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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

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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
\by N.~A.~Vavilov, A.~Yu.~Luzgarev
\paper The normalizer of Chevalley groups of type $\mathrm{E}_6$
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 5
\pages 37--64
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 5
\pages 699--718

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    This publication is cited in the following articles:
    1. N. A. Vavilov, A. K. Stavrova, “Osnovnye reduktsii v zadache opisaniya normalnykh podgrupp”, Voprosy teorii predstavlenii algebr i grupp. 16, Zap. nauchn. sem. POMI, 349, POMI, SPb., 2007, 30–52  mathnet  elib; N. A. Vavilov, A. K. Stavrova, “Basic reductions for the description of normal subgroups”, J. Math. Sci. (N. Y.), 151:3 (2008), 2949–2960  crossref  elib
    2. N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya gruppy Shevalle tipa $\mathrm F_4$”, Algebra i analiz, 20:4 (2008), 27–63  mathnet  mathscinet  zmath  elib; 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  crossref  isi
    3. A. Yu. Luzgarëv, “Opisanie nadgrupp $\mathrm F_4$ v $\mathrm E_6$ nad kommutativnym koltsom”, Algebra i analiz, 20:6 (2008), 148–185  mathnet  mathscinet  zmath; A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  crossref  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, “Avtomorfizmy grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s 1/2”, Fundament. i prikl. matem., 15:2 (2009), 35–59  mathnet  mathscinet; 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  crossref  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, “Gruppa Shevalle tipa $\mathrm E_7$ v 56-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 5–99  mathnet; 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  crossref
    10. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. II”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 242–264  mathnet; 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  crossref
    11. I. M. Pevzner, “Geometriya kornevykh elementov v gruppakh tipa $\mathrm E_6$”, Algebra i analiz, 23:3 (2011), 261–309  mathnet  mathscinet  zmath  elib; I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  crossref  isi  elib
    12. A. S. Ananevskii, N. A. Vavilov, S. S. Sinchuk, “O nadgruppakh $E(m,R)\otimes E(n,R)$. I. Urovni i normalizatory”, Algebra i analiz, 23:5 (2011), 55–98  mathnet  mathscinet  elib; 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  crossref  isi  elib
    13. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. I”, Algebra i analiz, 23:5 (2011), 155–198  mathnet  mathscinet  elib; 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  crossref  isi  elib
    14. N. A. Vavilov, “$\mathrm A_3$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$$\mathrm E_7$. II. Osnovnaya lemma”, Algebra i analiz, 23:6 (2011), 1–31  mathnet  mathscinet  elib; 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  crossref  isi  elib
    15. N. A. Vavilov, “Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25 years after”, Voprosy teorii predstavlenii algebr i grupp. 27, Zap. nauchn. sem. POMI, 430, POMI, SPb., 2014, 32–52  mathnet  mathscinet; J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  crossref
    16. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    17. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm E_7$”, Algebra i analiz, 27:6 (2015), 57–88  mathnet  mathscinet  elib; 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  crossref  isi
    18. M. M. Atamanova, A. Yu. Luzgarev, “Kubicheskie formy na prisoedinennykh predstavleniyakh isklyuchitelnykh grupp”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 9–23  mathnet  mathscinet; M. M. Atamanova, A. Yu. Luzgarev, “Cubic forms on adjoint representations of exceptional groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 370–379  crossref
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