Quantum permutation groups

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.