Flat matrix models for quantum permutation groups

TL;DR

This paper explores flat matrix models for quantum permutation groups, generalizing Pauli matrix constructions and proposing a universal representation inspired by the Sinkhorn algorithm, with conjectures on its faithfulness.

Contribution

It introduces a generalized construction of flat matrix models for quantum permutation groups and proposes a new universal representation inspired by classical algorithms.

Findings

Generalization of Pauli matrix construction at N=4

Construction of a universal representation of C(S_N^+)

Conjecture that the universal representation is inner faithful

Abstract

We study the matrix models $\pi:C(S_N^+)\to M_N(C(X))$ which are flat, in the sense that the standard generators of $C(S_N^+)$ are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at $N=4$, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of $C(S_N^+)$, inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.