On the Grothendieck ring of fusion categories of rank 5

Abstract

In this paper, we study fusion categories of rank 5 with nontrivial faithful grading, and give a partial classification of their Grothendieck rings. Specifically, we obtain five Grothendieck rings, three of them can be realized and the last two are new. Moreover, the new ones meet the Ostrik’s criterion for categorification and the Schur product criterion for categorification.