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

Abstract

Let \({\mathcal C}\) 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 \({\mathcal C}\), 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 \({\mathcal C}\) is an equivariantization under the action of a finite group, a \(2\)-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 \({\mathcal C}\) is any integral non-degenerate braided fusion category, then the prime graph of \({\mathcal C}\) has at most \(3\) connected components, and it has at most \(2\) connected components if \({\mathcal C}\) 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.