Classifying integral Grothendieck rings up to rank 5 and beyond

TL;DR

This paper classifies integral Grothendieck rings up to rank 5 and extends the classification to higher ranks using computational and theoretical methods, revealing new structures and exceptions in fusion categories.

Contribution

It provides the first complete classification of integral Grothendieck rings up to rank 5 and extends results to higher ranks, introducing new techniques and identifying novel fusion categories.

Findings

Complete classification of integral Grothendieck rings up to rank 5.

Identification of the first non-Isaacs integral fusion category.

Classification of integral noncommutative Drinfeld rings of rank 8.

Abstract

In this paper, we define a Grothendieck ring as a fusion ring categorifiable into a fusion category over the complex field. An integral fusion ring is called Drinfeld if all its formal codegrees are integers dividing the global Frobenius--Perron dimension. Every integral Grothendieck ring is necessarily Drinfeld. Using the fact that the formal codegrees of integral Drinfeld rings form an Egyptian fraction summing to 1, we derive a finite list of possible global FPdims for small ranks. Applying Normaliz, we classify all fusion rings with these candidate FPdims, retaining only those admitting a Drinfeld structure. To exclude Drinfeld rings that are not Grothendieck rings, we analyze induction matrices to the Drinfeld center, classified via our new Normaliz feature. Further exclusions and constructions involve group-theoretical fusion categories and Schur multipliers. Our main result is a complete classification of integral Grothendieck rings up to rank 5, extended to rank 7 in odd-dimensional and noncommutative cases using Frobenius--Schur indicators and Galois theory. Moreover, we show that any noncommutative, odd-dimensional, integral Grothendieck ring of rank at most 22 is pointed of rank 21. We also classify all integral 1-Frobenius Drinfeld rings of rank 6, identify the first known non-Isaacs integral fusion category (which turns out to be group-theoretical), classify integral noncommutative Drinfeld rings of rank 8, and integral 1-Frobenius MNSD Drinfeld rings of rank 9. Finally, we determine the smallest-rank exotic simple integral fusion rings: rank 4 in general, rank 6 in the Drinfeld case, and rank 7 in the 1-Frobenius Drinfeld case.