Burnside type results for fusion categories - Semester 1, Session 16

This video is the recording of Session 16 (of Semester 1) of the course introducing to Burnside type results for fusion categories covering the paper in reference below. This session was given at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) on 17/05/2024.
Webpage:
bimsa.net/activity/Burtypresf...
Slides and video:
bimsa.net/doc/notes/30841.pdf
bimsa.net/bimsavideo.html?id=...
Prerequisite
Be a bit familiar with the finite group representation theory, and the notions of fusion categories, fusion rings and hypergroup, but the definitions will be recalled.
Introduction
We extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We also treat the dual vanishing result. We show that any nilpotent fusion categories satisfy both Burnside's property and its dual. Using Drinfeld's map, we obtain that the Grothendieck ring of any weakly-integral modular tensor category satisfies both properties. As applications, we prove new identities that hold in the Grothendieck ring of any weakly-integral fusion category satisfying the dual-Burnside's property, thus providing new categorification criteria. We also prove some new results on the perfect fusion categories.
Reference
- S. Burciu, S. Palcoux. Burnside type results for fusion rings, arXiv:2302.07604, [in a future version, "rings" should be replaced by "categories"],
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories, volume 205. Mathematical Surveys and Monographs, AMS, Providence, RI, 2015,
- P. Etingof, D. Nikshych, and V. Ostrik. On fusion categories. Ann. of Math. (2), 162(2):581-642, 2005,
- P. Etingof, D. Nikshych, and V. Ostrik. Weakly group-theoretical and solvable fusion categories. Adv. Math., 226(1):176-205, 2011,
- S. Gelaki and D. Nikshych. Nilpotent fusion categories. Adv. in Math., 217(3):1053-1071, 2008.