Skip to main content
SpringerLink
Log in

Triples of long root subgroups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the 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 papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it is proved in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in 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 E l, we need a detailed justification of this list, which we could not extract from the published papers. That is why in the present paper, we produce a direct elementary proof based on the reduction to D 4, where the question is settled by straightforward matrix calculations. Bibliography: 73 titles.

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

References

  1. E. Artin, Geometric Algebra [Russian translation], Nauka, Moscow (1969).

    MATH  Google Scholar 

  2. E. L. Bashkirov, “On linear groups generated by two long root subgroups,” Sib. Mat. Zh., 34,No. 2, 15–23 (1993).

    MathSciNet  Google Scholar 

  3. E. L. Bashkirov, “Linear groups that contain a root subgroup,” Sib. Mat. Zh., 37,No. 6, 1238–1255 (1996).

    MathSciNet  Google Scholar 

  4. E. L. Bashkirov, “The groups of the general linear group of degree 4 over the quaternion skew field that contain the special unitary group of index 1,” Algebra Analiz, 13,No. 3, 18–42 (2001).

    MathSciNet  Google Scholar 

  5. E. L. Bashkirov, “The groups \(S_{pin_8 } \) and some subgroups of unitary groups of degree 4 over the quaternion skew field,” Algebra Analiz, 13,No. 3, 43–64 (2001).

    MathSciNet  Google Scholar 

  6. E. L. Bashkirov, “Linear groups over skew fields, which contain subgroups of quadratic unipotent elements,” Doctoral Thesis, Belarus State Univ. of Computer Science and Radio Electronics (2006).

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

  8. N. A. Vavilov, “The geometry of long root subgroups in Chevalley groups,” Vestn. Leningr. Gos. Univ., Ser. 1, No. 1, 5–10 (1988)

  9. N. A. Vavilov, “The Bruhat decomposition of long root elements in Chevalley groups,” in: Rings and Modules. Limit Theorems in Probability Theory, Leningrad (1988), pp. 18–39.

  10. N. A. Vavilov, “Subgroups of split orthogonal groups over a ring,” Sib. Mat. Zh., 29,No. 4, 31–43 (1988).

    MathSciNet  Google Scholar 

  11. N. A. Vavilov, “Semisimple root elements and triples of unipotent root subgroups in Chevalley groups,” in: Problems of Algebra, Vol. 4, Minsk (1989), pp. 162–173.

    MATH  MathSciNet  Google Scholar 

  12. N. A. Vavilov, “Subgroups of split classical groups,” Tr. Steklov Mat. Inst. Akad. Nauk SSSR, 183, 29–41 (1990).

    MATH  MathSciNet  Google Scholar 

  13. N. A. Vavilov, “The subgroups of Chevalley groups that contain a maximal torus,” Tr. Leningr. Mat. Obshch., 1, 64–109 (1990).

    MathSciNet  MATH  Google Scholar 

  14. N. A. Vavilov, “Subgroups of split orthogonal groups over a commutative ring,” Zap. Nauchn. Semin. POMI, 281, 35–59 (2001).

    Google Scholar 

  15. N. A. Vavilov, “Geometry of 1-tori in GLn,” Algebra Analiz, 19,No. 3, 120–151 (2007).

    MathSciNet  Google Scholar 

  16. N. A. Vavilov, “Weight elements of Chevalley groups,” Algebra Analiz, 19,No. 5 (2007).

  17. N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “Chevalley groups of type E 6 in the 27-dimensional representation,” Zap. Nauchn. Semin. POMI, 388, 5–68 (2007).

    MathSciNet  Google Scholar 

  18. N. A. Vavilov and V. V. Nesterov, “Pairs of microweight tori in Chevalley groups of type E6,” Algebra Analiz (to appear).

  19. N. A. Vavilov and A. A. Semenov, “The Bruhat decomposition of long root tori in Chevalley groups,” Zap. Nauchn. Semin. LOMI, 175, 12–23 (1989).

    Google Scholar 

  20. N. A. Vavilov and A. A. Semenov, “Long root semisimple elements in Chevalley groups,” Dokl. RAN, 338,No. 6, 725–727 (1994).

    Google Scholar 

  21. J. Dieudonné, Geometry of Classical Groups [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  22. A. E. Zalesskii, “Linear groups,” Usp. Mat. Nauk, 36,No. 6, 56-1–7 (1981).

