RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

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

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 These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques.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.

Full text: PDF file (323 kB)
References: PDF file   HTML file

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

Bibliographic databases:

MSC: 20G15, 20G35

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
\Bibitem{Vav07} \by N.~Vavilov \paper Geometry of 1-tori in $\mathrm{GL}(n,T)$ \jour Algebra i Analiz \yr 2007 \vol 19 \issue 3 \pages 119--150 \mathnet{http://mi.mathnet.ru/aa122} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2340708} \zmath{https://zbmath.org/?q=an:1202.20054} \elib{http://elibrary.ru/item.asp?id=9540304} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267653300004} 

• http://mi.mathnet.ru/eng/aa122
• http://mi.mathnet.ru/eng/aa/v19/i3/p119

 SHARE:

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
2. N. Vavilov, “Weight elements of Chevalley groups”, St. Petersburg Math. J., 20:1 (2009), 23–57
3. N. A. Vavilov, V. V. Nesterov, “Geometriya mikrovesovykh torov”, Vladikavk. matem. zhurn., 10:1 (2008), 10–23
4. I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635
5. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430
•  Number of views: This page: 249 Full text: 70 References: 25 First page: 5