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A new look at the decomposition of unipotents and the normal structure of Chevalley groups


Author: A. Stepanov
Translated by: the author
Original publication: Algebra i Analiz, tom 28 (2016), nomer 3.
Journal: St. Petersburg Math. J. 28 (2017), 411-419
MSC (2010): Primary 20G35
DOI: https://doi.org/10.1090/spmj/1456
Published electronically: March 29, 2017
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Abstract: The paper continues a series of publications on the decomposition of unipotents in a Chevalley group $ \mathrm {G} (\Phi ,R)$ over a commutative ring $ R$ with a reduced irreducible root system $ \Phi $. Fix $ h\in \mathrm {G} (\Phi ,R)$. An element $ a\in \mathrm {G} (\Phi ,R)$ is said to be ``good'' if it belongs to the unipotent radical of a parabolic subgroup and the conjugate to $ a$ by $ h$ lies in another parabolic subgroup (all parabolic subgroups are assumed to contain the same split maximal torus). The ``decomposition of unipotents'' method is a representation of an elementary root unipotent element as a product of ``good'' elements. Decomposition of unipotents implies a simple proof of normality for the elementary subgroup and the standardness for the normal structure of $ \mathrm {G} (\Phi ,R)$. However, such a decomposition is available not for all root systems. In the paper, it is shown that to prove the standardness of the normal structure it suffices to find one ``good'' element for the generic element of the group scheme $ \mathrm {G}(\Phi ,{\hbox to 1.5ex{\hrulefill }})$. Also, some ``good'' elements are constructed. The question as to whether and when good elements span the elementary subgroup will be considered in a subsequent article of the series.


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Additional Information

A. Stepanov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University; St. Petersburg State Electrotechnical University “LETI”
Email: stepanov239@gmail.com

DOI: https://doi.org/10.1090/spmj/1456
Keywords: Chevalley groups, parabolic subgroup, unipotent element, generic element, universal localization, normal structure
Received by editor(s): December 15, 2015
Published electronically: March 29, 2017
Additional Notes: Supported by RSF, (grant no. 14-11-00297).
Article copyright: © Copyright 2017 American Mathematical Society