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arXiv · 2306.11721

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

S. Burciu
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Abstract

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 $p^2q^2r^2d$ with $p<q<r$ prime numbers and $d$ a square-free integer is weakly-group theoretical.

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S. Burciu. 2023-06-20. On Harada's identity and some other consequences of Burnside's vanishing property. https://arxiv.org/abs/2306.11721

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