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Zap. Nauchn. Sem. POMI, 2007, Volume 343, Pages 54–83 (Mi znsl111)  

This article is cited in 8 scientific papers (total in 8 papers)

Triples of long root subgroups

N. A. Vavilov, I. M. Pevzner

Saint-Petersburg State University

Abstract: Let $G=G(\Phi,K)$ be a Chevalley group over a field $K$ of characteristic $\ne 2$. In the present paper, we classify subgroups of $G$ generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields this result is contained in the papers by B. Cooperstein on geometry of root subgroups, whereas for $\mathrm{SL} (n,K)$ it is proven in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in the deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type $\mathrm{E}_l$, we need detailed justification of this list, which we could not extract from the published works. This is why in the present paper, we produce a direct elementary proof based on reduction to $\mathrm{D}_4$ where the question is settled by straightforward matrix calculations.

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English version:
Journal of Mathematical Sciences (New York), 2007, 147:5, 7005–7020

Bibliographic databases:

UDC: 512.5
Received: 20.03.2007

Citation: N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, Problems in the theory of representations of algebras and groups. Part 15, Zap. Nauchn. Sem. POMI, 343, POMI, St. Petersburg, 2007, 54–83; J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020

Citation in format AMSBIB
\Bibitem{VavPev07}
\by N.~A.~Vavilov, I.~M.~Pevzner
\paper Triples of long root subgroups
\inbook Problems in the theory of representations of algebras and groups. Part~15
\serial Zap. Nauchn. Sem. POMI
\yr 2007
\vol 343
\pages 54--83
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl111}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2469413}
\elib{http://elibrary.ru/item.asp?id=9595466}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 147
\issue 5
\pages 7005--7020
\crossref{https://doi.org/10.1007/s10958-007-0526-2}
\elib{http://elibrary.ru/item.asp?id=13557967}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-36148952489}


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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, V. V. Nesterov, “Geometriya mikrovesovykh torov”, Vladikavk. matem. zhurn., 10:1 (2008), 10–23  mathnet  mathscinet  elib
    2. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. II”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 242–264  mathnet; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350  crossref
    3. 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
    4. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. I”, Algebra i analiz, 23:5 (2011), 155–198  mathnet  mathscinet  elib; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. I”, St. Petersburg Math. J., 23:5 (2012), 891–919  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
    6. I. M. Pevzner, “Shirina gruppy $\mathrm{GL}(6,K)$ otnositelno mnozhestva kvazikornevykh elementov”, Voprosy teorii predstavlenii algebr i grupp. 26, Zap. nauchn. sem. POMI, 423, POMI, SPb., 2014, 183–204  mathnet  mathscinet; I. M. Pevzner, “Width of $\mathrm{GL}(6,K)$ with respect to quasi-root elements”, J. Math. Sci. (N. Y.), 209:4 (2015), 600–613  crossref
    7. I. M. Pevzner, “Shirina ekstraspetsialnogo unipotentnogo radikala otnositelno mnozhestva kornevykh elementov”, Voprosy teorii predstavlenii algebr i grupp. 28, Zap. nauchn. sem. POMI, 435, POMI, SPb., 2015, 168–177  mathnet  mathscinet
    8. V. V. Nesterov, “Teoremy reduktsii dlya troek korotkikh kornevykh podgrupp v gruppakh Shevalle”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 106–132  mathnet  mathscinet
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