On the Brauer-Picard groups of fusion categories

TL;DR

This paper develops computational methods for Brauer-Picard groups of fusion categories and applies them to various classes, revealing connections to finite groups of Lie type.

Contribution

It introduces new methods for computing Brauer-Picard groups and applies them to specific fusion categories, uncovering their structure and relation to finite groups of Lie type.

Findings

Brauer-Picard groups computed for categories of prime power dimension

Many finite groups of Lie type appear as composition factors

Explicit calculations for elementary abelian groups, extra special p-groups, and Kac-Paljutkin algebra

Abstract

We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity constraint, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.