Flat matrix models for quantum permutation groups

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