Random Walks on Finite Quantum Groups

Abstract

In this paper we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e. the limit state if it exists. We note that the possible limits are any central idempotent state. We also look at cut-off phenomenon in the Sekine finite quantum groups. Keywords: convergence of random walks and finite quantum group and Sekine quantum groups and central idempotent state and representation theory

The Ergodic Theorem for Random Walks on Finite Quantum Groups

“…When stating it for the case of a Sekine quantum group, Baraquin (Prop. 3,[2]) all but wrote down the following corollary:…”

“…As will be seen in Section 5.4, as the representation theory generalises so well from classical to quantum, the upper bound lemma of Diaconis and Shahshahani can also be used to analyse random walks on quantum groups. The upper bound lemma has been used to analyse random walks on the dual symmetric group, S n [27]; Sekine quantum groups, Y n [2,27]; the Kac-Paljutkin quantum group, G 0 [2]; free orthogonal quantum groups, O + N [17]; free symmetric quantum groups, S + N [17]; the quantum automorphism group of (M N (C), tr) [17]; free unitary groups, U + N [18]; free wreath products Γ ≀ * S + N , including quantum reflection groups H s + N [18]; duals of discrete groups, Γ, including for Γ = F N the free group on N generators, [19].…”

“…A number of partial results, stated for Sekine quantum groups; a sufficient condition of Zhang for aperiodicity [41], and an ergodic theorem of Baraquin for central states [2], have been shown to hold more generally. As remarked above, as the detection of whether or not a random walk satisfies the conditions for ergodicity is non-trivial, partial results such as these are most welcome for any study of random walks on quantum groups.…”

“…Definition 1.1. The algebra of functions on a finite quantum group, is a unital C * -Hopf algebra A; that is a C * -algebra A with a *-homomorphism ∆, a counit ε, and an antipode S, satisfying the relations (2).…”