HAL CCSD
Clifford’s Theorem for Orbit Categories
Zimmermann, Alexander
Université de Picardie Jules Verne (UPJV)
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) ; Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 0927-2852
EISSN: 1572-9095
Applied Categorical Structures
Springer Verlag (Germany)
hal-04083225
https://u-picardie.hal.science/hal-04083225
https://u-picardie.hal.science/hal-04083225
Applied Categorical Structures, 2023, 31 (2), pp.17. ⟨10.1007/s10485-023-09721-4⟩
ARXIV: 2206.09394
info:eu-repo/semantics/altIdentifier/arxiv/2206.09394
DOI: 10.1007/s10485-023-09721-4
info:eu-repo/semantics/altIdentifier/doi/10.1007/s10485-023-09721-4
en
Clifford theory
Kleisli construction
Eilenberg-Moore construction
Orbit category
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
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 r 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 G.
2023-04