Quantum automorphism groups of homogeneous graphs

Associated to a finite graph X is its quantum automorphism group G. The main problem is to compute the Poincaré series of G, meaning the series f(z)=1+c1z+c2z2+⋯ whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives f for all homogeneous graphs having 8 vertices or less.