Fusion categories containing a fusion subcategory with maximal rank

TL;DR

This paper investigates fusion categories with a maximal rank fusion subcategory, establishing their spherical structure, classifying non-degenerate and symmetric cases, and providing a complete classification for rank 4.

Contribution

It introduces a classification framework for fusion categories containing a maximal rank subcategory, including spherical, non-degenerate, symmetric, and rank 4 cases.

Findings

Fusion categories with maximal rank subcategories admit spherical structures.

Classification of non-degenerate and symmetric cases achieved.

Complete classification of such categories at rank 4 provided.

Abstract

In this paper, we study fusion categories which contain a proper fusion subcategory with maximal rank. They can be viewed as generalizations of near-group fusion categories. We first prove that they admit spherical structure. We then classify those which are non-degenerate or symmetric. Finally, we classify such fusion categories of rank 4.