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

This article is cited in 9 scientific papers (total in 9 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
\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
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 147
\issue 5
\pages 7005--7020

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    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, “Width of groups of type $\mathrm E_6$ with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350  mathnet  crossref
    3. I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. 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  mathnet  crossref  mathscinet  isi  elib  elib
    5. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. 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  mathnet  crossref  mathscinet
    7. I. M. Pevzner, “Width of extraspecial unipotent radical with respect to root elements”, J. Math. Sci. (N. Y.), 219:4 (2016), 598–603  mathnet  crossref  mathscinet
    8. V. V. Nesterov, “Reduction theorems for triples of short root subgroups in Chevalley groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 437–452  mathnet  crossref  mathscinet
    9. I. M. Pevzner, “Suschestvovanie kornevoi podgruppy, kotoruyu dannyi element perevodit v protivopolozhnuyu”, Voprosy teorii predstavlenii algebr i grupp. 32, Zap. nauchn. sem. POMI, 460, POMI, SPb., 2017, 190–202  mathnet
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