Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

TL;DR

This paper generalizes the Diaconis-Shahshahani Upper Bound Lemma from finite groups to finite quantum groups, enabling analysis of convergence rates for ergodic quantum random walks.

Contribution

It extends the classical Upper Bound Lemma to finite quantum groups, providing a new tool for studying quantum ergodic random walks.

Findings

Derived an Upper Bound Lemma for finite quantum groups

Applied the lemma to analyze convergence on dual symmetric groups

Studied ergodic walks on Sekine quantum groups

Abstract

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 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 finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group $\widehat{S_n}$ as well as on the truly quantum groups of Sekine.