Normalizer of the Chevalley group of type ${\operatorname E}_7$
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N. A. Vavilov and A. Yu. Luzgarev
Translated by: N. A. Vavilov - St. Petersburg Math. J. 27 (2016), 899-921
- 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.References
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Bibliographic 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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 899-921
- MSC (2010): Primary 20G15
- DOI: https://doi.org/10.1090/spmj/1426
- MathSciNet review: 3589222
Dedicated: To St. Petersburg remarkable algebraist Sergei Vladimirovich Vostokov, a teacher, a colleague, and a friend