Quantum permutation groups

TL;DR

This paper reviews the properties and structures of quantum permutation groups, especially their subgroups, quantum symmetries of finite spaces, and specific quantum reflection groups, highlighting recent developments and open questions.

Contribution

It provides a comprehensive overview of quantum permutation groups, including their easiness, Weingarten calculus, subgroup structures, and quantum symmetries of finite graphs and spaces.

Findings

$S_N^+$ is infinite for $N\, ext{at least}\,4$

Isomorphism $S_4^+=SO_3^{-1}$ and its implications

Detailed analysis of quantum symmetry groups of finite graphs and spaces

Abstract

The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We review the known facts regarding $S_N^+$, and notably its easiness property, Weingarten calculus, and the isomorphism $S_4^+=SO_3^{-1}$ and its consequences. We discuss then the structure of the closed subgroups $G\subset S_N^+$, and notably of the quantum symmetry groups of finite graphs $G^+(X)\subset S_N^+$, with particular attention to the quantum reflection groups $H_N^{s+}$. We also discuss, more generally, the quantum symmetry groups $S_Z^+$ of the finite quantum spaces $Z$, and their closed subgroups $G\subset S_Z^+$, with particular attention to the quantum graph case, and to quantum reflection groups.