Skip to main content
SpringerLink
Log in

Net subgroups of Chevalley groups. II. Gauss decomposition

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

Abstract

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set б=(б∝),∝∈Φ, of ideals б∝ of R such that б∝бβ⊑б∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(б) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.

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. Z. I. Borevich, “Parabolic subgroups of linear groups over a semilocal ring,” Vestn. Leningr. Univ., No. 13, 16–24 (1976).

    Google Scholar 

  2. Z. I. Borevich, “Parabolic subgroups of the special linear group over a semilocal ring,” Vestn. Leningr. Univ., No. 19, 29–34 (1976).

    Google Scholar 

  3. Z. I. Borevich, “A description of the subgroups of the full linear group that contain the group of diagonal matrices,” J. Sov. Math.,17, No. 2 (1981).

  4. Z. I. Borevich, “Subgroups of linear groups rich in transvections,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,75, 22–31 (1978).

    Google Scholar 

  5. A. Borel, “Properties and linear representations of Chevalley groups,” in: Seminar on Algebraic Groups [Russian translation], Mir, Moscow (1973), pp. 9–59.

    Google Scholar 

  6. N. Bourbaki, Lie Groups and Lie Algebras [Russian translation], Chaps. IV–VI, Moscow (1972); Chaps. VII, VIII, Moscow (1978).

  7. N. A. Vavilov, “Arrangement of subgroups in linear groups over rings,” Candidate's Dissertation, Leningrad Univ, (1977).

  8. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a semilocal ring,” Zap. Nauch. Seminar. Leningrad. Otd. Mat. Inst. Akad. Nauk SSSR,75, 43–58 (1978).

    Google Scholar 

  9. N. A. Vavilov, “Parabolic subgroups of twisted Chevalley groups over a semilocal ring,” J. Sov. Math.,19, No. 1 (1982).

  10. N. A. Vavilov and E. G. Plotkin, “Net subgroups of Chevalley groups,” J. Sov. Math.,19, No. 1 (1982).

  11. E. V. Dybkova, “Some congruence subgroups of the symplectic group,” J. Sov. Math.,17, No. 2 (1981).

  12. Kourovka Notebook (Unsolved Problems of Group Theory) [in Russian], 7th ed., Novosibirsk (1980).

  13. E. B. Plotkin, “Parabolic subgroups of3D4(R),” J. Sov. Math.,24, No. 4 (1984).

  14. N. S. Romanovskii, “Subgroups of the general and special linear groups over a ring,” Mat. Zaraetki,9, No. 6, 699–708 (1971).

    Google Scholar 

  15. R. Steinberg, Lectures on Chevalley Groups, Yale University Lecture Notes (1967).

  16. E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” TÔkoku Math. J.,28, No. 2, 185–198 (1976).

    Google Scholar 

  17. N. Iwahori and H. Matsumoto, “On some Bruhat decomposition and structure of the Hecke rings of p-adic Chevalley groups,” Publ. Math. Inst. Haut. études Sci., No. 25, 5–48 (1965).

    Google Scholar 

  18. H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. école Norm. Sup., 4-ème sér.,2, 1–62 (1969).

    Google Scholar 

  19. M. R. Stein, “Subjective stability in demension O for K2 and related functors,” Trans. Am. Math. Soc.,178, No. 1, 165–191 (1973).

    Google Scholar 

  20. M. R. Stein, Stability theorems for k1, K2 and related functors modeled on Chevalley groups,” Jpn. J. Math.,4, No. 1, 77–108 (1978).

    Google Scholar 

  21. K. Suzuki, “On parabolic subgroups of Chevalley groups over local rings,” TÔhoku Math. J.,28, No. 1, 57–66 (1976).

    Google Scholar 

  22. K. Suzuki, “On parabolic subgroups of Chevalley groups over commutative rings,” Sci. Repts. Tokyo Kyoiku Daigaku,A13, Nos. 366–382, 225–232 (1977).

    Google Scholar 

Download references

Authors
  1. N. A. Vavilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. B. Plotkin
    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 AN SSSR, Vol. 114, pp. 62–76, 1982.

In conclusion, the authors would like to thank Z. I. Borevich for his interest in this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Plotkin, E.B. Net subgroups of Chevalley groups. II. Gauss decomposition. J Math Sci 27, 2874–2885 (1984). https://doi.org/10.1007/BF01410741

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01410741

Keywords

Search

Navigation