Groups of projectivities and Levi subgroups in spherical buildings of simply laced type
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Sira Busch
Abstract
We introduce the special and general projectivity groups attached to a simplex 𝐹 of a thick, irreducible, spherical building of simply laced type.
If the residue of 𝐹 is irreducible, we determine the permutation group of both projectivity groups of 𝐹, acting on the residue of 𝐹 and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue.
This reveals three special cases for the exceptional types
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC 2044 – 390685587
Funding source: Marsden Fund
Award Identifier / Grant number: UOA-2122
Funding statement: The first author is funded by the Claussen-Simon-Stiftung and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics–Geometry–Structure. All authors were supported by the New Zealand Marsden Fund grant UOA-2122 of the second author. This work is part of the PhD project of the first author.
Acknowledgements
The authors are grateful to Gernot Stroth for an illuminating discussion concerning the structure and action of Levi complements in Chevalley groups and to the referee for some very helpful comments and suggesting different approaches at various points, in particular the approach using algebraic groups in Section 8.1.
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Communicated by: Christopher W. Parker
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