McKay matrices and Markov chains

Quivers constructed by tensoring modules for finite groups or finite-dimensional Hopf algebras and their adjacency matrices (McKay matrices) can provide results on invariants, chip-firing, and Markov chains. The analysis to determine sharp rates of convergence to stationarity of these chains can be challenging, and the chains may exhibit unexpected phenomena, as in our joint work with Persi, Martin Liebeck, and Pham Tiep.