Journal of Mathematical Sciences

, Volume 180, Issue 3, pp 197–251 | Cite as

Chevalley groups of type E7 in the 56-dimensional representation



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Abe, “Automorphisms of Chevalley groups over commutative rings,” Algebra Analiz, 5, No. 2, 74–90 (1993).Google Scholar
  2. 2.
    A. Borel, “Properties and linear representations of Chevalley groups,” [Russian translation], in: Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 9–59.Google Scholar
  3. 3.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps. IV–VI, Mir, Moscow (1972).Google Scholar
  4. 4.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps. VII, VIII, Mir, Moscow (1972).Google Scholar
  5. 5.
    E. I. Bunina, “Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2,” Fundam. Prikl. Mat., 15, No. 2, 35–59 (2007).Google Scholar
  6. 6.
    E. I. Bunina, “Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2,” Algebra Logika, 48, No. 4, 443–470 (2007).MathSciNetGoogle Scholar
  7. 7.
    N. A. Vavilov, “How to see the signs of structure constants,” Algebra Analiz, 19, No. 4, 34–68 (2007).MathSciNetGoogle Scholar
  8. 8.
    N. A. Vavilov, “Calculations in exceptional groups,” Vestn. Samarsk. Univ., Est. Nauchn. Seriya, No. 7, 11–24 (2007).Google Scholar
  9. 9.
    N. A. Vavilov, “Numerology of square equations,” Algebra Analiz, 20, No. 3, 9–40 (2008).Google Scholar
  10. 10.
    N. A. Vavilov, “Structure of isotropic reductive groups,” Tr. Inst. Mat. NAN Belarus, 18, No. 1, 1–13 (2010).MathSciNetGoogle Scholar
  11. 11.
    N. A. Vavilov, “Some more exceptional numerology,” Zap. Nauchn. Semin. POMI, 375, 22–31 (2010).Google Scholar
  12. 12.
    N. A. Vavilov, “An A3-proof of sturcture theorems for Chevalley groups of types E6 and E7. II. The main lemma,” Algebra Analiz, 23, No. 6, 1–31 (2011).Google Scholar
  13. 13.
    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
  14. 14.
    N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “Structure of Chevalley groups: a proof from the Book,” Zap. Nauchn. Semin. POMI, 330, 36–76 (2006).MATHGoogle Scholar
  15. 15.
    N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E6,” Algebra Analiz, 19, 35–62 (2007).Google Scholar
  16. 16.
    N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E7,” (2011) to appear.Google Scholar
  17. 17.
    N. A. Vavilov and A. Yu. Luzgarev, “An A2-proof of structure theorems for Chevalley groups of type E8,” (2011) to appear.Google Scholar
  18. 18.
    N. A. Vavilov and 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 (2008).Google Scholar
  19. 19.
    N. A. Vavilov and S. I. Nikolenko, “An A2-proof of structure theorems for Chevalley groups of type F4,” Algebra Analiz, 20, No. 4, 27–63 (2008).MathSciNetGoogle Scholar
  20. 20.
    N. A. Vavilov and A. V. Stepanov, “Overgroups of semisimple groups,” Vestn. Samarsk. Univ., Est. Nauchn. Seriya, No. 3, 51–95 (2008).Google Scholar
  21. 21.
    A. Yu. Luzgarev, “On overgroups of E(E 6, R) and E(E7, R) in minimal representations,” Zap. Nauchn. Semin. POMI, 319, 213–243 (2004).Google Scholar
  22. 22.
    A. Yu. Louzgarev, “Description of overgroups o E(F4, R) and G(E 6, R) over a commutative ring,” Algebra Analiz, 20, No. 6, 148–185 (2008).Google Scholar
  23. 23.
    A. Yu. Luzgarev, “Overgroups of exceptional groups,” Ph. D. Thesis, S.-Peterburg. Univ. (2008).Google Scholar
  24. 24.
    A. Yu. Luzgarev, “Characteristic-independent invariants of fourth degree for G(E7, R),” (2011) to appear.Google Scholar
  25. 25.
    A. Yu. Luzgarev, “Equations determining the orbit of the highest weight vector in the adjoint representation,” (2011) to appear.Google Scholar
  26. 26.
