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Zap. Nauchn. Sem. POMI, 2007, Volume 349, Pages 30–52 (Mi znsl53)  

This article is cited in 9 scientific papers (total in 9 papers)

Basic reductions for the description of normal subgroups

N. A. Vavilov, A. K. Stavrova

Saint-Petersburg State University

Abstract: Classification of subgroups in a Chevalley group $G(\Phi,R)$ over a commutative ring $R$, normalised by the elementary subgroup $E(\Phi,R)$, is well known. However, for exceptional groups one cannot find in the available literature neither the parabolic reduction, nor the level reduction. This is due to the fact that the Abe–Suzuki–Vaserstein proof relied on localisation and reduction modulo Jacobson radical. Recently for the groups of types $\operatorname{E}_6$, $\operatorname{E}_7$ and $\operatorname{F}_4$ the first-named author, M. Gavrilovich and S. Nikolenko proposed an even more straightforward geometric approach to the proof of structure theorems, similar to the one used for classical cases. In the present work we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank $\ge 2$. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralisers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems.

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English version:
Journal of Mathematical Sciences (New York), 2008, 151:3, 2949–2960

UDC: 513.6
Received: 10.06.2007

Citation: N. A. Vavilov, A. K. Stavrova, “Basic reductions for the description of normal subgroups”, Problems in the theory of representations of algebras and groups. Part 16, Zap. Nauchn. Sem. POMI, 349, POMI, St. Petersburg, 2007, 30–52; J. Math. Sci. (N. Y.), 151:3 (2008), 2949–2960

Citation in format AMSBIB
\by N.~A.~Vavilov, A.~K.~Stavrova
\paper Basic reductions for the description of normal subgroups
\inbook Problems in the theory of representations of algebras and groups. Part~16
\serial Zap. Nauchn. Sem. POMI
\yr 2007
\vol 349
\pages 30--52
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2008
\vol 151
\issue 3
\pages 2949--2960

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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
    4. 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
    5. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    6. 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
    7. 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
    8. 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
    9. A. V. Stepanov, “Novyi vzglyad na razlozhenie unipotentov i normalnoe stroenie grupp Shevalle”, Algebra i analiz, 28:3 (2016), 161–173  mathnet  mathscinet  elib; A. V. Stepanov, “A new look at the decomposition of unipotents and the normal structure of Chevalley groups”, St. Petersburg Math. J., 28:3 (2017), 411–419  crossref  isi
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