A Higher Frobenius-Schur Indicator Formula for Group-Theoretical Fusion Categories

TL;DR

This paper derives formulas for higher Frobenius-Schur indicators of simple objects in group-theoretical fusion categories, linking these indicators to the underlying group and cohomological data, thus advancing understanding of their structure.

Contribution

It introduces explicit formulas for higher Frobenius-Schur indicators in group-theoretical fusion categories based on their defining group and cohomology data.

Findings

Formulas express indicators in terms of group cohomology data

Connects simple objects to projective representations of stabilizers

Enhances computational tools for fusion categories

Abstract

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $\omega$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $\omega$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $\omega$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius-Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.