Double coset Markov chains

Abstract

Let G be a finite group. Let H, K be subgroups of G and H\G/K the double coset space. If Q is a probability on G which is constant on conjugacy classes (Q(s−1ts) = Q(t)), then the random walk driven by Q on G projects to a Markov chain on H\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 GLn(q) onto a Markov chain on Sn via the Bruhat decomposition. The chain on Sn 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 with G a compact group are discussed.