random Walks on the Symmetric Group
• Cutoff Phenomenon and Limiting Profile of a Random Walk on the Symmetric Group
• On averages of randomized class functions on the symmetric groups and their asymptotics