On some integral properties of dimensions in Isaacs fusion categories

TL;DR

This paper investigates integral properties of object dimensions in Isaacs fusion categories, establishing new divisibility results that extend to modular fusion categories and relate to Ito-Michler type theorems.

Contribution

It introduces new integral divisibility properties for simple object dimensions in Isaacs and modular fusion categories, advancing understanding of their structural relationships.

Findings

Proves new integral properties for simple object dimensions in Isaacs fusion categories.

Establishes a stronger divisibility result for modular fusion categories.

Derives a converse of the Ito-Michler type theorem for modular fusion categories.

Abstract

For a fusion category, we prove some new integral properties concerning the dimension of a simple object that generates a Isaacs fusion subcategory. A stronger divisibility result is proven for any modular fusion category. This divisibility result implies the converse direction of a Ito-Michler type result for modular fusion categories, recently established by the author.