A Hopf-power Markov chain models the breaking and reassembling of combinatorial objects. Its transition matrix arises from applying a variant of the Doob transform to the coproduct-then-product operator on a combinatorial Hopf algebra. Key examples include the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, a model of tree-pruning, and the restriction-then-induction of representations of the symmetric group. I'll outline how Hopf algebra structure theory diagonalises a large class of these chains and determines their stationary distributions, and how Hopf morphisms produce projections of the chains. No knowledge of Hopf algebras is assumed.
The ideas in this talk had their origins in joint work with Persi Diaconis and Arun Ram, and in conversations with Marcelo Aguiar and with Sami Assaf.