    MathSciNet  Google Scholar 

  23. A. E. Zalesskii, “Linear groups,” in: Scientific Reviews. Fundamental Directions. Algebra, Vol. 4, Moscow (1989), pp. 114–228.

    MathSciNet  Google Scholar 

  24. A. E. Zalesskii and V. N. Serezhkin, “Linear groups generated by transvections,” Izv. Akad. Nauk SSSR, 10, 25–46 (1977).

    MATH  Google Scholar 

  25. A. S. Kondratiev, “Subgroups of finite Chevalley groups,” Usp. Mat. Nauk, 41,No. 1, 57–96 (1986).

    MathSciNet  Google Scholar 

  26. V. V. Nesterov, “Pairs of short root subgroups in a Chevalley group,” Dokl RAN, 357, 302–305 (1997).

    MATH  MathSciNet  Google Scholar 

  27. V. V. Nesterov, “Pairs of short root subgroups in a Chevalley group of type G2,” Zap. Nauchn. Semin. POMI, 281, 253–273 (2001).

    Google Scholar 

  28. V. V. Nesterov, “Pairs of short root subgroups in a Chevalley group,” Algebra Analiz, 16,No. 6, 172–208 (2004).

    MathSciNet  Google Scholar 

  29. V. A. Petrov, “Odd unitary groups,” Zap. Nauchn. Semin. POMI, 305, 195–225 (2003).

    Google Scholar 

  30. A. A. Semenov, “Bruhat decomposition of long root semisimple tori in Chevalley groups,” Ph. D. Thesis, St.Petersburg State Univ. (1991).

  31. R. Steinberg, Lectures in Chevalley Groups [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  32. A. A. Suslin and V. I. Kopeiko, “Quadratic modules and orthogonal groups over rings of polynomials,” Zap. Nauchn. Semin. LOMI, 71, 216–250 (1997).

    MathSciNet  Google Scholar 

  33. M. Aschbacher and G. M. Seitz, “Involutions in Chevalley groups over fields of odd order,” Nagoya Math. J., 63, 1–91 (1976).

    MATH  MathSciNet  Google Scholar 

  34. A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloquium, 7,No. 2, 159–196 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  35. E. L. Bashkirov, “Some completely reducible linear groups over a division ring, containing a root subgroup,” Comm. Algebra, 31,No. 12, 5727–5754 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  36. E. L. Bashkirov, “Irreducible linear groups of degree 3 over a quaternion division ring, containing a root subgroup,” Comm. Algebra, 32,No. 5, 1747–1763 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  37. E. L. Bashkirov, “Irreducible linear groups of degree 4 over a quaternion division algebra that contain a subgroup diag(T 3(K, Φ0), 1),” J. Algebra, 287,No. 2, 319–350 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  38. R. Brown and S. P. Humphries, “Orbits under symplectic transvections. I, II,” Proc. London Math. Soc., 52, 517–531, 532–556 (1986).

    Article  MathSciNet  Google Scholar 

  39. P. J. Cameron and J. I. Hall, “Some groups generated by transvection subgroups,” J. Algebra, 140, 184–209 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  40. R. W. Carter, Simple Groups of Lie Type, Wiley, London et al. (1972).

    MATH  Google Scholar 

  41. B. N. Cooperstein, “Subgroups of the group E6(q) which are generated by root subgroups,” J. Algebra, 46, 355–388 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  42. B. N. Cooperstein, “The geometry of root subgroups in exceptional groups. I, II,” Geom. Dedic., 8, 317–381 (1979); 15, 1–45 (1983).

    MATH  MathSciNet  Google Scholar 

  43. B. N. Cooperstein, “Geometry of long root subgroups in groups of Lie type,” Proc. Symp. Pure Math., 37, 243–248 (1980).

    MathSciNet  Google Scholar 

  44. B. N. Cooperstein, “Subgroups of exceptional groups of Lie type generated by long root elements. I, II,” J. Algebra, 70,No. 1, 270–282, 283–298 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  45. H. Cuypers, “The geometry of k-transvection groups,” Preprint Eindhoven Univ. Technology, 1–24 (1994).

  46. H. Cuypers, “Symplectic geometries, transvection subgroups, and modules,” J. Comb. Theory, Ser. A, 65, 39–59 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  47. H. Cuypers and A. I. Steinbach, “Linear transvection groups and embedded polar spaces,” Invent. Math., 137,No. 1, 169–198 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  48. H. Cuypers and A. I. Steinbach, “Special linear groups generated by transvections and embedded projective spaces,” J. London Math. Soc., 64,No. 3, 576–594 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  49. L. Di Martino and N. A. Vavilov, “(2, 3)-generation of SL(n, q). I, II. Cases n = 5, 6, 7,” Comm. Algebra, 22,No. 4, 1321–1347 (1994); 24, No. 3, 487–515 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, N.Y. et al. (1989).

