  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.
