Random Walks on Finite Quantum Groups: Diaconis-Shahshahani Theory for Quantum Groups

Abstract

The concept of a random walk on a finite group converging to random - and a way of measuring the distance to random after $k$ transitions - is generalised from the classical case to the case of random walks on finite quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis--Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups $\mathbb{Z}_n$, $\mathbb{Z}_2^n$, the dual group $\widehat{S_n}$ as well as the `truly' quantum groups of Kac & Paljutkin, and Sekine.