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 Poincare series of G, meaning the series f(z) = 1 + c(1)z + c(2)z(2) + (. . .) whose coefficients are multiplicities of I 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. (c) 2004 Elsevier Inc. All rights reserved.