On a question of Külshammer for representations of finite groups in reductive groups

Michael Bate; Benjamin Martin; Gerhard Rohrle

doi:10.1007/s11856-016-1337-2

On a question of Külshammer for representations of finite groups in reductive groups

Michael Bate, Benjamin Martin, Gerhard Rohrle

Mathematical Science

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Abstract

Let G be a simple algebraic group of type G2 over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ2 and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ2 are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H1(Γ, V) → H1(Γp, V) induced by restriction of 1-cocycles has an infinite fibre.

To Burkhard Külshammer on his sixtieth birthday

Original language	English
Pages (from-to)	463-470
Number of pages	8
Journal	Israel Journal of Mathematics
Volume	214
Issue number	1
DOIs	https://doi.org/10.1007/s11856-016-1337-2
Publication status	Published - 1 Jul 2016

Bibliographical note

Acknowledgments. The authors acknowledge the financial support of EPSRC
Grant EP/L005328/1, Marsden Grants UOC1009 and UOA1021, and the
DFG-priority programme SPP1388 “Representation Theory”. We are grateful
to the referee for helpful suggestions.

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Cite this

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abstract = "Let G be a simple algebraic group of type G2 over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ2 and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ2 are all conjugate. This answers a question of Burkhard K{\"u}lshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H1(Γ, V) → H1(Γp, V) induced by restriction of 1-cocycles has an infinite fibre.To Burkhard K{\"u}lshammer on his sixtieth birthday",

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AB - Let G be a simple algebraic group of type G2 over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ2 and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ2 are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H1(Γ, V) → H1(Γp, V) induced by restriction of 1-cocycles has an infinite fibre.To Burkhard Külshammer on his sixtieth birthday

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On a question of Külshammer for representations of finite groups in reductive groups

Abstract

Bibliographical note

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