On low rank fusion rings

Abstract

We present a method to generate all fusion rings of a specific rank and multiplicity. This method generated 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 construction generalises the Tambara–Yamagami (TY) and Haagerup-Izumi (HI) fusion rings. We give an example of two such rings that are categorifiable and 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. We also applied several categorifiability criteria to the fusion rings. We provide a table of all multiplicity-free fusion rings up to rank 9 with info on the categorifiability of each ring.