    MATH  Google Scholar 

  51. A. L. Harebov and N. A. Vavilov, “On the lattice of subgroups of Chevalley groups containing a split maximal torus,” Comm. Algebra, 24,No. 1, 109–133 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  52. W. M. Kantor, “Subgroups of classical groups generated by long root elements,” Trans. Amer. Math. Soc., 248,No. 2, 347–379 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  53. W. M. Kantor, “Generation of linear groups,” in: The Geometric Vein: Coxeter Festschrift, Springer-Verlag, Berlin (1981), pp. 497–509.

    Google Scholar 

  54. Li Shang Zhi, “Irreducible subgroups of SL(n, K) generated by root subgroups,” Geom. Dedic., 31, 41–44 (1989).

    Article  Google Scholar 

  55. M. W. Liebeck and G. M. Seitz, “Subgroups generated by root elements in groups of Lie type,” Ann. Math., 139, 293–361 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  56. V. A. Petrov, “Overgroups of unitary groups,” K-Theory, 29, 147–174 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  57. G. E. Röhrle, “On extraspecial parabolic subgroups,” Contemp. Math., 153, 143–155 (1993).

    Google Scholar 

  58. A. I. Steinbach, “Untergruppen von klassischen Gruppen, die von Transvektionen oder Siegel-Transvektionen erzeugt werden,” Ph.D. Thesis, Gießen (1995).

  59. A. I. Steinbach, “Subgroups of classical groups generated by transvections or Siegel transvections. I, II,” Geom. Dedic., 68, 281–322, 323–357 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  60. A. I. Steinbach, “Subgroups isomorphic to G2(L) in orthogonal groups,” J. Algebra, 205,No. 1, 77–90 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  61. A. I. Steinbach, “Groups of Lie type generated by long root elements in F4(K),” Habilitationsschriftr, Gießen (2000).

  62. A. I. Steinbach, “Subgroups of the Chevalley groups of type F4(K) arising from a polar space,” Adv. Geom., 3, 73–100 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  63. F. G. Timmesfeld, “Groups generated by k-transvections,” Invent. Math., 100, 167–206 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  64. F. G. Timmesfeld, “Groups generated by k-root subgroups,” Invent. Math., 106, 575–666 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  65. F. G. Timmesfeld, “Groups generated by k-root subgroups — a survey,” in: Groups, Combinatorics, and Geometry (Durham — 1990), Cambridge Univ. Press (1992), pp. 183–204.

  66. F. G. Timmesfeld, “Moufang planes and the groups E K6 and SL2(K), K a Cayley division algebra,” Forum Math., 6,No. 2, 209–231 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  67. F. G. Timmesfeld, “Subgroups generated by root elements of groups generated by k-root subgroups,” Geom. Dedic., 49, 293–321 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  68. F. G. Timmesfeld, “Abstract root subgroups and quadratic actions. With an appendix by A. E. Zalesskii,” Adv. Math., 142,No. 1, 1–150 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  69. F. G. Timmesfeld, Abstract Root Subgroups and Groups of Lie Type, Birkhäuser Verlag, Basel (2001), pp. 1–389.

    MATH  Google Scholar 

  70. N. A. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proceedings of the Conference on Nonassociative Algebras and Related Topics (Hiroshima — 1990), World Sci. Publ., London (1991), pp. 219–335.

    Google Scholar 

  71. N. A. Vavilov, “Intermediate subgroups in Chevalley groups,” in: Proceedings of the Conference on Groups of Lie Type and Their Geometries (Como — 1993), Cambridge Univ. Press (1995), pp. 233–280.

  72. N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–115 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  73. A. Wagner, “Groups generated by elations,” Abh. Math. Sem. Univ. Hamburg, 41, 199–205 (1974).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    N. A. Vavilov & I. M. Pevzner

Authors
  1. N. A. Vavilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. M. Pevzner
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 54–83.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Pevzner, I.M. Triples of long root subgroups. J Math Sci 147, 7005–7020 (2007). https://doi.org/10.1007/s10958-007-0526-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-007-0526-2

Keywords

Search

Navigation