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

This article is cited in 7 scientific papers (total in 7 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
\Bibitem{Vav78}
\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
\mathnet{http://mi.mathnet.ru/znsl3785}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=0503839}
\zmath{https://zbmath.org/?q=an:0612.20028|0448.20046}
\transl
\jour J. Soviet Math.
\yr 1987
\vol 37
\issue 2
\pages 942--952
\crossref{https://doi.org/10.1007/BF01089086}


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    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. O. I. Tavgen, “Ogranichennaya porozhdaemost grupp Shevalle nad koltsami $S$-tselykh algebraicheskikh chisel”, Izv. AN SSSR. Ser. matem., 54:1 (1990), 97–122  mathnet  mathscinet  zmath  adsnasa; O. I. Tavgen', “Bounded generation of Chevalley groups over rings of algebraic $S$-integers”, Math. USSR-Izv., 36:1 (1991), 101–128  crossref
    2. K. Yu. Lavrov, “Podgruppy ortogonalnykh grupp chetnogo poryadka nad lokalnym polem”, Voprosy teorii predstavlenii algebr i grupp. 12, Zap. nauchn. sem. POMI, 321, POMI, SPb., 2005, 240–250  mathnet  mathscinet  zmath; 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  crossref
    3. 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
    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, “Parabolicheskie podgruppy $\mathrm{SO}_{2l}$ nad dedekindovym koltsom arifmeticheskogo tipa”, Voprosy teorii predstavlenii algebr i grupp. 23, Zap. nauchn. sem. POMI, 400, POMI, SPb., 2012, 50–69  mathnet  mathscinet; 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  crossref
    7. 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
  • Записки научных семинаров ПОМИ
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