On the Grothendieck rings of equivariant fusion categories

Abstract

In this paper, we describe a Mackey type decomposition for group actions on abelian categories. This allows us to define new Mackey functors which associates to any subgroup the K-theory of the corresponding equivariantized abelian category. In the case of an action by tensor autoequivalences, the Mackey functor at the level of Grothendieck rings has a Green functor structure. As an application we give a description of the Grothendieck rings of equivariantized fusion categories under group actions by tensor autoequivalences on graded fusion categories. In this settings, a new formula for the tensor product of any two simple objects of an equivariantized fusion category is given, simplifying the fusion formula from Burciu and Natale [J. Math. Phys. 54, 013511 (2013)].