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Zap. Nauchn. Sem. POMI, 2006, Volume 330, Pages 36–76 (Mi znsl278)  

This article is cited in 29 scientific papers (total in 29 papers)

Structure of Chevalley groups: the proof from the Book

N. A. Vavilova, M. R. Gavrilovichb, S. I. Nikolenkoc

a Saint-Petersburg State University
b University of Oxford
c St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We describe and compare different geometric proofs of the main structure theorems for Chevalley groups over commutative rings. To warm up we sketch the known geometric proofs, published by I.Ż. Golubchik, N. A. Vavilov, A. V. Stepanov and E. B. Plotkin, such as the $A_2$ and $A_3$ proofs for classical groups, $A_5$ and $D_5$ proofs for $E_6$; $A_7$ and $D_6$ proofs for $E_7$, and $D_8$ proof for $E_8$. After that we expound in more details the $A_2$ proofs for exceptional groups of types $F_4$, $E_6$ and $E_7$, based on multiple commutation. This new proof, the Proof from the Book, gives better bounds than any previously known. Moreover, unlike all previously known proofs it does not use results for fields, factorisation modulo radical, or any specific information concerning structure constants and equations defining exceptional Chevalley groups.

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English version:
Journal of Mathematical Sciences (New York), 2007, 140:5, 626–645

Bibliographic databases:

UDC: 512.5
Received: 10.12.2005

Citation: N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, Problems in the theory of representations of algebras and groups. Part 13, Zap. Nauchn. Sem. POMI, 330, POMI, St. Petersburg, 2006, 36–76; J. Math. Sci. (N. Y.), 140:5 (2007), 626–645

Citation in format AMSBIB
\Bibitem{VavGavNik06}
\by N.~A.~Vavilov, M.~R.~Gavrilovich, S.~I.~Nikolenko
\paper Structure of Chevalley groups: the proof from the Book
\inbook Problems in the theory of representations of algebras and groups. Part~13
\serial Zap. Nauchn. Sem. POMI
\yr 2006
\vol 330
\pages 36--76
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl278}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2253566}
\zmath{https://zbmath.org/?q=an:1162.20032}
\elib{http://elibrary.ru/item.asp?id=9161492}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 140
\issue 5
\pages 626--645
\crossref{https://doi.org/10.1007/s10958-007-0003-y}
\elib{http://elibrary.ru/item.asp?id=13534553}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846125905}


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    This publication is cited in the following articles:
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    2. 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
    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
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    7. 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
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    11. 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
    12. E. I. Bunina, “Avtomorfizmy grupp Shevalle tipa $B_l$ nad lokalnymi koltsami s 1/2”, Fundament. i prikl. matem., 15:7 (2009), 3–46  mathnet  mathscinet  elib; E. I. Bunina, “Automorphisms of Chevalley groups of type $B_l$ over local rings with 1/2”, J. Math. Sci., 169:5 (2010), 557–588  crossref  elib
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    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. Klyachko A.A., “Automorphisms and isomorphisms of Chevalley groups and algebras”, J. Algebra, 324:10 (2010), 2608–2619  crossref  mathscinet  zmath  isi  elib
    17. Hazrat R., Petrov V., Vavilov N., “Relative subgroups in Chevalley groups”, J. K-Theory, 5:3 (2010), 603–618  crossref  mathscinet  zmath  isi  elib
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    19. 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
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    23. 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
    24. 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
    25. Bardini C., “Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two”, Chin. Ann. Math. Ser. B, 33:5 (2012), 783–800  crossref  mathscinet  zmath  isi  elib
    26. 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
    27. 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
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    29. V. A. Petrov, “Razlozhenie transvektsii: algebro-geometricheskii podkhod”, Algebra i analiz, 28:1 (2016), 150–157  mathnet  mathscinet  elib; V. A. Petrov, “Decomposition of transvections: an algebro-geometric approach”, St. Petersburg Math. J., 28:1 (2017), 109–114  crossref  isi
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