G - functors arising from categorical group actions on abelian categories

TL;DR

This paper develops a Mackey type decomposition for group actions on abelian categories, leading to new Mackey functors that connect subgroup actions with K-theory and Green functor structures, with applications to fusion categories.

Contribution

It introduces a novel Mackey functor framework for abelian categories under group actions, extending to Green functors in tensor autoequivalence cases, and applies to fusion categories.

Findings

Defined Mackey functors associating subgroups with K-theory of equivariantized categories.

Established Green functor structures on Grothendieck rings for tensor autoequivalence actions.

Described Grothendieck rings of equivariantized fusion categories under group actions.

Abstract

A Mackey type decomposition for group actions on abelian categories is described. 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.