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. © 2005 Elsevier Inc. All rights reserved.