Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories

TL;DR

This paper extends the computation of Frobenius-Schur indicators to specific fusion categories generated by a single non-invertible object, using modular data and tube algebra techniques.

Contribution

It provides explicit formulas for Frobenius-Schur indicators in near-group and Haagerup-Izumi fusion categories, expanding computational tools in fusion category theory.

Findings

Derived formulas for indicators in near-group categories

Derived formulas for indicators in Haagerup-Izumi categories

Utilized Evans-Gannon tube algebra computations

Abstract

Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariants of finite-depth Murray-von Neumann subfactors. We utilize the Evans-Gannon computation of the tube algebras to obtain formulae for the Frobenius-Schur indicators of objects in both of these families.