    A. Yu. Luzgarev and A. K. Stavrova, “The elementary subgroup of an isotropic reductive group is perfect,” Algebra Analiz, 23, No. 55, 140–154 (2011).Google Scholar
  27. 27.
    Yu. I. Manin, Cubic Forms: Algebra, Geometry, and Arithmetic [in Russian], Nauka, Moscow (1972).Google Scholar
  28. 28.
    I. M. Pevzner, “Root elements in exceptional groups,” Ph. D. Thesis, S. Peterburg. Univ. (2008).Google Scholar
  29. 29.
    I. M. Pevzner, “Geometry of root subgroups in groups of type E6,” Algebra Analiz, 23, No. 3, 261–309 (2011).MathSciNetGoogle Scholar
  30. 30.
    I. M. Pevzner, “Width of groups of type E6 with respect to the set of root elements. I,” Algebra Analiz, 23, No. 5, 155–198 (2008).Google Scholar
  31. 31.
    V. A. Petrov and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups,” Algebra Analiz, 20, No. 4, 160–188 (2008).MathSciNetGoogle Scholar
  32. 32.
    E. B. Plotkin, “Surjective stability of the K 1-functor for some exceptional Chevalley groups,” Zap. Nauchn. Semin. POMI, 198, 65–88 (1991).Google Scholar
  33. 33.
    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
  34. 34.
    A. K. Stavrove, “Structure of isotropic reductive groups,” Ph. D. Thesis, S. Peterburg. Univ. (2009).Google Scholar
  35. 35.
    R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).Google Scholar
  36. 36.
    D. Humphrey, Linear Algebraic Groups [Russian translation], Nauka, Moscow (1981).Google Scholar
  37. 37.
    D. Humphrey, Introduction to the Theory of Lie Algebras and Their Representation [Russian translation], Moscow (2003).Google Scholar
  38. 38.
    R. Hartshorn, Algebraic Geometry [Russian translation], Mir, Moscow (1981).Google Scholar
  39. 39.
    E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J., 21, No. 3, 474–494 (1969).MATHCrossRefGoogle Scholar
  40. 40.
    E. Abe, “Whitehead groups of Chevalley groups over polynomial rings,” Comm. Algebra, 11, No. 12, 1271–1307 (1983).MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    E. Abe, “Chevalley groups over commutative rings,” in: Radical Theory (Sendai, 1988) (Tokyo), Uchida Rokakuho (1989), pp. 1–23Google Scholar
  42. 42.
    E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” in: Algebraic K-theory and Algebraic Number Theory (Honolulu, HI, 1987), Providence, Rhode Island (1989), pp. 1–17.Google Scholar
  43. 43.
    E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 2, 185–198 (1976).MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    M. Aschbacher, “Some multilinear forms with large isometry groups,” Geom. Dedicata, 25, Nos. 1–3, 417–465 (1988).MATHMathSciNetGoogle Scholar
  45. 45.
    H. Azad, M. Barry, and G. M. Seitz, “On the structure of parabolic subgroups,” Comm. Algebra, 18, No. 2, 551–562 (1990).MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    J. Baez, “The octonions,” Bull. Amer. Math. Soc., 39, 145–205 (2002).MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K1 is nilpotent,” J. Pure Appl. Algebra, 213, 1075–1085 (2009).MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    M. Brion and V. Lakshmibai, “A geometric approach to standard monomial theory,” J. Representation Theory, 7, 651–680 (2003).MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance Regular Graphs, Springer Verlag, New York (1989).MATHGoogle Scholar
  50. 50.
    R. B. Brown, “A new type of nonassociative algebras,” Proc. Nat. Acad. Sci. USA, 50, No. 5, 947–949 (1963).MATHCrossRefGoogle Scholar
  51. 51.
    R. B. Brown, “A minimal representation for the Lie algebra E7,” Ill. J. Math., 12, No. 1, 190–200 (1968).MATHGoogle Scholar
  52. 52.
    R. B. Brown, “Groups of type E7,” J. reine angew. Math., 236, 79–102 (1969).MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    R. Brylinski and B. Kostant, “Minimal representations of E6, E7, and E8 and the generalized Capelli identity,” Proc. Nat. Acad. Sci. U.S.A., 91, No. 7, 2469–2472 (1994).MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    N. Burgoyne and C. Williamson, “Some computations involving simple Lie algebras,” in: Symposium on Symbolic and Algebraic Manipulation, Ass. Comp. Mach., New York (1971), pp. 162–171.Google Scholar
  55. 55.
