Clifford's theorem for orbit categories

TL;DR

This paper extends Clifford's theorem from group representation theory to orbit categories derived from Krull-Schmidt categories with group actions, clarifying the decomposition of indecomposables in this broader context.

Contribution

It generalizes Clifford's theorem to Krull-Schmidt orbit categories with finite group actions, linking it to monad theory and providing new insights into object decompositions.

Findings

Clifford's theorem is valid for Krull-Schmidt orbit categories.

The decomposition of indecomposables is clarified in the orbit category setting.

The adjoint functors form the Kleisli category of a natural monad.

Abstract

Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford's theorem for Krull-Schmidt orbit categories with respect to a finite group $\Gamma$ of automorphisms, clarifying this way how the image of an indecomposable object in the original category decomposes in the orbit category. The pair of adjoint functors appears as the Kleisli category of the naturally appearing monad given by $\Gamma$.