Burnside-type results for fusion categories

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 [V. Ostrik and Z. Yu, On the minimal extension and structure of braided weakly group-theoretical fusion categories, Adv. Math. 419 (2023) 108961, Theorem 4.5] as follows: A weakly integral modular fusion category of [Formula: see text] [Formula: see text] with [Formula: see text] square-free coprime with [Formula: see text] and [Formula: see text] for every simple object [Formula: see text], has a pointed modular fusion subcategory of [Formula: see text] [Formula: see text]. We also present new results on perfect modular fusion categories, including a Cauchy-type theorem.