Topological generation and matrix models for quantum reflection groups

Abstract

We establish several new topological generation results for the quantum permutation groups S + N and the quantum reflection groups H s+ N . We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for S + N are inner faithful.

Quantum Increasing Sequences Generate Quantum Permutation Groups

“…In both cases, we conclude that S + n−1 ⊂ I + k,n . Together with the first observation, this yields S n , S + n−1 ⊂ I + k,n ⊂ S + n , but from [7,Theorem 3.3] it follows that all inclusions are equalities.…”

“…The conjecture found its positive answer for n = 4 at [3], and only recently for n = 5 by providing a deep connection to the subfactor theory, see [2]. The latter was further utilised to argue a inductive-type generation result for quantum permutation groups in [7].…”

“…This enables us to put ourselves in the inductive generation framework: knowing that I + k,n−1 = I + k−1,n−1 = S + n−1 and I k,n−1 = S n , we deduce that S + n−1 , S n ⊂ I + k,n . From [7], one knows that S + n−1 , S n = S + n . The latter relies heavily on the solution of Banica's conjecture for n = 5.…”

“…The sequence (2<3<5<6<8) ∈ I 5,9 , drawn with full circles and segments, is completed, with the aid of empty circles and dashed segments, to the permutation(1,2,3,5,8,7,4,6).…”

Free wreath products with amalgamation

“…Proof. It is clear that a direct product of residually finite compact quantum groups is residually finite, and both Γ and S + N are residually finite, the first one by assumption and the second one by [8,Thm 3.6]. Moreover, it was proven in [8, Thm 3.11 and Rem 3.21] that the free wreath product of a residually finite discrete group by S + N is residually finite.…”