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

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

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
\Bibitem{Vav07} \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 \mathnet{http://mi.mathnet.ru/znsl52} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2742852} \elib{http://elibrary.ru/item.asp?id=13077201} \transl \jour J. Math. Sci. (N. Y.) \yr 2008 \vol 151 \issue 3 \pages 2937--2948 \crossref{https://doi.org/10.1007/s10958-008-9020-8} \elib{http://elibrary.ru/item.asp?id=13581263} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-49249101527} 

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This publication is cited in the following articles:
1. 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
2. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430
3. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195
4. A. V. Shchegolev, “Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results”, J. Math. Sci. (N. Y.), 222:4 (2017), 516–523
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