Computing Higher Frobenius-Schur Indicators in Fusion Categories Constructed from Inclusions of Finite Groups

TL;DR

This paper develops a formula to compute higher Frobenius-Schur indicators in a specific class of fusion categories derived from finite groups and their subgroups, with explicit examples involving symmetric groups.

Contribution

It introduces a new formula for calculating higher Frobenius-Schur indicators in group-theoretical fusion categories based on group combinatorics and representation theory.

Findings

Derived a formula for higher Frobenius-Schur indicators

Applied the formula to symmetric group inclusions

Provided explicit computational examples

Abstract

We consider a subclass of the class of group-theoretical fusion categories: To every finite group $G$ and subgroup $H$ one can associate the category of $G$-graded vector spaces with a two-sided $H$-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.