Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

TL;DR

This paper classifies Grothendieck rings of complex fusion categories with multiplicity one up to rank six, identifying 47 categorifiable rings and discovering 6 new Grothendieck rings through a localization approach.

Contribution

It provides a complete classification of certain fusion rings and introduces a new method for categorification using localization of the Pentagon Equation.

Findings

47 fusion rings admit unitary complex categorification

25 rings filtered out by categorification criteria

6 new Grothendieck rings discovered

Abstract

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.