Diaconis–Shahshahani Upper Bound Lemma for Finite 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 S n as well as on the truly quantum groups of Sekine.2000 Mathematics Subject Classification. 46L53 (60J05, 20G42).

A state-space approach to quantum permutations

“…F (G); an F (G)-analogue of the (inclusion of the) identity is the counit ε : F (G) → C, ε(f ) := ev e (f ) = f (e); and an F (G)-analogue of the inverse is an antihomomorphism called the antipode, S : F (G) → F (G), Sf (σ) = f (σ −1 ). A most leisurely introduction to how these maps, and the F (G)-analogues of associativity (coassociativity, (1.3)), of the identity axiom (the counital property), and of the inverse axiom (the antipodal property), are F (G)-analogues of the (finite) group axioms is given in Section 1.1, [33].…”

“…Quasi-subgroups. In the Gelfand-Birkhoff picture a random walk on a quantum permutation group is a sequence (ς ⋆k ) k≥0 in G (see [33] for more). Of particular interest are ergodic random walks, those random walks such that the sequence converges in the weak-* topology to the Haar state, h G .…”

Cutoff profiles for quantum Lévy processes and quantum random transpositions

“…One very useful tool to prove that such a phenomenon occurs is the following lemma originally due to P. Diaconis and M. Shahshahani in [14] for finite groups and to J.P. McCarthy in [23] for finite quantum groups. A proof for compact quantum groups can be found in [19, Lem 2.7], but we simply state it in our particular case.…”

Random Walks on Finite Quantum Groups

“…In[12], McCarthy gives another example of a random walk in Sekine family where there is no cut-off, but with a state which formally does not depend on n and which is not the Fourier transform of an element in the central algebra. Note that in[7] and[8], Freslon finds cut-off examples in compact quantum groups.…”

“…If we look at the characterizations(11) and(12), we have also the following equivalences:∀l ∈ Z n , a ρ + l = 1 and |a α | < d α for all the others α ∈ I(KP n ) ⇐⇒ µ = h {0}×Z n ,1,(1) n−and |a α | < d α for all the others α ∈ I(KP n )⇐⇒ µ = h Z n ×2Z n ,0 a ρ + 0 and |a α | < d α for all the others α ∈ I(KP n ) ⇐⇒ µ = h Z n ×2Z n ,∀i ∈ Z n 2 , a σ + 2i or ∀i ∈ Z n 2 , a σ − 2iand |a α | < d α for all the others α ∈ I(KP n ) ⇐⇒ µ = h n 2 Z n ×2Z n ,l,τ Thanks to Remark 3.2 we get that the random walk diverges if and only if there exists α in I(KP n ) such that |a α | = d α but a α = d α . In this case, the random walk can be cyclic or not.Example 3.2.…”