On Harada's identity and some other consequences

Abstract

We establish several consequences of Burnside's vanishing property \cite{b-galois}. One well-known implication of this property is Harada's identity, which concerns the product of all conjugacy class sums of a finite group and is derived from the vanishing property of characters. Expanding on this idea, we present a similar formula applicable to any weakly-integral fusion category. As an application we show that any modular integral tensor category of dimension $p^2q^2r^2d$ with $p\prec q\prec r$ prime numbers and $d$ a square-free integer is weakly-group theoretical.