Clifford's theorem for orbit categories
Résumé
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 Γ 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 Γ
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)