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Zap. Nauchn. Sem. POMI, 2001, Volume 281, Pages 35–59 (Mi znsl1488)  

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

Subgroups of split orthogonal groups over a commutative ring

N. A. Vavilov

Saint-Petersburg State University

Abstract: We describe subgroups of the split orthogonal group $\Gamma=\mathrm{GO}(n,R)$ of degree $n$ over a commutative ring $R$ such that $2\in R^*$, which contain the elementary subgroup of a regularly embedded semi-simple group $F$. We show that if the ranks of all irreducible components of $F$ are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup $H$ there exists a unique orthogonal net of ideals such that $H$ lies between the corresponding net subgroup and its normalaser in $\Gamma$. A similar result for subgroups of the general linear group $\mathrm{GL}(n,R)$ with irreducible components of ranks at least 2 was obtained by Z. I. Borevich and the present author. We construct examples which show that if $F$ has irreducible components of ranks 2 or 3, then the standard description does not hold. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick.

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English version:
Journal of Mathematical Sciences (New York), 2004, 120:4, 1501–1512

Bibliographic databases:

UDC: 513.6
Received: 21.05.2001

Citation: N. A. Vavilov, “Subgroups of split orthogonal groups over a commutative ring”, Problems in the theory of representations of algebras and groups. Part 8, Zap. Nauchn. Sem. POMI, 281, POMI, St. Petersburg, 2001, 35–59; J. Math. Sci. (N. Y.), 120:4 (2004), 1501–1512

Citation in format AMSBIB
\by N.~A.~Vavilov
\paper Subgroups of split orthogonal groups over a~commutative ring
\inbook Problems in the theory of representations of algebras and groups. Part~8
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 281
\pages 35--59
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 120
\issue 4
\pages 1501--1512

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    This publication is cited in the following articles:
    1. N. A. Vavilov, V. A. Petrov, “O nadgruppakh $\mathrm{Ep}(2l,R)$”, Algebra i analiz, 15:4 (2003), 72–114  mathnet  mathscinet  zmath; N. A. Vavilov, V. A. Petrov, “On supergroups of $\mathrm{Ep}(2l,R)$”, St. Petersburg Math. J., 15:4 (2004), 515–543  crossref
    2. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$ i $\mathrm E_7$”, Algebra i analiz, 16:4 (2004), 54–87  mathnet  mathscinet  zmath; N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-proof of structure theorem for Chevaller groups of type $\mathrm E_6$ and $\mathrm E_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672  crossref
    3. N. A. Vavilov, V. A. Petrov, “O nadgruppakh $\mathrm{EO}(n,R)$”, Algebra i analiz, 19:2 (2007), 10–51  mathnet  mathscinet  zmath  elib; N. A. Vavilov, V. A. Petrov, “Overgroups of $\mathrm{EO}(n,R)$”, St. Petersburg Math. J., 19:2 (2008), 167–195  crossref  isi
    4. 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
    5. N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020  mathnet  crossref  mathscinet  elib  elib
    6. 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
    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
    8. 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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