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Zap. Nauchn. Sem. POMI, 2010, Volume 375, Pages 5–21 (Mi znsl3604)  

This article is cited in 1 scientific paper (total in 1 paper)

Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a Dedekind ring of arithmetic type

A. V. Alexandrov, N. A. Vavilov

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: Let $R$ be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup $P$ in $G=\mathrm{SL}(n,R)$, $n\ge3$, or in $G=\mathrm{Sp}(2l,R)$, $l\ge2$, containing Borel subgroup $B$, the following alternative holds. Either $P$ contains a relative elementary subgroup $E_I$ for some ideal $I\neq0$, or $H$ is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows, under some mild additional assumptions on units, to completely describe overgroups of $B$ in $G$. Bibl. – 30 titles.

Key words and phrases: special linear group, symplectic group, transvections, parabolic subgroups, Dedekind ring of arythmetic type.

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English version:
Journal of Mathematical Sciences (New York), 2010, 171:3, 307–316

Document Type: Article
UDC: 513.6
Received: 31.03.2010

Citation: A. V. Alexandrov, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a Dedekind ring of arithmetic type”, Problems in the theory of representations of algebras and groups. Part 19, Zap. Nauchn. Sem. POMI, 375, POMI, St. Petersburg, 2010, 5–21; J. Math. Sci. (N. Y.), 171:3 (2010), 307–316

Citation in format AMSBIB
\Bibitem{AleVav10}
\by A.~V.~Alexandrov, N.~A.~Vavilov
\paper Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a~Dedekind ring of arithmetic type
\inbook Problems in the theory of representations of algebras and groups. Part~19
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 375
\pages 5--21
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3604}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2010
\vol 171
\issue 3
\pages 307--316
\crossref{https://doi.org/10.1007/s10958-010-0135-3}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78649446442}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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
  • Записки научных семинаров ПОМИ
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