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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Normalizer of the Chevalley group of type $ {\mathrm E}_6$


Authors: N. A. Vavilov and A. Yu. Luzgarev
Translated by: the authors
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 699-718
MSC (2000): Primary 20G15
Published electronically: June 25, 2008
MathSciNet review: 2381940
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Abstract: We consider the simply connected Chevalley group $ G(\operatorname{E}_6,R)$ of type $ \operatorname{E}_6$ in a 27-dimensional representation. The main goal is to establish that the following four groups coincide: the normalizer of the Chevally group $ G(\operatorname{E}_6,R)$ itself, the normalizer of its elementary subgroup $ E(\operatorname{E}_6,R)$, the transporter of $ E(\operatorname{E}_6,R)$ in $ G(\operatorname{E}_6,R)$, and the extended Chevalley group $ \overline G(\operatorname{E}_6,R)$. This is true over an arbitrary commutative ring $ R$, all normalizers and transporters being taken in $ \operatorname{GL}(27,R)$. Moreover, $ \overline G(\operatorname{E}_6,R)$ is characterized as the stabilizer of a system of quadrics. This result is classically known over algebraically closed fields; in the paper it is established that the corresponding scheme over $ \mathbb{Z}$ is smooth, which implies that the above characterization is valid over an arbitrary commutative ring. As an application of these results, we explicitly list equations a matrix $ g\in\operatorname{GL}(27,R)$ must satisfy in order to belong to $ \overline G(\operatorname{E}_6,R)$. These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.


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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198504, Russia
Email: nikolai-vavilov@yandex.ru

A. Yu. Luzgarev
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198504, Russia

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01016-9
PII: S 1061-0022(08)01016-9
Keywords: Chevalley groups, elementary subgroups, normal subgroups, standard description, minimal module, parabolic subgroups, decomposition of unipotents, root elements, orbit of the highest weight vector, the proof from the Book
Received by editor(s): May 20, 2007
Published electronically: June 25, 2008
Additional Notes: Supported by RFBR (grant no. 03-01-00349) and by INTAS (grant nos. 00-566 and 03-51-3251). Part of the work was carried out during the authors’ stays at the University of Bielefeld
Dedicated: Dedicated to the centenary of the birth of Dmitriĭ Konstantinovich Faddeev
Article copyright: © Copyright 2008 American Mathematical Society