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Algebra i Analiz, 2007, Volume 19, Issue 3, Pages 119–150 (Mi aa122)  

This article is cited in 5 scientific papers (total in 5 papers)

Research Papers

Geometry of 1-tori in $\mathrm{GL}(n,T)$

N. Vavilov

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We describe the orbits of the general linear group $\mathrm{GL}(n,T)$ over a skew field $T$ acting by simultaneous conjugation on pairs of 1-tori, i.e., subgroups conjugate to $\operatorname{diag}(T^*,1,…, 1)$ 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 These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques.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 $\mathrm{GL}(n,T)$ 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.

Keywords: General linear group, unipotent root subgroups, pseudoreflections, one-dimensional tori, diagonal subgroup, orbitals.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:3, 407–429

Bibliographic databases:

MSC: 20G15, 20G35
Received: 10.10.2006

Citation: N. Vavilov, “Geometry of 1-tori in $\mathrm{GL}(n,T)$”, Algebra i Analiz, 19:3 (2007), 119–150; St. Petersburg Math. J., 19:3 (2008), 407–429

Citation in format AMSBIB
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\paper Geometry of 1-tori in $\mathrm{GL}(n,T)$
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\pages 119--150
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\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 3
\pages 407--429
\crossref{https://doi.org/10.1090/S1061-0022-08-01004-2}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. N. Vavilov, “Vesovye elementy grupp Shevalle”, Algebra i analiz, 20:1 (2008), 34–85  mathnet  mathscinet  zmath  elib; N. Vavilov, “Weight elements of Chevalley groups”, St. Petersburg Math. J., 20:1 (2009), 23–57  crossref  isi
    3. N. A. Vavilov, V. V. Nesterov, “Geometriya mikrovesovykh torov”, Vladikavk. matem. zhurn., 10:1 (2008), 10–23  mathnet  mathscinet  elib
    4. I. M. Pevzner, “Geometriya kornevykh elementov v gruppakh tipa $\mathrm E_6$”, Algebra i analiz, 23:3 (2011), 261–309  mathnet  mathscinet  zmath  elib; I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  crossref  isi  elib
    5. N. A. Vavilov, A. A. Semenov, “Dlinnye kornevye tory v gruppakh Shevalle”, Algebra i analiz, 24:3 (2012), 22–83  mathnet  mathscinet  zmath  elib; N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  crossref  isi  elib
  • Алгебра и анализ St. Petersburg Mathematical Journal
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