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Zap. Nauchn. Sem. POMI, 2007, Volume 349, Pages 5–29 (Mi znsl52)  

This article is cited in 4 scientific papers (total in 4 papers)

On subgroups of symplectic group containing a subsystem subgroup

N. A. Vavilov

Saint-Petersburg State University

Abstract: Let $\Gamma=\operatorname{GSp}(2l,R)$ be the general symplectic group of rank $l$ over a commutative ring $R$ such, that $2\in R^*$, and $\nu$ be a symmetric equivalence relation on the index set $\{1,\ldots,l,-l,\ldots,1\}$, all of whose classes contain at least 3 elements. In the present paper we prove that if a subgroup $H$ of $\Gamma$ contains the group $E_{\Gamma}(\nu)$ of elementary block diagonal matrices of type $\nu$, then $H$ normalises the subgroup generated by all elementary symplectic transvections $T_{ij}(\xi)\in H$. Combined with the previous results, this completely describes overgroups of subsystem subgroups in this case. Similar results for subgroups of $\operatorname{GL}(n,R)$ were established by Z. I. Borewicz and the author in early 1980-ies, while for $\operatorname{GSp}(2l,R)$ and $\operatorname{GO}(n,R)$ they have been announced by the author in late 1980-ies, but the complete proof for the symplectic case has not been published before.

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English version:
Journal of Mathematical Sciences (New York), 2008, 151:3, 2937–2948

Bibliographic databases:

UDC: 513.6
Received: 20.06.2007

Citation: N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, Problems in the theory of representations of algebras and groups. Part 16, Zap. Nauchn. Sem. POMI, 349, POMI, St. Petersburg, 2007, 5–29; J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948

Citation in format AMSBIB
\by N.~A.~Vavilov
\paper On subgroups of symplectic group containing a~subsystem subgroup
\inbook Problems in the theory of representations of algebras and groups. Part~16
\serial Zap. Nauchn. Sem. POMI
\yr 2007
\vol 349
\pages 5--29
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2008
\vol 151
\issue 3
\pages 2937--2948

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    This publication is cited in the following articles:
    1. A. S. Ananevskii, N. A. Vavilov, S. S. Sinchuk, “O nadgruppakh $E(m,R)\otimes E(n,R)$. I. Urovni i normalizatory”, Algebra i analiz, 23:5 (2011), 55–98  mathnet  mathscinet  elib; A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$. I”, St. Petersburg Math. J., 23:5 (2012), 819–849  crossref  isi  elib
    2. N. A. Vavilov, A. A. Semenov, “Dlinnye kornevye tory v gruppakh Shevalle”, Algebra i analiz, 24:3 (2012), 22–83  mathnet  mathscinet  zmath  elib; N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  crossref  isi  elib
    3. 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
    4. A. V. Schegolev, “Nadgruppy blochno-diagonalnykh podgrupp giperbolicheskoi unitarnoi gruppy nad kvazi-konechnym koltsom: osnovnye rezultaty”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 222–233  mathnet  mathscinet
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