Some quasitensor autoequivalences of Drinfeld doubles of finite groups

TL;DR

This paper explores two classes of autoequivalences of the category of Yetter-Drinfeld modules over finite groups, related to the r-th power operation, which preserve certain invariants but are not monoidal.

Contribution

It introduces new autoequivalences of the Drinfeld center of finite group representations, extending known operations and analyzing their properties.

Findings

Autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugation.

Both autoequivalences preserve tensor products but are not monoidal.

One autoequivalence applies more generally to group-theoretical fusion categories.

Abstract

We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the $r$-th power operation, with $r$ relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.