Higher frobenius-schur indicators for drinfeld doubles of finite groups through characters of centralizers

TL;DR

This paper introduces an efficient conjugacy class-based method for computing higher Frobenius-Schur indicators of Drinfeld doubles of finite groups, enabling large-scale computations and rationality testing.

Contribution

It provides a novel, practical formula for indicators that improves computational efficiency and allows rationality analysis without explicit calculation.

Findings

Indicators are non-negative for symmetric groups up to S_{18}

Most simple groups have rational indicators, with few exceptions

Efficient implementation in GAP enables large group analysis

Abstract

We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{\"a}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $2 \cdot 10^8$. The approach also allows us to test whether all indicators over the double of a given group are rational , without computing them. Among simple groups of order up to about $5 \cdot 10^{11}$ an inspection yields exactly one example (of order about $5 \cdot 10^9$) where irrational indicators occur.