Double Coset Markov Chains

TL;DR

This paper studies Markov chains derived from random walks on finite groups, focusing on their projections onto double coset spaces, with applications to algebraic and combinatorial structures like Gaussian elimination.

Contribution

It introduces a framework for analyzing double coset Markov chains using representation theory, including new examples and connections to algebraic structures.

Findings

Projection of random walks onto double coset spaces simplifies analysis.

The chain on $S_n$ exhibits Mallows distribution and unique mixing behavior.

Representation of transvections sum in the Hecke algebra links algebraic and probabilistic aspects.

Abstract

Let $G$ be a finite group. Let $H, K$ be subgroups of $G$ and $H \backslash G / K$ the double coset space. Let $Q$ be a probability on $G$ which is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$). The random walk driven by $Q$ on $G$ projects to a Markov chain on $H \backslash G /K$. This allows analysis of the lumped chain using the representation theory of $G$. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets. Some extensions and examples of double coset Markov chains with $G$ a compact group are discussed.