2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/95786Associated 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 $f(z)=1+c_1z+c_2z^2+...$ 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.30 pagesQuantum Algebratext