Álvaro Gutierrez Caceres

Shuffles that frustrate card-counters

A shuffle of a deck is an example of a Markov chain in the Cayley graph of a finite group G.
In a game such as blackjack, a player only observes a walk on the set of left cosets of G by a subgroup, in this case Sym(50) / (Sym(50) × Sym(2)). We say "card counters are frustrated" if the observations are independent from the history of past observations. That is, if the walk on cosets is a Markov chain. In probability, we say that the random walk lumps (weakly) to left cosets.

We give a new characterisation of weak lumping of random walks on groups. The results reveal interesting connections between group theory, probability theory, and representation theory.

(joint work with Edward Crane, Erin Russell, and Mark Wildon)