$FSZ$-groups and Frobenius-Schur Indicators of Quantum Doubles

TL;DR

This paper investigates the higher Frobenius-Schur indicators of quantum doubles of finite groups, identifying which groups have all integer indicators and providing examples of both groups with and without this property.

Contribution

It characterizes groups with integer Frobenius-Schur indicators for their quantum doubles, including many classical groups and a counterexample among nilpotent groups.

Findings

Many classical groups have all integer indicators

Some nilpotent groups have non-integer indicators

The property is linked to specific group-theoretic features

Abstract

We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, PSL_2(q), M_{11}, M_{12} and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5^6 with non-integer indicators.