On fusion rules and solvability of a fusion category

TL;DR

This paper investigates whether the solvability of a fusion category can be deduced solely from its fusion rules, providing positive results for certain non-solvable examples and analyzing invariants in spherical fusion categories.

Contribution

It proves that fusion rules determine solvability for specific non-solvable categories from Hopf algebra representations and examines the $S$-matrix invariant in group-theoretical spherical fusion categories.

Findings

Fusion rules determine solvability for some non-solvable categories.

The $S$-matrix invariant determines solvability in group-theoretical spherical categories.

Positive results for categories arising from symmetric and alternating groups.

Abstract

We address the question whether the condition on a fusion category being solvable or not is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the $S$-matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.