Fusion rules of equivariantizations of fusion categories

TL;DR

This paper derives the fusion rules for equivariantizations of fusion categories, linking them to the original category and group actions, with applications to pointed and braided group-theoretical categories.

Contribution

It provides explicit formulas for fusion rules of equivariantized fusion categories based on group actions and fusion data, advancing understanding of their structure.

Findings

Fusion rules expressed via original category and group data

Formula for pointed fusion categories' fusion rules

Description of fusion rules in braided group-theoretical categories

Abstract

We determine the fusion rules of the equivariantization of a fusion category $\mathcal{C}$ under the action of a finite group $G$ in terms of the fusion rules of $\mathcal{C}$ and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.