Green equivalences in equivariant mathematics

Balmer, Paul; Dell’Ambrogio, Ivo

doi:10.1007/s00208-021-02145-2

Cited by 8 publications

(4 citation statements)

References 15 publications

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“…To outline another application, let us simply mention that the general Green correspondence of [3] is most useful for Mackey 2-functors which are cohomological, for which it gives rise to a ‘ p -local theory’ as in modular representation theory. See [3, §§ 6-7] for details.…”

Section: Cohomological Mackey 2-functorsmentioning

confidence: 99%

Cohomological Mackey 2-Functors

Balmer¹,

Dell’Ambrogio²

2022

J. Inst. Math. Jussieu

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We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

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Section: Cohomological Mackey 2-functorsmentioning

confidence: 99%

Cohomological Mackey 2-Functors

Balmer¹,

Dell’Ambrogio²

2022

J. Inst. Math. Jussieu

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“…Remark. To outline another application, let us simply mention that the general Green correspondence of [BD21] is most useful for Mackey 2-functors which are cohomological, for which it gives rise to a 'p-local theory' as in modular representation theory. See [BD21, § § 6-7] for details.…”

Section: Theorem If M Is Any Cohomological Mackey 2-functor Its Value...mentioning

confidence: 99%

Mackey 2-Functors and Mackey 2-Motives

Balmer

Dell’Ambrogio

2020

EMS Monographs in Mathematics

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We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [BD20], obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

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“…Alternative approaches were given by Benson, Carlson, Grime, Peng, Wheeler, Wang and Zhang in a series of joint papers [8,22,6,9,31]. Another different approach is given by Balmer and del-Ambrogio [5] in the context of tensor triangulated categories and Mackey 2-functors.…”

Section: Introductionmentioning

confidence: 99%

Clifford's theorem for orbit categories

Zimmermann¹

2022

Preprint

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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.

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Green equivalences in equivariant mathematics

Cited by 8 publications

References 15 publications

Cohomological Mackey 2-Functors

Cohomological Mackey 2-Functors

Mackey 2-Functors and Mackey 2-Motives

Clifford's theorem for orbit categories

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