Frobenius-Perron Dimensions of Conjugacy Classes and an Ito-Michler-Type Result in Modular Fusion Categories

TL;DR

This paper extends classical group theory results to modular fusion categories, showing how arithmetic conditions on conjugacy class sizes influence the structure of these categories.

Contribution

It establishes an Ito-Michler-type theorem for modular fusion categories, linking conjugacy class sizes to the category structure, a novel generalization from finite groups.

Findings

Proves an Ito-Michler-type result in modular fusion categories.

Connects conjugacy class sizes with fusion category structure.

Provides new insights into the arithmetic properties of fusion categories.

Abstract

The influence of certain arithmetic conditions on the sizes of conjugacy classes of a finite group on the group structure has been extensively studied in recent years. In this paper, we explore analogous properties for fusion categories. In particular, we establish an Ito-Michler-type result for modular fusion categories.