Classifying integral Grothendieck rings up to rank 5 and beyond

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 candidate global FPdims for small ranks, covering all that occur. Using Normaliz (an open-source software for computing algebraic polyhedra), we classify all integral 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 feature on Normaliz. 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), and classify integral noncommutative Drinfeld rings of rank 8 and integral 1-Frobenius MNSD Drinfeld rings of rank 9. Finally, we determine the smallest rank simple exotic integral fusion rings: rank 4 in general, rank 6 in the Drinfeld case, and rank 7 in the 1-Frobenius Drinfeld case.