On Grothedieck rings of rank $4$ self-dual fusion categories

TL;DR

This paper classifies certain Grothendieck rings associated with rank 4 self-dual fusion categories, identifying three new families with explicit and parameterized structures.

Contribution

It introduces three new families of Grothendieck rings for rank 4 self-dual fusion categories, including a fully determined case and two parameterized families.

Findings

One family of Grothendieck rings is completely classified.

Two additional families are described with multiple parameters.

The results expand understanding of fusion category structures.

Abstract

Let $\C$ be a self-dual fusion category of rank $4$ which has a nontrivial proper fusion subcategory. We identify three new families of Grothendieck rings for $\C$: one of them is completely determined, the other two are parameterized by several non-negative integers.