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Hypergraph Characterization of Fusion Rings

Paul Bruillard, Kathleen Nowak, Stephen J. Young

TL;DR

This work develops a digraph/hypergraph framework to encode the Grothendieck rings of multiplicity-free, self-dual fusion categories, enabling graph-theoretic classification. By establishing a concrete correspondence between fusion rules and a pair (D,H) of a digraph and a 3-uniform hypergraph, the authors reduce fusion-ring enumeration to combinatorial graph problems solvable with nauty. They provide a complete triangle-free, undirected-graph classification, showing that the associated fusion categories are Grothendieck equivalent to $\mathrm{Fib}\,\mathrm{PSU}(3)_2$, $\mathrm{PSU}(2)_6$, or $\mathrm{Rep}(G)$ with $G \cong (\mathbb{Z}/2\mathbb{Z})^k$, and they enumerate all multiplicity-free, self-dual fusion rings up to rank $8$ using a computational graph-theoretic approach. The results highlight sharp structural constraints (e.g., Steiner triple systems for empty graphs and diameter/minor restrictions for nontrivial components) and provide a practical pathway to catalog fusion rings via combinatorial methods. Overall, the paper bridges fusion-ring theory with graph/hypergraph techniques to enable complete classifications in low-rank regimes and modular constructions via graph products.

Abstract

We present a correspondence between multiplicity-free, self-dual, fusion rings and a digraph, hypergraph pair $(D,H)$. This correspondence is used to provide a complete characterization of all fusion rings corresponding to graphical properties of $D$. Further, we exploit this correspondence to provide a complete list of all non-isomorphic, self-dual, multiplicity-free fusion rings of rank at most 8.

Hypergraph Characterization of Fusion Rings

TL;DR

This work develops a digraph/hypergraph framework to encode the Grothendieck rings of multiplicity-free, self-dual fusion categories, enabling graph-theoretic classification. By establishing a concrete correspondence between fusion rules and a pair (D,H) of a digraph and a 3-uniform hypergraph, the authors reduce fusion-ring enumeration to combinatorial graph problems solvable with nauty. They provide a complete triangle-free, undirected-graph classification, showing that the associated fusion categories are Grothendieck equivalent to , , or with , and they enumerate all multiplicity-free, self-dual fusion rings up to rank using a computational graph-theoretic approach. The results highlight sharp structural constraints (e.g., Steiner triple systems for empty graphs and diameter/minor restrictions for nontrivial components) and provide a practical pathway to catalog fusion rings via combinatorial methods. Overall, the paper bridges fusion-ring theory with graph/hypergraph techniques to enable complete classifications in low-rank regimes and modular constructions via graph products.

Abstract

We present a correspondence between multiplicity-free, self-dual, fusion rings and a digraph, hypergraph pair . This correspondence is used to provide a complete characterization of all fusion rings corresponding to graphical properties of . Further, we exploit this correspondence to provide a complete list of all non-isomorphic, self-dual, multiplicity-free fusion rings of rank at most 8.
Paper Structure (11 sections, 10 theorems, 29 equations, 3 figures)

This paper contains 11 sections, 10 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fusion Rings
  4. Graphs and Hypergraphs
  5. Fusion Ring/Digraph Correspondence
  6. Fusion Rings Generated by Undirected Graphs
  7. Empty Graphs
  8. Simple Components
  9. Non-simple Components
  10. Fusion Rings Generated by Triangle-Free Graphs
  11. Low Rank Fusion Rings

Key Result

Theorem 1

The multiplicity-free, self-dual braided fusion categories generated by an undirected, triangle-free graph $G$ are the Grothendieck equivalent to one of the following:

Figures (3)

  • Figure 1: Presentations of the Fusion Ring $\mathop{\mathrm{Rep}}\nolimits\left(\mathbb{Z}_{3}^{2}\rtimes\mathbb{Z}_{2}\right)$.
  • Figure 2: Initial edges and hyperedges required by vertex of degree 3.
  • Figure 3: Edges and hyperedges required by neighbors of vertex of degree 3.

Theorems & Definitions (19)

  • Theorem
  • Remark 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Remark 1
  • ...and 9 more