Embedding a Markov Chain into a Random Walk on a Permutation Group

Using representation theory, we obtain a necessary and sufficient condition for a discrete–time Markov chain on a finite state space E to be representable as ΨnΨn−1 · · ·Ψ1z, n ≥ 0, for any z ∈ E, where the Ψi are independent, identically distributed random permutations taking values in some given transitive group of permutations on E. The condition is particularly simple when the group is 2-transitive on E. We also work out the explicit form of our condition for the dihedral group of symmetries of a regular polygon.