Fusion categories arise in the study of conformal field theory While Rep(G) for G a finite group is the primary example of a fusion category, categories which are Morita equivalent to Rep (G) ( the so-called group theoretical fusion categories or GTCs) are also an important class of examples. We study some invariants of these categories. In order to study fusion rings of group theoretical fusion categories, we develop a character theory for these categories (similar to characters in Rep(G)) and compute fusion rings for GTCs of dimension < 21. We find counterexamples to a conjecture about Frobenius-Schur indicators of Morita equivalent fusion categories.  Drinfeld centers of certain GTCs (aka twisted group doubles) are not distinguished by their modular data. We study a topological invariant called the B-tensor and show that it can distingush all twisted group doubles where the group is of odd square-free order.
https://indico.math.cnrs.fr/event/6498/