Journal of Mathematical Sciences

, Volume 180, Issue 3, pp 338–350

Width of groups of type E6 with respect to root elements. II


The simply connected and adjoint groups of type E6 over fields are considered. Let K be a field such that every polynomial of degree at most 6 has a root in K. We prove that every element of the adjoint group of type E6 over K can be written as a product of at most seven root elements. Bibliography: 59 titles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Borel, “Properties and linear representations of Chevalley groups” [Russian translation], in: Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 9–59.Google Scholar
  2. 2.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps. IV-VI, Mir, Moscow (1972).Google Scholar
  3. 3.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps. VII-VIII, Mir, Moscow (1978).Google Scholar
  4. 4.
    N. A. Vavilov, “How to see the signs of structure constants,” Algebra Analiz, 19, No. 4, 34–68 (2007).MathSciNetGoogle Scholar
  5. 5.
    N. A. Vavilov and M. R. Gavrilovich, “An A2-proof of structure theorems for Chevalley groups of types E6 and E7,” Algebra Analiz, 16, No. 4, 54–87 (2004).MathSciNetGoogle Scholar
  6. 6.
    N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “The structure of Chevalley groups: a proof from the Book,” Zap. Nauchn. Semin. POMI, 330, 36–76 (2006).MATHGoogle Scholar
  7. 7.
    N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E6,” Algebra Analiz, 19, 35–62 (2007).Google Scholar
  8. 8.
    N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “The Chevalley group of type E6 in the 27-dimensional representation,” Zap. Nauchn. Semin. POMI, 338, 5–68 (2006).MATHGoogle Scholar
  9. 9.
    N. A. Vavilov and I. M. Pevzner, “Triples of long root subgroups,” Zap. Nauchn. Semin. POMI, 343, 54–83 (2007).MathSciNetGoogle Scholar
  10. 10.
    E. B. Vinberg and A. L. Onishchik, Seminar on Lie Groups and Algebraic groups [in Russian], Nauka, Moscow (1988).Google Scholar
  11. 11.
    A. Yu. Luzgarev, “On overgroups of E(E6, R) and E(E7, R) in minimal representations,” Zap. Nauchn. Semin. POMI, 319, 216–243 (2004).MATHGoogle Scholar
  12. 12.
    A. Yu. Luzgarev, “Description of overgroups of F4 in E6 over a commutative ring,” Algebra Analiz, 20, No. 6, 148–185 (2008).MathSciNetGoogle Scholar
  13. 13.
    O. O’Meara, “Lectures on linear groups” [Russian translation], in: Automorphisms of Classical Groups, Mir, Moscow (1976), pp. 57–167.Google Scholar
  14. 14.
    O. O’Meara, Lectures on Symplectic Groups [Russian translations], Mir, Moscow (1979).Google Scholar
  15. 15.
    I. M. Pevzner, “Geometry of root elements in groups of type E6,” Algebra Analiz (2011).Google Scholar
  16. 16.
    I. M. Pevzner, “Width of groups of type E6 with respect to root elements. I,” Algebra Analiz (2011).Google Scholar
  17. 17.
    T. A. Springer, “Linear algebraic groups” [Russian translation], in: Algebraic Geometry. 4, Advances in Science and Technique, 55, VINITI, Moscow (1989), pp. 5–136.Google Scholar
  18. 18.
    R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).Google Scholar
  19. 19.
    J. Humphrey, Linear Algebraic Groups [Russian translation], Nauka, Moscow (1980).Google Scholar
  20. 20.
    J. Humphrey, Introduction to the Theory of Lie Algebras and Their Representations [Russian translation], Moscow (2003).Google Scholar
  21. 21.
    M. Aschbacher, “The 27-dimensional module for E6. I,” Invent. Math., 89, No1, 159–195 (1987).CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    M. Aschbacher, “The 27-dimensional module for E6. II,” J. London Math. Soc., 37, 275–293 (1988).CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    M. Aschbacher, “The 27-dimensional module for E6. III,” Trans. Amer. Math. Soc., 321, 45–84 (1990).CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    M. Aschbacher, “The 27-dimensional module for E6. IV,” J. Algebra, 131, 23–39 (1990).CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    M. Aschbacher, “Some multilinear forms with large isometry groups,” Geom. Dedicata, 25, No. 1–3, 417–465 (1988).MathSciNetMATHGoogle Scholar
  26. 26.
    M. Aschbacher, “The geometry of trilinear forms,” Finite Geometrics, Buildings and Related Topics, Oxford Univ. Press, Oxford (1990), pp. 75–84.Google Scholar
  27. 27.
    R. W. Carter, Simple Groups of Lie Type, Wiley, London (1989).MATHGoogle Scholar
  28. 28.
    C. Chevalley and R. D. Schafer, “The exceptional simple Lie algebras F4 and E6,” Proc. Nat. Acad. Sci. USA, 36, 137–141 (1950).CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    A. M. Cohen, M. W. Liebeck, J. Saxl, and G. M. Seitz, “The local maximal subgroups of exceptional groups of Lie type, finite and algebraic,” Proc. London Math. Soc., 3-64, No. 1, 21–48 (1992).CrossRefMathSciNetGoogle Scholar
  30. 30.
