On Harada's identity and some other consequences of Burnside's vanishing property
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2023
- DOI:
- 10.48550/arXiv.2306.11721
- arXiv:
- arXiv:2306.11721
- Bibcode:
- 2023arXiv230611721B
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematics - Representation Theory
- E-Print:
- 8 pages. Comments are welcome!