Graphs attached to simple Frobenius-Perron dimensions of an integral fusion category

Abstract

Let (Formula presented.) be an integral fusion category. We study some graphs, called the prime graph and the common divisor graph, related to the Frobenius-Perron dimensions of simple objects in the category (Formula presented.) , that extend the corresponding graphs associated to the irreducible character degrees and the conjugacy class sizes of a finite group. We describe these graphs in several cases, among others, when (Formula presented.) is an equivariantization under the action of a finite group, a (Formula presented.) -step nilpotent fusion category, and the representation category of a twisted quantum double. We prove generalizations of known results on the number of connected components of the corresponding graphs for finite groups in the context of braided fusion categories. In particular, we show that if (Formula presented.) is any integral non-degenerate braided fusion category, then the prime graph of (Formula presented.) has at most (Formula presented.) connected components, and it has at most (Formula presented.) connected components if (Formula presented.) is in addition solvable. As an application we prove a classification result for weakly integral braided fusion categories all of whose simple objects have prime power Frobenius-Perron dimension.