On Harada's identity and some other consequences of Burnside's vanishing property

In this short note, we prove some consequences of Burnside’s vanishing property \cite{b-pa}. It is known that Harada’s identity concerning the product of all conjugacy classes of a finite group is a consequence of Burnside’s vanishing property of characters. We prove a similar formula for any weakly-integral fusion category. In the second part, we prove some structural results concerning nilpotent and modular fusion categories. As an application we show that any fusion category of dimension $p^2{q}^2{r}^{2}d$ with $p<q<r$ prime numbers and $d$ a square-free integer is weakly-group theoretical.