Lam-Williams Markov chains on symmetric groups

Abstract

This paper reviews the current state of the Lam-Williams conjectures on a multivariate Markov chain on the symmetric group S_n. We start with Lam's work on random core partitions which led to a remarkable Markov chain on Weyl groups. Then we give a discussion on a multivariate generalization of this Markov chain in type A and state a series of related conjectures, due to Lam and Williams. Recent developments in proving the conjectures, due to Ayyer and Linusson, are presented. Finally, a different perspective is offered, in which the stationary distributions are mappings from a certain simplex into a real projective space, and a formula is given for such stationary distributions.