Normalizer of the Chevalley group of type ${\mathrm E}_6$
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N. A. Vavilov and A. Yu. Luzgarev
Translated by: the authors - St. Petersburg Math. J. 19 (2008), 699-718
- DOI: https://doi.org/10.1090/S1061-0022-08-01016-9
- Published electronically: June 25, 2008
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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.References
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Bibliographic 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
- 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
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 699-718
- MSC (2000): Primary 20G15
- DOI: https://doi.org/10.1090/S1061-0022-08-01016-9
- MathSciNet review: 2381940
Dedicated: Dedicated to the centenary of the birth of Dmitriĭ Konstantinovich Faddeev