    R. W. Carter, Simple Groups of Lie Type, John Wiley & Sons, London New York Sydney (1972).MATHGoogle Scholar
  56. 56.
    R. W. Carter, Finite groups of Lie type: Conjugacy Classes and Complex Characters, John Wiley & Sons, London et al. (1985).MATHGoogle Scholar
  57. 57.
    P. E. Chaput, L. Manivel, and N. Perrin, “Quantum cohomology of minuscule homogeneous varieties,” ccsd-0086927, 28 Sep 2006, 1–34.Google Scholar
  58. 58.
    Y. Choi and S. Yoon, “Homology of the double and the triple loop spaces of E6, E7 and E8,” Manuscripta Math., 103, 101–116 (2000).MATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    V. Chernousov, “The kernel of the Rost invariant, Serre’s conjecture II and the Hasse principle for quasi-split groups D4, E6, E7,” Math. Ann., 326, No. 2, 297–330 (2003).MATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    A. M. Cohen and R. H. Cushman, “Gröbner bases and standard monomail theory,” in: Computational Algebraic Geometry (Nice, 1992), Birkhäuser, Boston (1993), pp. 41–60.CrossRefGoogle Scholar
  61. 61.
    A. M. Cohen, S. H. Murray, and D. E. Taylor, “Computing in groups of Lie type,” Math. Comput., 73, 1477–1498 (2004).MATHMathSciNetGoogle Scholar
  62. 62.
    B. N. Cooperstein, “The fifty-six-dimensional module for E7. I. A four form for E7,” J. Algebra, 173, No. 2, 361–389 (1995).MATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    T. De Medts, “A characterization of quadratic forms of type E6, E7, and E8,” J. Algebra, 252, No. 2, 394–410 (2002).MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    L. E. Dickson, “The configurations of the 27 lines on a cubic surface and the 28 bitangents to a quartic curve,” Amer. Math. Soc. Bull., 8, 63–70 (1901).MATHCrossRefGoogle Scholar
  65. 65.
    D. Ðoković, “Explicit Cayley triples in real forms of E7,” Pacific J. Math., 191, 1–23 (1999).MathSciNetCrossRefGoogle Scholar
  66. 66.
    D. Ðoković, “The closure diagram for nilpotent orbits of the split real form of E7,” Reprent. Theory, 5, 284–316 (2001).CrossRefGoogle Scholar
  67. 67.
    J. R. Faulkner, “A geometry for E7,” Trans. Amer. Math. Soc., 167, 49–58 (1972).MATHMathSciNetGoogle Scholar
  68. 68.
    J. R. Faulkner and J. C. Ferrar, “Exceptional Lie algebras and related algebraic and gemetric structures,” Bull. London Math. Soc., 9, No. 1, 1–35 (1977).MATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    J. C. Ferrar, “Strictly regular elements in Freudenthal triple systems,” Trans. Amer. Math. Soc., 174, 313–331 (1972).MathSciNetCrossRefGoogle Scholar
  70. 70.
    J. C. Ferrar, “On the classification of Freudenthal triple systems and Lie algebras of type E7,” J. Algebra, 62, 276–282 (1980).MATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    I. B. Frenkel and V. Kac, “Basic representations of affine Lie algebras and dual resonance models,” Invent. Math., 62, No. 2, 23–66 (1980).MATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, New York (1988).MATHGoogle Scholar
  73. 73.
    H. Freudenthal, “Sur des invariantes caractéristiques des groupes semi-simples,” Proc. Nederl. Akad. Wetensch. Ser. A, 56, 90–94 (1953).MATHMathSciNetGoogle Scholar
  74. 74.
    H. Freudenthal, “Sur le groupe exceptionnel E7,” Proc. Nederl. Akad. Wetensch. Ser. A, 56, 81–89 (1953).MATHMathSciNetGoogle Scholar
  75. 75.
    H. Freudenthal, “Zur ebenen Oktavengeometrie,” Indag. Math., 15, 195–200 (1953).MathSciNetGoogle Scholar
  76. 76.
