Burnside type results for fusion categories

TL;DR

This paper generalizes Burnside's vanishing results from finite group character tables to fusion rings and hypergroups, establishing new properties for nilpotent and weakly-integral fusion categories with applications to categorification.

Contribution

It extends classical Burnside results to fusion categories and hypergroups, and derives new identities and criteria for weakly-integral and perfect modular fusion categories.

Findings

Nilpotent unitary fusion categories satisfy Burnside's property and its dual.

Grothendieck rings of weakly-integral modular fusion categories satisfy these properties.

New categorification criteria and a Cauchy-type theorem for perfect modular fusion categories.

Abstract

In this paper, we extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We also treat the dual vanishing result. We show that any nilpotent unitary fusion categories satisfy both Burnside's property and its dual. Using Drinfeld's map, we obtain that the Grothendieck ring of any weakly-integral modular fusion category satisfies both properties. As applications, we prove new identities that hold in the Grothendieck ring of any weakly-integral fusion category satisfying the dual-Burnside's property, thus providing new categorification criteria. In particular we improve [OY23, Theorem 4.5] as follows: A weakly integral modular fusion category of FPdim md with d square-free coprime with m and FPdim(X)^2 for every simple object X, has a pointed modular fusion subcategory of FPdim d. We also present new results on perfect modular fusion categories, including a Cauchy-type theorem.