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Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)

Victor Ostrik

TL;DR

The paper provides a near-complete classification of fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero, identifying five explicit families (including Ising and $S_3$-related cases) and tightly constraining other rank-3 possibilities via the induction functor to the Drinfeld center, formal codegrees, and a pseudo-unitary inequality. It develops new toolkits—most notably a decomposition result for $I(\mathbf 1)$ and a powerful inequality on formal codegrees—that prune vast swathes of potential rank-3 based rings, and combines these with cyclotomic and $d$-number tests to isolate categorifiable structures. The appendix extends these methods to near-group categories, establishing restrictions on their Grothendieck rings and demonstrating their group-theoretical nature in the integral case. The work connects to subfactor theory (e.g., $E_6$-related categories) and quantum group examples, illuminating the landscape of small fusion categories and guiding future searches for exotic rank-3 examples. Overall, it significantly advances understanding of how simple rank-3 fusion categories can arise and interact with center, sphericality, and modular data.

Abstract

We classify all fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero. Also in the Appendix (joint with D.Nikshych) we give some restrictions on Grothendieck rings of near-group categories.

Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)

TL;DR

The paper provides a near-complete classification of fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero, identifying five explicit families (including Ising and -related cases) and tightly constraining other rank-3 possibilities via the induction functor to the Drinfeld center, formal codegrees, and a pseudo-unitary inequality. It develops new toolkits—most notably a decomposition result for and a powerful inequality on formal codegrees—that prune vast swathes of potential rank-3 based rings, and combines these with cyclotomic and -number tests to isolate categorifiable structures. The appendix extends these methods to near-group categories, establishing restrictions on their Grothendieck rings and demonstrating their group-theoretical nature in the integral case. The work connects to subfactor theory (e.g., -related categories) and quantum group examples, illuminating the landscape of small fusion categories and guiding future searches for exotic rank-3 examples. Overall, it significantly advances understanding of how simple rank-3 fusion categories can arise and interact with center, sphericality, and modular data.

Abstract

We classify all fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero. Also in the Appendix (joint with D.Nikshych) we give some restrictions on Grothendieck rings of near-group categories.

Paper Structure

This paper contains 28 sections, 36 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fusion categories and based rings
  4. The induction functor $I : \mathcal{C} \to \mathcal{Z}(\mathcal{C})$
  5. Formal codegrees and decomposition of $I(\mathbf{1})$
  6. Pseudo-unitary inequality
  7. Categorifiable simple based rings of rank 3
  8. Based rings of rank 3
  9. Notations
  10. Pivotal categories of rank 3 are pseudo-unitary
  11. The generic case
  12. Small cases
  13. The case $b<50$
  14. The case $c-4b\le 4$
  15. The ring $K(2,1,2,1)$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a fusion category of rank 3 admitting a pivotal structure over an algebraically closed field of characteristic zero. Then $\mathcal{C}$ is equivalent to one of the following: (i) pointed category with underlying group ${\mathbb Z}/3{\mathbb Z}$; (ii) even part of the category

Theorems & Definitions (85)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • Theorem 2.7
  • Proposition 2.8
  • ...and 75 more