Grothendieck rings of near-Tambara--Yamagami fusion categories

Abstract

We determine the Grothendieck rings of fusion categories over $\mathcal{C}$ containing a nonpointed Tambara--Yamagami fusion subcategory and exactly one further simple object. The subcategory necessarily has rank five, its invertible objects form the Klein four-group, and the coefficient $\kappa$ of the additional simple object in its tensor square belongs to $\{0,2,7\}$. Each of these three based rings is categorifiable. For $\kappa =0$, every category belongs, up to tensor equivalence, to the rank-one odd-component extension list of Choi--Lu--Sun. For $\kappa =2$ we construct an explicit unitary group-theoretical realization, and for $\kappa =7$ a realization is $\mathrm{Rep}(\mathrm{PSU}(3,2))$.

arXiv comment31 pages