    H. Freudenthal, “Beziehungen der E7 und E8 zur Oktavenebene I–XI,” Proc. Nederl. Akad. Wetensch. Ser. A, 57, 218–230, 363–368 (1954); 58, 151–157, 277–285 (1955); 62, 165–201, 447–474 (1959); 66, 457–487 (1963).MATHGoogle Scholar
  77. 77.
    H. Freudenthal, “Oktaven, Ausnahmegruppen und Oktavengeometrie,” Geom. Dedic., 19, 7–63 (1985).MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    R. S. Garibaldi, “Sturcturable algebras and groups of type E6 and E7,” J. Algebra, 236, No. 2, 651–691 (2001).MATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    R. S. Garibaldi, “Groups of type E7 over arbitrary fields,” Comm. Algebra, 29, No. 6, 2689–2710 (2001).MATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    R. S. Garibaldi, “Cohomological invariants: exceptional groups and Spin groups,” Preprint Emory Univ. Atlanta (2006).Google Scholar
  81. 81.
    Ph. Gille, “Le problème de Kneser Tits,” Sèminaire Bourbaki, No. 983, 2–39 (2007).Google Scholar
  82. 82.
    P. B. Gilkey and G. M. Seitz, “Some representations of exceptional Lie algebras,” Geom. Dedicata, 25, Nos. 1–3, 407–416 (1988).MATHMathSciNetGoogle Scholar
  83. 83.
    D. Ginzburg, “On standard L-functions for E6 and E7,” J. Reine Angew. Math., 465, 101–131 (1995).MATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    N. Gonciulea and V. Lakshmibai, Gröbner Bases and Standard Monomial Bases (2001).Google Scholar
  85. 85.
    R. Griess and A. Ryba, “Embeddings of U 3 (8), Sz (8) and the Rudvalis group on algebraic groups of type E7,” Invent. Math, 116, 215–141 (1994).MATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    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).MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    S. J. Haris, “Some irreductible representations of exceptional algebraic groups,” Amer. J. Math., 93, 75–106 (1971)MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    R. Hazrat, V. Petrov, and N. Vavilov, “Relative subgroups in Chevalley groups,” J. K-theory, 5, 603–618 (2010).MATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    R. Hazrat, N. Vavilov, and Zhang Zuhong, “Relative commutator calculus in Chevalley groups.”J. Algebra, to appear.Google Scholar
  90. 90.
    J.-Y. Hèe, “Groupes de Chevalley et groupes classiques,” in: Seminar of Finite Groups, Vol II, Publ. Math. Univ. Paris VII, vol. 17, Univ. Paris VII, Paris (1984), pp. 1–54.Google Scholar
  91. 91.
    R. B. Howlett, L.J. Rylands, and D. E. Taylor, “Matrix generators for exceptional groups in Lie type,” J. Symb. Comput., 11, 1–17 (2000).Google Scholar
  92. 92.
    H. Kaji and O. Yasukura, “Projective geometry of Freudenthal’s varieties of certain type,” Michigan Math. J., 52, No. 3, 515–542 (2004).MATHMathSciNetCrossRefGoogle Scholar
  93. 93.
    V. Kac, Infinite Dimensional Lie Algebras, Cambridge Univ. Press, 2nd ed. (1985).Google Scholar
  94. 94.
    P. Kleidman and A. Ryba, “Kostant’s conjecture holds for \( {{\text{E}}_7}:{L_2}\left( {37} \right) < {{\text{E}}_7}\left( \mathbb{C} \right) \),” J. Algebra, 161, 535–540 (1993).MATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    P. Kleidman, U. Meierfrankengeld, and A. Ryba, “HS < E7 (5),” J. London Math. Soc., 60, 95–107 (1999).MATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    V. Lakshmibai, P. Littelmann, and P, Magyar, “Standard monomial theory and applications,” in: Representation Tueory and Geometry, Kluwer Acad. Publ., Dordrecht et al., (1998), pp. 319–364.Google Scholar
  97. 97.
    V. Lakshmibai and C. S. Seshadri, “Standard monomial theory,” in Hyderabad Conference on Algebraic Groups, Madras, Manoj Prakashan (1991), pp. 279–323.Google Scholar
  98. 98.
