Modeling questions for quantum permutations

Abstract

Given a quantum permutation group [Formula: see text], with orbits having the same size [Formula: see text], we construct a universal matrix model [Formula: see text], having the property that the images of the standard coordinates [Formula: see text] are projections of rank [Formula: see text]. Our conjecture is that this model is inner faithful under suitable algebraic assumptions, and is in addition stationary under suitable analytic assumptions. We prove this conjecture for the classical groups, and for several key families of group duals.