    D. I. Deriziotis and A. P. Fakiolas, “The maximal tori in the finite Chevalley groups of type E6, E7 and E8,” Commun. Algebra, 19, No. 3, 889–903 (1991).CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    J. Dieudonné, “Sur les générateurs des groupes classiques,” Summa Brasil. Math., 3, 149–179 (1955).MathSciNetGoogle Scholar
  32. 32.
    D. Ž. Djoković and J. G. Malzan, “Products of reflections in the general linear group over a division ring,” Linear Algebra Appl., 28, 53–62 (1979).CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    D. Ž. Djoković and J. G. Malzan, Products of Reflections in U(p, q), Providence, Rhode Island (1982).Google Scholar
  34. 34.
    R. H. Dye, “Scherk’s theorem on orthogonalities revisited,” Geom. Dedicata, 20, No. 3, 349–356 (1986).CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    E. W. Ellers, “Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries,” Abh. Math. Sem. Univ. Hamburg, 46, 97–127 (1977).CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    E. W. Ellers and R. Frank, “Products of quasireflections and transvections over local rings,” J. Geom., 31, Nos. 1–2, 69–78 (1988).CrossRefMathSciNetMATHGoogle Scholar
  37. 37.
    E. W. Ellers and H. Ishibashi, “Factorization of transformations over a local ring,” Linear Algebra Appl., 85, 12–27 (1987).CrossRefMathSciNetGoogle Scholar
  38. 38.
    E. W. Ellers and H. Lausch, “Length theorems for the general linear group of a module over a local ring,” J. Austral. Math. Soc. Ser. A, 46, No. 1, 122–131 (1989).CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    E. W. Ellers and H. Lausch, “Generators for classical groups of modules over local rings,” J. Geom., 39, Nos. 1–2, 60–79 (1990).CrossRefMathSciNetMATHGoogle Scholar
  40. 40.
    P. Gilkey and G. M. Seitz, “Some representations of exceptional Lie algebras,” Geom. Dedicata, 25, Nos. 1–3, 407–416 (1988).MathSciNetMATHGoogle Scholar
  41. 41.
    M. Götzky, “Unverkürzbare Produkte und Relationen in unitären Gruppen,” Math. Z., 104, 1–15 (1968).CrossRefMathSciNetMATHGoogle Scholar
  42. 42.
    M. Götzky, “Über die Erzeugenden der engeren unitären Gruppen,” Arch. Math., 19, 383–389 (1968).CrossRefMATHGoogle Scholar
  43. 43.
    H. Ishibashi, “Generators of orthogonal groups over a local valuation domain,” J. Algebra, 55, No. 2, 302–307 (1978).CrossRefMathSciNetMATHGoogle Scholar
  44. 44.
    H. Ishibashi, “Generators of Spn(V) over a quasisemilocal semihereditary ring,” J. Pure Appl. Algebra, 22, No. 2, 121–129 (1981).CrossRefMathSciNetMATHGoogle Scholar
  45. 45.
    H. Ishibashi, “Generators of orthogonal groups over valuation rings,” Canad. J. Math., 33, No. 1, 116–128 (1981).CrossRefMathSciNetMATHGoogle Scholar
  46. 46.
    G. Malle, J. Saxl, and T. S. Weigel, “Generation of classical groups,” Geom. Dedicata, 49, No. 1, 85–116 (1994).CrossRefMathSciNetMATHGoogle Scholar
  47. 47.
    H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup., 4ème Sér. 2, No. 1, 1–62 (1969).MATHGoogle Scholar
  48. 48.
    Ch. Parker and G. E. Röhrle, “Miniscule Representations,” Preprint Universität Bielefeld, No. 72 (1993).Google Scholar
  49. 49.
    E. B. Plotkin, A. A. Semenov, and N. A. Vavilov, “Visual basic representations: an atlas,” Int. J. Algebra Computations, 8, No. 1, 61–97 (1998).CrossRefMathSciNetMATHGoogle Scholar
  50. 50.
    U. Spengler and H. Wolff, “Die Länge einer symplektischen Abbildung,” J. reine angew. Math., 274–275, 150–157 (1975).MathSciNetGoogle Scholar
  51. 51.
    T. A. Springer, Linear Algebraic Groups, Birkhäuser Boston Inc., Boston (1998).CrossRefMATHGoogle Scholar
  52. 52.
    C. Stanley-Albarda, “A comparison of length definitions for maps of modules over local rings,” J. Geom., 53, Nos. 1–2, 191–200 (1995).CrossRefMathSciNetMATHGoogle Scholar
  53. 53.
    J. Tits, “Sure les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples,” Publ. Math. Inst. Hautes Et. Sci., No. 31, 21–58 (1966).Google Scholar
  54. 54.
    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 et al. (1991), pp. 219–335.Google Scholar
  55. 55.
    N. A. Vavilov, “A third look at weight diagrams,” Rend. Sem. Mat. Univ. Padova, 204, 1–45 (2000).MathSciNetGoogle Scholar
  56. 56.
    N. A. Vavilov, “Do it yourself structure constants for Lie algebras of type E l,” Zap. Nauchn. Semin. POMI, 281, 60–104 (2001).Google Scholar
  57. 57.
    N. A. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7,” Int. J. Algebra Computations, 17, Nos. 5–6, 1283–1298 (2007).CrossRefMathSciNetMATHGoogle Scholar
  58. 58.
    N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Applicandae Math., 45, 73–115 (1996).CrossRefMathSciNetMATHGoogle Scholar
  59. 59.
    L. G. Zhou, “Scherk’s theorem on orthogonal groups over a local ring. I. Expressing orthogonal transformations as the product of symmetries and semi-symmetry,” Donghei Shida Xuebao, 2, 17–24 (1985).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Russian State A. I. Gertsen Pedagogical UniversitySt. PetersburgRussia

Personalised recommendations