    J. M. Landsberg and L. Manivel, “The projective geometry of Freudenthal’s magic square,” J. Algebra, 239, No. 2, 477–512 (2001).MATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    W. Lichtenstein, “A system of quadrics describing the orbit of the highest weight vector,” Proc. Amer. Math. Soc., 84, No. 4, 605–608 (1982).MATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    P. Littelmann, “Contracting modules and standard monomial theory for symmetrisable Kac-Moody algebras,” J. Amer. Math. Soc., 11, 551–567 (1998).MATHMathSciNetCrossRefGoogle Scholar
  101. 101.
    P. Littelmann, “The path model, the quantum Frobenius map and standard monomial theory,” in: Algebraic Groups and Their Representations, Kluwer Acad. Publ., Dordrecht et al., (1998), pp. 175–212.Google Scholar
  102. 102.
    J. Lurie, “On simply laced Lie algebras and their minuscule representations,” Comment. Math. Helv.., 166, 515–575 (2001).MathSciNetCrossRefGoogle Scholar
  103. 103.
    A. Luzgarev, V. Petrov, and N. Vavilov, “Explicit equations on the orbit of the highest wight vector” (2001), to appear.Google Scholar
  104. 104.
    J. G. M. Mars, “Les nombres de Tamagawa de certains groupes exceptionnels,” Bull. Soc. Math. France, 94, 97–140 (1966).MATHMathSciNetGoogle Scholar
  105. 105.
    H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup., (4) 2, 1–62 (1969).MATHGoogle Scholar
  106. 106.
    K. Mizuno, “The conjugate classes of unipotent elements of the Chevalley groups E7 and E8,” Tokyo J. Math., 3, No. 2, 391–461 (1980).MATHMathSciNetCrossRefGoogle Scholar
  107. 107.
    C. Parker and G. E. Röhrle, Minuscule Representations. Google Scholar
  108. 108.
    V. Petrov, N. Semenov, and K. Zainoulline, “Zero cycles on a twisted Cayley plane,” Canad.Math. Bull., 51, No. 1, 114–124 (2008).MATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    V. Petrov and A. Stavrova, “Tits indices over semilocal rings,” Transformation Groups, 16, No. 1, 193–217 (2011).MATHMathSciNetCrossRefGoogle Scholar
  110. 110.
    V. Petrov, A. Stavrova, and N. Vavilov, “Relative elementary subgroups in isotropic reductive groups,” (2011), to appear.Google Scholar
  111. 111.
    E. B. Plotkin, “On the stability of the K1-functor for Chevalley groups of type E7, J. Algebra, 210, No. 1, 67–85 (1998).MATHMathSciNetCrossRefGoogle Scholar
  112. 112.
    E. B. Plotkin, A. A. Semenov, and N. A. Vavilov, “Visual basic representations: an atlas,” Internat. J. Algebra Comput., 8, No. 1, 61–95 (1998).MATHMathSciNetCrossRefGoogle Scholar
  113. 113.
    R. Richardson, G. E. Röhrle, and R. Steinberg, “Parabolic subgroups with abelian unipotent radical,” Invent. Math., 110, No. 3, 649–671 (1992).MATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    G. E. Röhrle, “On the structure of parabolic subgroups in algebraic groups,” J. Algebra, 157, No. 1, 80–115 (1993).MATHMathSciNetCrossRefGoogle Scholar
  115. 115.
    G. E. Röhrle, “On extraspecial parabolic subgroups,” Contemp. Math., 153, 143–155 (1993).Google Scholar
  116. 116.
    H. Rubenthaler, “The (A2, G2) duality in E6, octonions and the triality principle,” Trans. Amer. Math. Soc., 360, No. 1, 347–367 (2008).MATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    A. J. E. Ryba, “Identification of matrix generators of a Chevalley group.” J. Algebra, 309, 484–496 (2007).MATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    L. J. Rylands and D. E. Taylor, “Construction for octonion and exceptional Jordan algebras, “Preprint Univ. Sydney (2000).Google Scholar
  119. 119.
    J. Sekiguchi, “Configurations of seven lines on the real projective plane and the root system of type E7,” J. Math. Soc. Japan, 51, No. 4, 987–1013 (1999).MATHMathSciNetCrossRefGoogle Scholar
  120. 120.
    C. S. Seshadri, “Geometry of G/P. I Standard monomial theory for minuscule P,” in: C. P. Ramanujam” a Tribute, Tata Press, Bombay (1978), pp. 207–239.Google Scholar
  121. 121.
