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Subgroups of a spinor group containing a maximal split torus. I

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Abstract

This paper studies subgroups of the split spinor group G=Spin (n, k),n ≥6, over a field K containing a maximal split torus T=T(n,K)assuming that charK≠2and ||K ||≥7.The main result of the first pan of the paper is the reduction of the proof that subgroups containing T are standard to the study of subgroups not containing root unipotent elements.

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References

  1. E. Artin, Geometric Algebra, Interscience Publishers Inc., New York-London (1957).

    Google Scholar 

  2. Z. I. Borevich, “Parabolic subgroups in linear groups over a semilocal ring,” Vestnik Leningrad. Univ., No. 13, 16–24 (1976).

    Google Scholar 

  3. Z. I. Borevich, “Description of the subgroups of the general linear group that contain the group of diagonal matrices,” Zap. Nauchn. Sem. Leningrad. Old. Mat. Inst. Steklov. (LOMI),64, 12–29 (1976).

    Google Scholar 

  4. Z. I. Borevich, “Subgroups of linear groups that are full of transvections,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),75, 22–31 (1978).

    Google Scholar 

  5. Z. I. Borevich and N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Trudy Mat. Inst. Steklov,148, 43–57 (1978).

    Google Scholar 

  6. Z. I. Borevich and N. A. Vavilov, ”Definition of the net subgroup,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),132, 26–33 (1983).

    Google Scholar 

  7. Z. I. Borevich and N. A. Vavilov, “Arrangement of subgroups in the general linear group over a commutative ring,” Trudy Mat. Inst. Steklov,165, 24–42 (1984),

    Google Scholar 

  8. A. Borel, Linear Algebraic Groups, Notes Taken by Hyman Bass, W. A. Benjamin Inc., New York-Amsterdam (1969).

    Google Scholar 

  9. A. Borel and J. Tits, “Groupes réductifs,” Inst. Hautes Études Sci. Publ. Math.,27, 55–150 (1965).

    Google Scholar 

  10. H. Bourbaki, Lie Groups and Lie Algebras [Russian translation], Chs. IV–VI, Mir, Moscow (1972); Chs. VII, VIII (1978).

    Google Scholar 

  11. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over semilocal rings,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),75, 43–58 (1977).

    Google Scholar 

  12. N. A. Vavilov, “On the conjugacy of subgroups of the general linear group containing the group of diagonal matrices,” Usp. Mat. Nauk,34, No. 5, 216–217 (1979).

    Google Scholar 

  13. N. A. Vavilov, “The Bruhat decomposition for subgroups containing the group of diagonal matrices,rd Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),103, 20–30 (1980);114, 50–61 (1982).

    Google Scholar 

  14. N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring containing the group of diagonal matrices,” Vestnik Leningrad. Univ., No. 1, 10–15 (1981).

    Google Scholar 

  15. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a commutative ring,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),116, 20–43 (1982).

    Google Scholar 

  16. N. A. Vavilov, “Subgroups of Chevalley groups over a field that contain a maximal torus,” in: Seventeenth All-Union Algebraic Conference, Theses [in Russian], Minsk (1983), Part 1, pp. 38–39.

  17. N. A. Vavilov, “Subgroups of the special linear group containing the group of diagonal matrices,” Vestnik Leningrad. Univ. Ser.1, No. 22, 3–7 (1985); No. 1,10–15 (1986); No. 2, 3–8 (1987); No. 3, 10–15 (1988).

    Google Scholar 

  18. N. A. Vavilov, The Structure of Chevally Groups of Type E6 [in Russian], Leningrad (1986), (Manuscripts deposited in VINITI on April 22, 1986 and July 17, 1986) Dep. Nos. N.2962-B and N.5228-B.

  19. N. A. Vavilov, “Maximal subgroups of Chevalley groups containing a maximal split torus,” in: Z. I. Borevich and V. V. Petrov (eds.), Rings and Modules. Limit Theorems of Probability Theory, I [in Russian], Izd. Lenigrad. Univ., Leningrad (1986), pp. 67–75.

    Google Scholar 

  20. N. A. Vavilov, “The Bruhat decomposition of one-dimensional transformations,” Vestnik Leningrad. Univ. Ser. 1, No. 3, 14–20 (1986).

    Google Scholar 

  21. N. A. Vavilov, “On a problem of G. Seitz,” Theses of the Tenth All-Union Symposium on Group Theory [in Russian], Gomel' (1986).

  22. N. A. Vavilov, Subgroups of Split Classical Groups [in Russian], Doctorate of Sciences Dissertation, Leningrad (1987).

  23. N. A. Vavilov, “Weight elements of Chevalley groups,” Dokl. Akad. Nauk SSSR,298, No. 3, 524–527 (1988).

    Google Scholar 

  24. N. A. Vavilov, “Conjugacy theorems for subgroups of extended Chevalley groups containing a maximal split torus,” Dokl. Akad. Nauk SSSR,299, No. 2, 269–272 (1988).

    Google Scholar 

  25. N. A. Vavilov, “Subgroups of split orthogonal groups,” Sib. Mat. Zh.,29, No. 3, 12–25 (1988).

    Google Scholar 

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

    Google Scholar 

  27. N. A. Vavilov, “Root semisimple elements and triples of root unipotent subgroups of Chevalley groups,” Works of the Tenth All-Union Symposium on Group Theory, Problems in Algebra, No. 4 [in Russian], Minsk (1989), pp. 162–173.

