Finite Group Modular Data

TL;DR

This paper explores the properties of modular data associated with finite groups, comparing it to affine algebra data, and explicitly computes twisted modular matrices using group cohomology.

Contribution

It provides the first explicit formulas for twisted modular S and T matrices for finite groups, expanding understanding of their modular data properties.

Findings

Finite group modular data shares some properties with affine algebra data.

Explicit formulas for twisted modular matrices are derived.

The study highlights differences and similarities between finite and affine modular data.

Abstract

In a remarkable variety of contexts appears the modular data associated to finite groups. And yet, compared to the well-understood affine algebra modular data, the general properties of this finite group modular data has been poorly explored. In this paper we undergo such a study. We identify some senses in which the finite group data is similar to, and different from, the affine data. We also consider the data arising from a cohomological twist, and write down, explicitly in terms of quantities associated directly with the finite group, the modular S and T matrices for a general twist, for what appears to be the first time in print.