# Models of quantum permutations

## TL;DR

This paper constructs operator algebraic models for quantum permutation groups using *-homomorphisms into C*-algebras, leading to a new family of quantum groups that interpolate between classical and free quantum permutations.

## Contribution

It introduces a series of *-homomorphisms that model quantum permutation matrices and constructs an inverse limit quantum group, extending understanding of quantum permutation groups for larger N.

## Key findings

- Constructed *-homomorphisms for N≥4
- Defined a compact matrix quantum group G between S_N and S_N^+
- Open problem for N≥6 regarding the nature of G

## Abstract

For $N\ge 4$ we present a series of *-homomorphisms $\varphi_n:C(S_N^+)\rightarrow B_n$ where $S_N^+$ is the quantum permutation group. They are not necessarily representations of the quantum group $S_N^+$ but they yield good operator algebraic models of quantum permutation matrices. The C*-algebras $B_n$ allow the construction of an inverse limit $B_{\infty}$ which defines a compact matrix quantum group $S_N\subsetneq G\subseteq S_N^+$. We know $G=S_N^+$ for $N=4,5$ from Banica's work, but we have to leave open the case $N\ge 6$.