Representation and character theory of finite categorical groups

TL;DR

This paper explores the representation theory of finite categorical groups, focusing on gerbal representations, categorical characters, and their relation to twisted Drinfeld doubles, using diagrammatic methods.

Contribution

It introduces a framework for understanding gerbal representations and categorical characters of finite categorical groups through diagrammatic formalism and their connection to twisted Drinfeld doubles.

Findings

Categorical characters form modules over twisted Drinfeld doubles.

Gerbal representations relate to module categories over $Vec_G^eta$.

The approach uses adapted string diagram formalism.

Abstract

We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a module over the twisted Drinfeld double $D^\alpha(G)$. We interpret this twisted Drinfeld double in terms of the inertia groupoid of a categorical group.