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Overgroups of $ E(m,R){\otimes}E(n,R)$ I. Levels and normalizers


Authors: A. S. Anan′evskiĭ, N. A. Vavilov and S. S. Sinchuk
Translated by: N. Vavilov
Original publication: Algebra i Analiz, tom 23 (2011), nomer 5.
Journal: St. Petersburg Math. J. 23 (2012), 819-849
MSC (2010): Primary 20G35
DOI: https://doi.org/10.1090/S1061-0022-2012-01219-7
Published electronically: July 10, 2012
MathSciNet review: 2918424
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Abstract: The study of subgroups $ H$ such that

$\displaystyle E(m,R)\otimes E(n,R)\le H\le G=\operatorname {GL}(mn,R) $

is started, provided that the ring $ R$ is commutative and $ m,n\ge 3$. The principal results of this part can be summarized as follows. The group $ \operatorname {GL}_m\otimes \operatorname {GL}_n$ is described by equations, and it is proved that the elementary subgroup $ E(m,R)\otimes E(n,R)$ is normal in $ (\operatorname {GL}_m\otimes \operatorname {GL}_n)(R)$. Moreover, when $ m\neq n$, the normalizers of all three subgroups $ E(m,R)\otimes e$, $ e\otimes E(n,R)$, and $ E(m,R)\otimes E(n,R)$ in $ \operatorname {GL}(mn,R)$ coincide with $ (\operatorname {GL}_m\otimes \operatorname {GL}_n)(R)$. With each such intermediate subgroup $ H$, a uniquely defined level $ (A,B,C)$ is associated where $ A,B,C$ are ideals in $ R$ such that $ mA,A^2\le B\le A$ and $ nA,A^2\le C\le A$. Conversely, a level determines a perfect intermediate subgroup $ \operatorname {EE}(m,n,R,A,B,C)$. It is shown that each intermediate subgroup contains a unique largest subgroup of this type. Next, the normalizer $ N_G(\operatorname {EE}(m,n,R,A))$ of these perfect intermediate subgroups is calculated completely in the crucial case, where $ A=B=C$. The standard answer to the above problem can now be stated as follows. Every such intermediate subgroup $ H$ is contained in the normalizer $ N_G(\operatorname {EE}(m,n,R,A,B,C))$. In the special case where $ n\ge m+2$, such a standard description will be established in the second part of the present work.

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Additional Information

A. S. Anan′evskiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: alseang@gmail.com

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: nikolai-vavilov@yandex.ru

S. S. Sinchuk
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: sinchukss@yandex.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01219-7
Keywords: General linear group, elementary subgroup, tensor product, affine group schemes, intermediate subgroups, standard description, elementary transvections, lower level, form parameters, normalizer, automorphisms
Received by editor(s): June 10, 2010
Published electronically: July 10, 2012
Additional Notes: The research of the second author was started in the framework of the RFBR projects 09-01-00878 “Overgroups of reductive groups in algebraic groups over rings” and 09-01-90304 “Structure theory of classical and algebraic groups”. Apart from that, at the initial stage of the work he was supported by EPSRC EP/D03695X/1 (first grant scheme of Roozbeh Hazrat) and SFB-701 at the Uni. Bielefeld, and at the final stage he was supported by the RFBR projects 08-01-00756, 09-01-00762, 09-01-00784, 09-01-91333, and 10-01-90016. The third author acknowledges support of the RFBR project 10-01-92651 “Higher composition laws, algebraic $K$-theory, and exceptional groups”.
Article copyright: © Copyright 2012 American Mathematical Society