The ergodic theorem for random walks on finite quantum groups

A state-space approach to quantum permutations

“…It seems like this observation could help achieve some first partial results in the extension of the finite quantum group random walk ergodic theorem [34] to the case of quantum permutation groups, alas there are many quantum permutations ς = h G such that Φ(ς) = Φ(h G ). There are even examples of random permutations whose associated random walk is not ergodic, for example ν ∈ M p (S 3 ) given by:…”

“…The one dimensional factors give deterministic permutations, f 1 = ev e , f 2 = ev (34) , f 3 = ev (12) and f 4 = ev (12) (34) , so that…”

“…Quasi-subgroups of finite quantum groups behave very much like quantum subgroups: they are closed under the quantum group law (Prop. 3.12, [34]), they contain the identity ε, and they are closed under precomposition with the antipode ( [29]).…”

“…In the case of finite quantum groups, an idempotent state φ S is associated to a grouplike projection 1 S , which is also its support projection (see [34] for more including original references), and therefore it is tenable to define: Definition 6.1. Let G be a finite quantum permutation group with an idempotent state φ S ∈ G and associated group-like projection 1 S ∈ F (G).…”

“…In the finite case, assuming that a random walk is not concentrated on a quasi-subgroup, the support of ς for a non-ergodic random walk is concentrated on a cyclic coset of quasi-subgroup: Definition 6.2. [34] Let G be a finite quantum permutation group. A quantum permutation ς ∈ G is supported on a cyclic coset of a proper quasi-subgroup if there exists a pair of projections p 0 , p 1 , such that p 0 p 1 = 0, p 0 + p 1 ≤ 1 G , ς(p 1 ) = 1, p 0 is a group-like projection, (ς ⊗ I F (G) )∆(p 1 ) = p 0 , and there exists d > 1 such that ((ς…”