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Algebra i Analiz, 2008, Volume 20, Issue 4, Pages 27–63 (Mi aa521)  

This article is cited in 24 scientific papers (total in 24 papers)

Research Papers

$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$

N. A. Vavilova, S. I. Nikolenkob

a St. Petersburg State University, Department of Mathematics and Mechanics
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: A new geometric proof is given for the standard description of subgroups in the Chevalley group $G=G(\mathrm{F}_4,R)$ of type $\mathrm{F}_4$ over a commutative ring $R$ that are normalized by the elementary subgroup $E(\mathrm{F}_4,R)$. There are two major approaches to the proof of such results. Localization proofs (Quillen, Suslin, Bak) are based on reduction in dimension. The first proofs of this type for exceptional groups were given by Abe, Suzuki, Taddei and Vaserstein, but they invoked the Chevalley simplicity theorem and reduction modulo the radical. At about the same time, the first author, Stepanov, and Plotkin developed a geometric approach, decomposition of unipotents, based on reduction in the rank of the group. This approach combines the methods introduced in the theory of classical groups by Wilson, Golubchik, and Suslin with ideas of Matsumoto and Stein coming from representation theory and K-theory. For classical groups in vector representations, the resulting proofs are quite straightforward, but their generalizations to exceptional groups required the explicit knowledge of the signs of action constants, and of equations satisfied by the orbit of the highest weight vector. They depend on the presence of high rank subgroups of types $\mathrm{A}_l$ or $\mathrm{D}_l$, such as $\mathrm{A}_5\le\mathrm{E}_6$ and $\mathrm{A}_7\le\mathrm{E}_7$. The first author and Gavrilovich introduced a new twist to the method of decomposition of unipotents, which made it possible to give an entirely elementary geometric proof (the proof from the Book) for Chevalley groups of types $\Phi=\mathrm{E}_6,\mathrm{E}_7$. This new proof, like the proofs for classical cases, relies upon embedding of $\mathrm{A}_2$. Unlike all previous proofs, neither results pertaining to the field case, nor explicit knowledge of structure constants and defining equations are ever used. In the present paper we show that, with some additional effort, we can make this proof work also for the case of $\Phi=\mathrm{F}_4$. Moreover, we establish some new facts about Chevalley groups of type $\mathrm{F}_4$ and their 27-dimensional representation.

Keywords: Chevalley group, elementary subgroup, normal subgroups, standard description, minimal module, parabolic subgroups, decomposition of unipotents, root element, orbit of the highest weight vector, the proof from the Book

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English version:
St. Petersburg Mathematical Journal, 2009, 20:4, 527–551

Bibliographic databases:

MSC: 20G15, 20G35
Received: 25.10.2006

Citation: N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, Algebra i Analiz, 20:4 (2008), 27–63; St. Petersburg Math. J., 20:4 (2009), 527–551

Citation in format AMSBIB
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\by N.~A.~Vavilov, S.~I.~Nikolenko
\paper $\mathrm A_2$-proof of structure theorems for Chevalley groups of type~$\mathrm F_4$
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 4
\pages 27--63
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\transl
\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 4
\pages 527--551
\crossref{https://doi.org/10.1090/S1061-0022-09-01060-7}
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    This publication is cited in the following articles:
    1. N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Gruppa Shevalle tipa $\mathrm E_6$ v 27-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 5–68  mathnet  mathscinet  zmath  elib; N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation”, J. Math. Sci. (N. Y.), 145:1 (2007), 4697–4736  crossref  elib
    2. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Stroenie grupp Shevalle: Dokazatelstvo iz Knigi”, Voprosy teorii predstavlenii algebr i grupp. 13, Zap. nauchn. sem. POMI, 330, POMI, SPb., 2006, 36–76  mathnet  mathscinet  zmath  elib; N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, J. Math. Sci. (N. Y.), 140:5 (2007), 626–645  crossref  elib
    3. 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
    4. N. A. Vavilov, “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68  mathnet  mathscinet  zmath; N. A. Vavilov, “Can one see the signs of structure constants?”, St. Petersburg Math. J., 19:4 (2008), 519–543  crossref  isi
    5. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm{E}_6$”, Algebra i analiz, 19:5 (2007), 37–64  mathnet  mathscinet  zmath; N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718  crossref  isi
    6. N. A. Vavilov, “O podgruppakh simplekticheskoi gruppy, soderzhaschikh subsystem subgroup”, Voprosy teorii predstavlenii algebr i grupp. 16, Zap. nauchn. sem. POMI, 349, POMI, SPb., 2007, 5–29  mathnet  mathscinet  elib; N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948  crossref  elib
    7. Vavilov N., “An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$”, Internat. J. Algebra Comput., 17:5–6 (2007), 1283–1298  crossref  mathscinet  zmath  isi  elib
    8. N. A. Vavilov, “Numerologiya kvadratnykh uravnenii”, Algebra i analiz, 20:5 (2008), 9–40  mathnet  mathscinet  zmath; N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707  crossref  isi
    9. 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
    10. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95  mathscinet  zmath
    11. 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
    12. Bak A., Hazrat R., Vavilov N., “Localization-completion strikes again: relative $K_1$ is nilpotent by abelian”, J. Pure Appl. Algebra, 213:6 (2009), 1075–1085  crossref  mathscinet  zmath  isi  elib
    13. N. Vavilov, A. Luzgarev, A. Stepanov, “Calculations in exceptional groups over rings”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XVII, Zap. nauchn. sem. POMI, 373, POMI, SPb., 2009, 48–72  mathnet; J. Math. Sci. (N. Y.), 168:3 (2010), 334–348  crossref
    14. Hazrat R., Vavilov N., “Bak's work on the $K$-theory of rings”, J. K-Theory, 4:1 (2009), 1–65  crossref  mathscinet  zmath  isi  elib
    15. N. A. Vavilov, V. G. Kazakevich, “More variations on decomposition of transvections”, J. Math. Sci. (N. Y.), 171:3 (2010), 322–330  mathnet  crossref
    16. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    17. 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
    18. 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
    19. 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
    20. N. A. Vavilov, A. V. Stepanov, “Lineinye gruppy nad obschimi koltsami I. Obschie mesta”, Voprosy teorii predstavlenii algebr i grupp. 22, Zap. nauchn. sem. POMI, 394, POMI, SPb., 2011, 33–139  mathnet  mathscinet; N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  crossref
    21. N. A. Vavilov, A. V. Schegolev, “Nadgruppy subsystem subgroups v isklyuchitelnykh gruppakh: urovni”, Voprosy teorii predstavlenii algebr i grupp. 23, Zap. nauchn. sem. POMI, 400, POMI, SPb., 2012, 70–126  mathnet  mathscinet; N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  crossref
    22. Hazrat R. Vavilov N. Zhang Z., “Relative Commutator Calculus in Chevalley Groups”, J. Algebra, 385 (2013), 262–293  crossref  mathscinet  zmath  isi  elib
    23. 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
    24. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
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