On Low Rank Fusion Rings

TL;DR

This paper introduces a comprehensive method for generating and classifying fusion rings of specific ranks, including new non-commutative examples, and provides computational tools and data resources for further research.

Contribution

It develops a systematic approach to generate all fusion rings of given rank and multiplicity, introduces a new class of non-commutative fusion rings, and provides computational tools and data for the community.

Findings

Generated exhaustive lists of fusion rings up to order 9.

Introduced a new class of non-commutative fusion rings based on group actions.

Provided a categorifiable example outside TY and HI types.

Abstract

We present a method to generate all fusion rings of a specific rank and multiplicity. This method was used to generate exhaustive lists of fusion rings up to order 9 for several multiplicities. We introduce a class of non-commutative fusion rings based on a group with transitive action on a set. This generalizes the Tambara-Yamagami (TY) and Haagerup- Izumi (HI) fusion rings. We give an example of such a ring which is categorifiable and is not of TY or HI type. The structure of non-commutative fusion rings with a subgroup is reviewed, and the one-and two-particle extensions of groups are classified. A website containing data on fusion rings is introduced, and an overview of a Wolfram Language package for working with these rings is given.