Topological generation and matrix models for quantum reflection groups

TL;DR

This paper proves new topological generation results for quantum permutation and reflection groups, demonstrating they have many matrix models and residual finiteness, advancing understanding of their algebraic and analytical properties.

Contribution

It introduces novel topological generation results for quantum groups and constructs matrix models, showing these groups are residually finite and hyperlinear.

Findings

Quantum permutation groups are topologically generated by specific subgroups.

Quantum reflection groups admit many matrix models, including flat models for $S_N^+$.

These quantum groups have residually finite duals and are hyperlinear.

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.