On some properties of quantum doubles of finite groups

TL;DR

This paper investigates properties of quantum doubles of finite groups, proving an integrality theorem for Frobenius-Schur indicators in certain wreath product groups and establishing a lower bound on the number of irreducible representations based on conjugacy classes.

Contribution

It introduces a new integrality theorem for Frobenius-Schur indicators in wreath product groups and provides a lower bound on irreducible representations of quantum doubles.

Findings

Proved integrality of higher Frobenius-Schur indicators for wreath product groups S_N#A^N.

Established a lower bound for the number of irreducible representations based on conjugacy classes.

Answered a previously open question about the maximum number of irreducible representations.

Abstract

We prove two results about quantum doubles of finite groups over the complex field. The first result is the integrality theorem for higher Frobenius-Schur indicators for wreath product groups S_N#A^N, where A is a finite abelian group. A proof of this result for A=1 appears in arXiv:1208.4153. The second result is a lower bound for the largest possible number of irreducible representations of the quantum double of a finite group with at most n conjugacy classes. This answers a question asked to me by Eric Rowell.