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.

arXiv comment17 pages, Theorem 1.8 has been strengthened (we now classify all nonpointed Burnside fusion rings of rank four), and streamlined Section 3. Comments are welcome!