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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Geometry of 1-tori in $ \mathrm{GL}_n$

Author: N. A. Vavilov
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 407-429
MSC (2000): Primary 20G15, 20G35
Published electronically: March 21, 2008
MathSciNet review: 2340708
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Abstract: We describe the orbits of the general linear group $ \operatorname{GL}(n,T)$ over a skew field $ T$ acting by simultaneous conjugation on pairs of $ 1$-tori, i.e., subgroups conjugate to $ \operatorname{diag}(T^*,1,\ldots,1)$, and identify the corresponding spans. We also provide some applications of these results to the description of intermediate subgroups and generation. These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques. As another application of our methods, we enumerate the orbits of $ \operatorname{GL}(n,T)$ on pairs of a $ 1$-torus and a root subgroup, and identify the corresponding spans. This paper constitutes an elementary invitation to a series of much more technical works by the author and V. Nesterov, where similar results are established for microweight tori in Chevalley groups over a field.

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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198504, Russia

Keywords: General linear group, unipotent root subgroups, pseudoreflections, one-dimensional tori, diagonal subgroup, orbitals
Received by editor(s): October 10, 2006
Published electronically: March 21, 2008
Additional Notes: The present work was supported by AvH-Stiftung, SFB-343 and Universitaire Instelling Antwerpen. At the final stage the author was supported by the projects RFFI 03-01-00349 (POMI RAN), INTAS 00-566, INTAS 03-51-3251 and by an express grant of the Russian Ministry of Education “Overgroups of semisimple groups” E02-1.0-61
Article copyright: © Copyright 2008 American Mathematical Society