On Harada's identity and some other consequences

We establish several consequences of Burnside's vanishing property {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 p²q²r²d with p q r prime numbers and d a square-free integer is weakly-group theoretical.