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Zap. Nauchn. Sem. LOMI, 1978, Volume 75, Pages 43–58 (Mi znsl3785)  

This article is cited in 8 scientific papers (total in 8 papers)

Parabolic subgroups of Chevalley groups over a semilocal ring

N. A. Vavilov

Abstract: Let $G$ be the Chevalley group over a commutative semilocal ring $R$ which is associated with a root system $\Phi$. The parabolic subgroups of $G$ are described in the work. A system $\sigma=(\sigma_\alpha)$ of ideals $\sigma_\alpha$ in $R$ ($\alpha$ runs through all roots of the system $\Phi$) is called a net of ideals in the commutative ring $R$ if $\sigma_\alpha\sigma_\beta\subset\sigma_{\alpha+\beta}$ for all those roots $\alpha$ and $\beta$ for which $\alpha+\beta$ is also a root. A net $\sigma$ is called parabolic if $\sigma_\alpha=R$ for $\alpha>0$. The main theorem: under minor additional assumptions all parabolic subgroups of $G$ are in bijective correspondence with all parabolic nets $\sigma$. The paper is related to two works of K. Suzuki in which the parabolic subgroups of $G$ are described under more stringent conditions. Bibl. 19 titles.

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English version:
Journal of Soviet Mathematics, 1987, 37:2, 942–952

Bibliographic databases:

Document Type: Article
UDC: 513.6

Citation: N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a semilocal ring”, Rings and linear groups, Zap. Nauchn. Sem. LOMI, 75, "Nauka", Leningrad. Otdel., Leningrad, 1978, 43–58; J. Soviet Math., 37:2 (1987), 942–952

Citation in format AMSBIB
\by N.~A.~Vavilov
\paper Parabolic subgroups of Chevalley groups over a~semilocal ring
\inbook Rings and linear groups
\serial Zap. Nauchn. Sem. LOMI
\yr 1978
\vol 75
\pages 43--58
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\jour J. Soviet Math.
\yr 1987
\vol 37
\issue 2
\pages 942--952

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    This publication is cited in the following articles:
    1. O. I. Tavgen', “Bounded generation of Chevalley groups over rings of algebraic $S$-integers”, Math. USSR-Izv., 36:1 (1991), 101–128  mathnet  crossref  mathscinet  zmath  adsnasa
    2. K. Yu. Lavrov, “Subgroups of the orthogonal groups of even degree over a local field”, J. Math. Sci. (N. Y.), 136:3 (2006), 3966–3971  mathnet  crossref  mathscinet  zmath
    3. N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948  mathnet  crossref  mathscinet  elib  elib
    4. A. V. Alexandrov, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a Dedekind ring of arithmetic type”, J. Math. Sci. (N. Y.), 171:3 (2010), 307–316  mathnet  crossref
    5. Yakov N. Nuzhin, “Faktorizatsiya kovrovykh podgrupp grupp Shevalle nad kommutativnymi koltsami”, Zhurn. SFU. Ser. Matem. i fiz., 4:4 (2011), 527–535  mathnet
    6. K. O. Batalkin, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SO}_{2l}$ over a Dedekind ring of arithmetic type”, J. Math. Sci. (N. Y.), 192:2 (2013), 154–163  mathnet  crossref  mathscinet
    7. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  mathnet  crossref  mathscinet
    8. V. A. Koibaev, S. K. Kuklina, A. O. Likhachëva, Ya. N. Nuzhin, “Podgruppy grupp Shevalle nad lokalno konechnym polem, opredelyaemye naborom additivnykh podgrupp”, Matem. zametki, 102:6 (2017), 857–865  mathnet  crossref  elib
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