Double coset Markov chains

Abstract

AbstractLetGbe a finite group. Let$H, K$be subgroups ofGand$H \backslash G / K$the double coset space. IfQis a probability onGwhich is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$), then the random walk driven byQonGprojects to a Markov chain on$H \backslash G /K$. This allows analysis of the lumped chain using the representation theory ofG. 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, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains withGa compact group are discussed.