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 Zap. Nauchn. Sem. POMI, 2016, Volume 443, Pages 222–233 (Mi znsl6265)

Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results

A. V. Shchegolev

St. Petersburg State University, St. Petersburg, Russia

Abstract: Let $H$ be a subgroup of the hyperbolic unitary group $\operatorname U(2n,R,\Lambda)$ that contains the elementary block-diagonal subgroup $\operatorname{EU}(\nu,R,\Lambda)$ of type $\nu$. Assume that all self-conjugate blocks of $\nu$ are of size at least 6 (at least 4 if the form parameter $\Lambda$ satisfies the condition $R\Lambda+\Lambda R=R$) and that all non-self-conjugate blocks are of size at least 5. Then there exists a unique major exact form net of ideals $(\sigma,\Gamma)$ such that $\operatorname{EU}(\sigma,\Gamma)\le H\le\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$, where $\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$ stands for the normalizer in $\operatorname U(2n,R,\Lambda)$ of the form net subgroup $\operatorname U(\sigma,\Gamma)$ of level $(\sigma,\Gamma)$ and $\operatorname{EU}(\sigma,\Gamma)$ denotes the corresponding elementary form net subgroup. The normalizer $\operatorname N_{\operatorname U(2n,R,\Lambda)}(\operatorname U(\sigma,\Gamma))$ is described in terms of congruences.

Key words and phrases: hyperbolic unitary group, elementary subgroup, transvections, parabolic subgroups, standard automorphisms, block-diagonal subgroups, localization.

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English version:
Journal of Mathematical Sciences (New York), 2017, 222:4, 516–523

Bibliographic databases:

Document Type: Article
UDC: 513.6

Citation: A. V. Shchegolev, “Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results”, Problems in the theory of representations of algebras and groups. Part 29, Zap. Nauchn. Sem. POMI, 443, POMI, St. Petersburg, 2016, 222–233; J. Math. Sci. (N. Y.), 222:4 (2017), 516–523

Citation in format AMSBIB
\Bibitem{Shc16}
\by A.~V.~Shchegolev
\paper Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results
\inbook Problems in the theory of representations of algebras and groups. Part~29
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 443
\pages 222--233
\publ POMI
\mathnet{http://mi.mathnet.ru/znsl6265}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3507773}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 222
\issue 4
\pages 516--523
\crossref{https://doi.org/10.1007/s10958-017-3319-2}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85014794187}