    E. Shult, “Embeddings and hyperplanes of the Lie incidence geometry of type E7,1,” J. Geom., 59, 152–172 (1997).MATHMathSciNetCrossRefGoogle Scholar
  122. 122.
    T. A. Springer, “Some groups of type E7,” Nagoya Math. J., 182, 259–284 (2006).MATHMathSciNetGoogle Scholar
  123. 123.
    T. A. Springer and F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer-Verlag, Berlin (2000).MATHGoogle Scholar
  124. 124.
    A. Stavrova, “Normal structure of maximal parabolic subgroups in Chevalley groups over commutative rings,” Algebra Coll., 16, No. 4, 631–648 (2009).MATHMathSciNetGoogle Scholar
  125. 125.
    M. R. Stein, “Generators, relations and coverings of Chevelly groups over commutative rings,” Amer. J. Math., 93, 965–1004 (1971).MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    M. R. Stein, “Stability theorems for K1, K2 and related functors modeled on Chevalley groups,” Japan. J. Math. (N. S.), 4, No. 1, 77–108 (1978).MathSciNetGoogle Scholar
  127. 127.
    A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,” Israel J. Math., 1–20 (2010), to appear.Google Scholar
  128. 128.
    G. Taddei, “Mormaliteé des groupes élémentaire dans les groupes de Chevalley sur un anneau,” in: Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), Providence, Rhode Island (1986), pp. 693–710.Google Scholar
  129. 129.
    J. Tits, “Le plan projectif des octaves et les groupes de Lie exceptionnels,” Acad. Roy, Belg. Bull. Cl. Sci., 39, 309–329 (1953).MathSciNetGoogle Scholar
  130. 130.
    J. Tits, “Le plan projectif des octaces et les groupes exceptionnels E6 et E7,” Acad. Roy. Belg. Bull. Cl. Sci., 40, 29–40 (1954).MathSciNetGoogle Scholar
  131. 131.
    J. Tits, “Algèbres alternatives, algebras de Jordan et algebras de Lie exceptionnelles. I. Construction,” Indag. Math., No. 28, 223–237 (1966).Google Scholar
  132. 132.
    L. N. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J. (2), 38, No. 2, 219–230 (1986).MATHMathSciNetCrossRefGoogle Scholar
  133. 133.
    N. A. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., New York (1991), pp. 219–335.Google Scholar
  134. 134.
    N. A. Vavilov, “A third look at weight diagrams,” Rend Sem. Mat. Univ. Padova, 104, 201–250 (2000).MATHMathSciNetGoogle Scholar
  135. 135.
    N. A. Vavilov, “Do it yourself structure constants for Lie algebras of type El,” Zap Nauchn. Semin POMI, 281, 60–104 (2001).Google Scholar
  136. 136.
    N. A. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7,” Int. J. Algebra Comput., 17, Nos. 5–6, 1283–1298 (2007).MATHMathSciNetCrossRefGoogle Scholar
  137. 137.
    N. A. Vavilov, A. Yu. Luzgarev, and A. V. Stepanov, “Calculations in exceptional groups over rings,” Zap. Nauchn. Semin. POMI, 373, 48–72 (2009).Google Scholar
  138. 138.
    N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, No. 1, 73–113 (1996).MATHMathSciNetCrossRefGoogle Scholar
  139. 139.
    R. Weiss, “Moufang quadrangles of type E6 and E7,” J. Reine Angew. Math., 590, 189–226 (2006).MATHMathSciNetCrossRefGoogle Scholar
  140. 140.
    Xu Xiaoping, “Polynomial representation of F4 and a new combinatorial identity about twenty-four,” arXiv:0810.4670[math.RT], 26 Oct 2008, 1–18.Google Scholar
  141. 141.
    Xu Xiaoping, “Polynomial representation of E6 and its combinatorial and PDE implications,” arXiv:0811.1399[math.RT], 10 Nov 2008, 1–24.Google Scholar
  142. 142.
    Xu Xiaoping, “Polynomial representation of E7 and its combinatorial and PDE implications,” arXiv:0812.1432[math.RT], 8 Dec 2008, 1–37.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Einstein Institute of MathematicsHebrew University of JerusalemJerusalemIsrael

Personalised recommendations