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

Jingcheng Dong

TL;DR

The paper addresses the classification of Grothendieck rings for rank $4$ self-dual fusion categories with a nontrivial subcategory, by reducing to a rank $2$ subcategory $\mathcal{D}$ that is either pointed or Fibonacci. It derives three new Grothendieck-ring families, one fully determined and two parameterized by nonnegative integers, through explicit analysis of fusion rules, FP-dimensions, and commutativity constraints, with realizations in constructions related to Fibonacci-type products. These results advance the rank-$4$ fusion-category program and provide concrete algebraic data that inform connections to subfactor theory and modular categories. The work also clarifies how near-group and Fibonacci-type structures interact in low-rank extensions, offering a framework for further classification and braided/unitary explorations.

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.

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

TL;DR

The paper addresses the classification of Grothendieck rings for rank self-dual fusion categories with a nontrivial subcategory, by reducing to a rank subcategory that is either pointed or Fibonacci. It derives three new Grothendieck-ring families, one fully determined and two parameterized by nonnegative integers, through explicit analysis of fusion rules, FP-dimensions, and commutativity constraints, with realizations in constructions related to Fibonacci-type products. These results advance the rank- fusion-category program and provide concrete algebraic data that inform connections to subfactor theory and modular categories. The work also clarifies how near-group and Fibonacci-type structures interact in low-rank extensions, offering a framework for further classification and braided/unitary explorations.

Abstract

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

Paper Structure

This paper contains 5 sections, 3 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Grothendieck rings
  4. ${\mathcal{D}}$ is pointed
  5. ${\mathcal{D}}$ is a Fibonacci fusion category

Key Result

Proposition 3.1

Let ${\mathcal{C}}$ be a self-dual fusion category of rank $4$. Assume that $\operatorname{Irr}({\mathcal{C}})=\{\textbf{1},g,X,Y\}$ and $\{\textbf{1},g\}$ generates ${\mathcal{C}}_{pt}$. If the action $g\otimes-$ on $\operatorname{Irr}({\mathcal{C}})/G({\mathcal{C}})$ is fixed-point free then ${\ma Moreover, (eq02) is realized by $\rm Fib\otimes \operatorname{Vec}_{\mathbb{Z}_2}^{\omega}$ for som

Theorems & Definitions (6)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof