RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Algebra i Analiz:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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.

Full text: PDF file (311 kB)
References: PDF file   HTML file

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
\Bibitem{VavLuz07}
\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
\mathnet{http://mi.mathnet.ru/aa135}
\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}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267421000002}


Linking options:
  • http://mi.mathnet.ru/eng/aa135
  • http://mi.mathnet.ru/eng/aa/v19/i5/p37

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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
    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
  • Алгебра и анализ St. Petersburg Mathematical Journal
    Number of views:
    This page:335
    Full text:99
    References:20
    First page:10

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2017