On Harada's identity and some other consequences (Q6614473)

The author explores certain implications of Burnside's vanishing property. Notably, Harada's identity, which relates to the product of all conjugacy classes in a finite group, arises as a direct consequence of this property. Extending this idea, he establishes an analogous formula for any weakly-integral fusion category. Additionally, he presents structural results pertaining to nilpotent and modular fusion categories. More concretely, he proves that, if \({\mathcal C}\) is a weakly-integral fusion category, \({\mathcal C}^j\) is the set of all its conjugacy classes and \(C_j\) their associated class sums, then one has\N\[\N (\prod_{j=0}^m \frac{C_j}{\mathrm{FPdim}(\mathcal C^j)})^2= \frac{\mathrm{FPdim}({\mathcal C}_{\mathrm{pt}})}{\mathrm{FPdim}({\mathcal C})}\big(\sum_{j\in {\mathcal J}_{{\mathcal C}_{\mathrm{pt}}}}C_j\big). \N\]\NAs a consequence, if \({\mathcal C}\) is a weakly-integral modular category and we suppose that \({\mathcal C}\) is of type \((d_1,n_1;d_2,n_2;\dots d_r,n_r)\) with \(d_1\)=1, then he proves that when \(\mathrm{FPdim}({\mathcal C})=nd\) with \((n,d)=1\) and \(d\) a square-free integer we have that \(d\big| n_i\) for all \(i\geq 1\). Finally, as an application, he obtains that, if \({\mathcal C}\) is an integral modular fusion category of dimension \(p^2q^2r^2d\) with \(d\) a square-free number and \(p<q<r\) three prime numbers, then \({\mathcal C}\) is weakly group-theoretical and can be written as a Deligne product \({\mathcal A}\boxtimes {\mathcal B}\) with \({\mathcal A}\simeq {\mathcal C}(\mathbb Z_q, d)\) and \({\mathcal B}\) a weakly group theoretical fusion category of dimension \(p^2q^2r^2\).