Skip to main content
SpringerLink
Log in

Bruhat decomposition for long root tori in Chevalley groups

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

It is proved that all elements of a given long root torus (i.e., a one-parameter subgroup of long root semisimple elements) in a Chevalley group over a field, except for at most three, belong to the same Bruhat decomposition class.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. N. Bourbaki, Groups and Lie Algebras [Russian translation], Mir, Moscow, Chaps. 4–6 (1972); Chaps. 7, 8 (1978).

    Google Scholar 

  2. N. A. Vavilov, “The Bruhat decomposition for one-dimensional transformations,” Vestn. Leningr. Univ, Ser. 1, No. 3, 14–20 (1986).

    Google Scholar 

  3. N. A. Vavilov, “Weight elements of Chevalley groups,” Dokl. Akad. Nauk SSSR,298, No. 3, 524–527 (1986).

    Google Scholar 

  4. N. A. Vavilov, “The geometry of long root subgroups in Chevalley groups,” Vestn. Leningr. Univ., Ser. 1, No. 1, 8–11 (1988).

    Google Scholar 

  5. N. A. Vavilov, “The conjugacy theorem for subgroups of solvable Chevalley groups that contain maximal split tori,” Dokl. Akad. Nauk SSSR,299, No. 2, 269–272 (1988).

    Google Scholar 

  6. N. A. Vavilov, “The Bruhat decomposition for long root semisimple elements in Chevalley groups,” in: Rings and Modules. Limit Theorems of Probability Theory [in Russian], No. 2, Leningr. State Univ., Leningrad (1988), pp. 18–39.

    Google Scholar 

  7. N. A. Vavilov, “The Bruhat decomposition for two-dimensional transformations,” Vestn. Leningr. Univ., Ser. 1, No. 3, 5–10 (1989).

    Google Scholar 

  8. E. B. Plotkin, “Net subgroups of Chevalley groups and stabilization problems,” Candidate's Dissertation, Leningrad (1985).

  9. A. A. Semenov, “On the Bruhat decomposition of roots of semisimple subgroups,” in: Abstracts of the 19th All-Union Conference on Algebra, Part 1 [in Russian], L'vov (1987).

  10. A. A. Semenov, “The Bruhat decomposition of the roots of semisimple root subgroups of the special linear group,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad Nauk SSSR,160, 239–246 (1987).

    Google Scholar 

  11. R. Steinberg, Lectures on Chevalley Groups, Yale Univ., New Haven (1972).

    Google Scholar 

  12. R. W. Carter, Simple Groups of Lie Type, Wiley, London (1972).

    Google Scholar 

  13. H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. Ecole Norm. Supér., Ser. 4,2, 1–62 (1969).

    Google Scholar 

  14. G. M. Seitz, “Properties of the known simple groups,” in: Proc. Symp. Pure Math.,37 (1980), pp. 231–237.

    Google Scholar 

  15. M. R. Stein, “Stability theorems for K1, K2, and related functors modelled on Chevalley groups,” Jpn. J. Math.,4, No. 1, 77–108 (1978).

    Google Scholar 

Download references

Authors
  1. N. A. Vavilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 12–23, 1989.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Semenov, A.A. Bruhat decomposition for long root tori in Chevalley groups. J Math Sci 57, 3453–3458 (1991). https://doi.org/10.1007/BF01100112

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01100112

Keywords

Search

Navigation