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Normalizer of the Chevalley group of type $ {\operatorname E}_7$


Authors: N. A. Vavilov and A. Yu. Luzgarev
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 899-921
MSC (2010): Primary 20G15
DOI: https://doi.org/10.1090/spmj/1426
Published electronically: September 30, 2016
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Abstract: The simply connected Chevalley group $ G(\mathrm {E}_7,R)$ of type $ \mathrm {E}_7$ is considered in the 56-dimensional representation. The main objective is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group $ E(\mathrm {E}_7,R)$, the normalizer of the Chevalley group $ G(\mathrm {E}_7,R)$ itself, the transporter of $ E(\mathrm {E}_7,R)$ into $ G(\mathrm {E}_7,R)$, and the extended Chevalley group $ \bar {G}(\mathrm {E}_7,R)$. This holds over an arbitrary commutative ring $ R$, with all normalizers and transporters being calculated in $ \mathrm {GL}(56,R)$. Moreover, $ \bar {G}(\mathrm {E}_7,R)$ is characterized as the stabilizer of a system of quadrics. This last result is classically known over algebraically closed fields, here it is proved that the corresponding group scheme is smooth over $ \mathbb{Z}$, which implies that it holds over arbitrary commutative rings. These results are a key step in a subsequent paper, devoted to overgroups of exceptional groups in minimal representations.


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Additional Information

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

A. Yu. Luzgarev
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, University prospect 28, Petrodvorets, 198504 St. Petersburg, Russia
Email: a.luzgarev@spbu.ru

DOI: https://doi.org/10.1090/spmj/1426
Keywords: Chevalley groups, elementary subgroups, minimal modules, invariant forms, decomposition of unipotents, root elements, highest weight orbit
Received by editor(s): May 11, 2015
Published electronically: September 30, 2016
Additional Notes: The main results of the present paper were proven in the framework of the RSF project 14-11-00297
Dedicated: To St. Petersburg remarkable algebraist Sergei Vladimirovich Vostokov, a teacher, a colleague, and a friend
Article copyright: © Copyright 2016 American Mathematical Society