On Low Rank Fusion Rings

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.

31 pages, 2 figures, 5 tables Changes since v1: * Updated references * Added comment on term Abelian in table in appendix * Changed "commutative fusion rings" -> "fusion rings" in section 4 * FIXED ERROR: the ring FR_8_1_2_4 has no categorifications, FR_9_1_4_3 has. * Replaced CategorificationsFR_9_1_4_3.wdx by file which also contains trivial F-symbols