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Algebra i Analiz, 2007, Volume 19, Issue 5, Pages 37–64 (Mi aa135)  

This article is cited in 18 scientific papers (total in 18 papers)

Research Papers

The normalizer of Chevalley groups of type $\mathrm{E}_6$

N. A. Vavilov, A. Yu. Luzgarev

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We consider the simply connected Chevalley group $G(\mathrm{E}_6,R)$ of type $\mathrm{E}_6$ in a 27-dimensional representation. The main goal is to establish that the following four groups coincide: the normalizer of the Chevally group $G(\mathrm{E}_6,R)$ itself, the normalizer of its elementary subgroup $E(\mathrm{E}_6,R)$, the transporter of $E(\mathrm{E}_6,R)$ in $G(\operatorname{E}_6,R)$, and the extended Chevalley group $\overline G(\mathrm{E}_6,R)$. This is true over an arbitrary commutative ring $R$, all normalizers and transporters being taken in $\mathrm{GL}(27,R)$. Moreover, $\overline G(\mathrm{E}_6,R)$ 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 $g\in\mathrm{GL}(27,R)$ must satisfy in order to belong to $\overline G(\mathrm{E}_6,R)$. These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.

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.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:5, 699–718

Bibliographic databases:

MSC: 20G15
Received: 20.05.2007

Citation: N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, Algebra i Analiz, 19:5 (2007), 37–64; St. Petersburg Math. J., 19:5 (2008), 699–718

Citation in format AMSBIB
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\paper The normalizer of Chevalley groups of type $\mathrm{E}_6$
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 5
\pages 37--64
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2381940}
\zmath{https://zbmath.org/?q=an:1206.20054}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 5
\pages 699--718
\crossref{https://doi.org/10.1090/S1061-0022-08-01016-9}
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    This publication is cited in the following articles:
    1. N. A. Vavilov, A. K. Stavrova, “Basic reductions for the description of normal subgroups”, J. Math. Sci. (N. Y.), 151:3 (2008), 2949–2960  mathnet  crossref  elib  elib
    2. N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, St. Petersburg Math. J., 20:4 (2009), 527–551  mathnet  crossref  mathscinet  zmath  isi  elib
    3. A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  mathnet  crossref  mathscinet  zmath  isi
    4. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95  mathscinet  zmath
    5. A. S. Ananievskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$”, J. Math. Sci. (N. Y.), 161:4 (2009), 461–473  mathnet  crossref  elib
    6. E. I. Bunina, “Automorphisms of Chevalley groups of types $A_l$, $D_l$, or $E_l$ over local rings with 1/2”, J. Math. Sci., 167:6 (2010), 749–766  mathnet  crossref  mathscinet  elib
    7. N. A. Vavilov, “Some more exceptional numerology”, J. Math. Sci. (N. Y.), 171:3 (2010), 317–321  mathnet  crossref
    8. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    9. N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251  mathnet  crossref
    10. I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350  mathnet  crossref
    11. I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    12. A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$. I”, St. Petersburg Math. J., 23:5 (2012), 819–849  mathnet  crossref  mathscinet  isi  elib  elib
    13. I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. I”, St. Petersburg Math. J., 23:5 (2012), 891–919  mathnet  crossref  mathscinet  isi  elib  elib
    14. N. A. Vavilov, “An $\mathrm A_3$-proof of the structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$. II. The main lemma”, St. Petersburg Math. J., 23:6 (2012), 921–942  mathnet  crossref  mathscinet  isi  elib  elib
    15. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  mathnet  crossref  mathscinet
    16. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    17. N. A. Vavilov, A. Yu. Luzgarev, “Normaliser of the Chevalley group of type $\mathrm E_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921  mathnet  crossref  mathscinet  isi  elib
    18. M. M. Atamanova, A. Yu. Luzgarev, “Cubic forms on adjoint representations of exceptional groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 370–379  mathnet  crossref  mathscinet
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