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Classification of vanishing fusion rings of rank four
Abstract
A fusion ring is said to be vanishing if one of its multiplication matrices has determinant zero. We classify all vanishing fusion rings of rank four. We then provide a more explicit classification in the integral case, which leads to new consequences: there are infinitely many nonisomorphic simple integral vanishing fusion rings of rank four and no simple integral fusion ring of rank four is 1-Frobenius. We also classify all Burnside fusion rings of rank four.
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BibTeX
@misc{dong2026classification,
title = {{Classification of vanishing fusion rings of rank four}},
author = {Jingcheng Dong and Sebastien Palcoux and Arnaud Plessis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22000},
eprint = {2610.22000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22000},
}
RIS
TY - UNPB
TI - Classification of vanishing fusion rings of rank four
AU - Jingcheng Dong
AU - Sebastien Palcoux
AU - Arnaud Plessis
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.22000
UR - https://arxiv.org/abs/2610.22000
AB - A fusion ring is said to be vanishing if one of its multiplication matrices has determinant zero. We classify all vanishing fusion rings of rank four. We then provide a more explicit classification in the integral case, which leads to new consequences: there are infinitely many nonisomorphic simple integral vanishing fusion rings of rank four and no simple integral fusion ring of rank four is 1-Frobenius. We also classify all Burnside fusion rings of rank four.
ER -