Sebastian Burciu (Simion Stoilow Institute of Mathematics of the Romanian Academy)

Abstract: We describe a Mackey type decomposition for group actions on abelian categories. 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. It is shown that these Grothendieck rings have a ring structure similar to the (double) Burnside rings of finite groups and some other rings obtained by Bouc and Witherspoon.