  28. N. A. Vavilov, “Subgroups of split classical groups,” Trudy Mat. Inst. Steklov,183, 29–42 (1990).

    Google Scholar 

  29. N. A. Vavilov, “Subgroups of Chevalley groups containing a split maximal torus,” Trudy Leningrad. Mat Obshch.,1, 64–109 (1990).

    Google Scholar 

  30. N. A. Vavilov and E. V. Dybkova, “Subgroups of the general symplectic group containing the group of diagonal matrices I & II,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),103, 31–47 (1980);132, 44–56 (1983).

    Google Scholar 

  31. N. A. Vavilov and E. B. Plotkin, “Net subgroups of Chevalley groups,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),94, 40–49 (1979);114, 62–76 (1982).

    Google Scholar 

  32. E. V. Dybkova, Arrangement of Subgroups in Symplectic Groups [in Russian], Candidate Dissertation, Leningrad (1986).

  33. E. B. DynKin, “Semisimple subalgebras of semisimple Lie algebras,” Mat. Sb.,30, No. 2, 349–462 (1952).

    Google Scholar 

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

    Google Scholar 

  35. A. E. Zalesskii, “Linear groups,” Usp. Mat. Nauk,36, No. 5, 56–107 (1981).

    Google Scholar 

  36. A. E. Zalesskii, “Linear groups,” Itogi Nauki i Tekhniki. Algebra. Topologiya. Geometriya, No. 21, 135–182.

  37. V. A. Koibaev, “Examples of nonmonomial linear groups without Transvections,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),71, 153–154 (1977).

    Google Scholar 

  38. V. A. Koibaev, “Subgroups of the special linear group over a field of five elements containing the group of diagonal matrices,” Theses of Reports of the Ninth All-Union Symposium on Group Theory [in Russian], Moscow (1984), pp. 210–211.

  39. V. A. Koibaev, “Subgroups of the special linear group over a field of four elements containing the group of diagonal matrices,” Theses of Reports of the Eighteenth All-Union Conference on Algebra [in Russian], Kishinev (1985), p. 264.

  40. V. A. Koibaev, “Intermediate subgroups in the special linear group of order 6 over a field of four elements,” Theses of Reports of the Tenth All-Union Symposium on Group Theory [in Russian], Minsk (1986), p. 115.

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

    Google Scholar 

  42. T. A. Springer and R. Steinberg, “Conjugacy classes,” in: Seminar on Algebraic Groups, Lecture Notes in Mathematics, No. 131, Springer-Verlag, Berlin-New York (1970).

    Google Scholar 

  43. R. Steinberg, Lectures on Chevalley Groups, Yale Univ. Press, New Haven (1967).

    Google Scholar 

  44. D. A. Suprunenko, Matrix Groups [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  45. J. Tits, “Groupes semi-simples isotropes,” Colloq. Théorie de Groupe Algébriques, Bruxelles 1962, Gauthier-Villars, Paris (1962), pp. 137–147.

    Google Scholar 

  46. J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York (1975).

    Google Scholar 

  47. H. Azad, “Root groups,” J. Algebra,76, No. 1, 211–213 (1982).

    Google Scholar 

  48. H. Bass, “Clifford algebras and spinor norms over a commutative ring,” Amer. J. Math.,96, No. 1, 156–206 (1974).

    Google Scholar 

  49. S. Berman and R. Moody, “Extensions of Chevalley groups,” Israel J. Math.,22, No. 1, 42–51 (1975).

    Google Scholar 

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

    Google Scholar 

  51. R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, London et al. (1985).

    Google Scholar 

  52. M. Golubitsky, “Primitive actions and maximal subgroups of Lie groups,” J. Diff. Geom.,7, No. 1–2, 175–191 (1972).

    Google Scholar 

  53. A. J. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin et al. (1989).

    Google Scholar 

  54. J.-Y. Hée, “Groupes de Chevalley et groupes classiques,” Publ. Math. Univ. Paris VII,17, 1–54 (1984).

    Google Scholar 

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

    Google Scholar 

  56. O. H. King, “On subgroups of the special linear group containing the special orthogonal group,” J. Algebra,96, No. 1, 178–193 (1985).

    Google Scholar 

  57. O. H. King, “Subgroups of the special linear group containing the diagonal subgroup,” J. Algebra,132, No. 1, 198–204 (1990).

    Google Scholar 

  58. Li Shang Zhi, “Maximal subgroups containing root subgroups in finite classical groups,” Kexue Tongbao,29, No. 1, 14–18 (1984).

    Google Scholar 

  59. Li Shang Zhi, “Maximal subgroups of PΩ(n,F,Q) with root subgroups,” Scientia Sinica Ser. A,28, No. 8, 826–838 (1985).

    Google Scholar 

  60. Li Shang Zhi, “Maximal subgroups in classical groups over arbitrary fields,” Proc. Symp. Pure Math.,47, Part 2, 487–493 (1987).

    Google Scholar 

  61. G. M. Seitz, “Small rank permutation representations of finite Chevalley groups,” J. Algebra,28, No. 3, 508–517 (1974).

    Google Scholar 

  62. G. M. Seitz, “Subgroups of finite groups of Lie type,” J. Algebra,61, No. 1, 16–27 (1979).

    Google Scholar 

  63. G. M. Seitz, ldProperties of the known simple groups,” Proc. Symp. Pure Math.,37, 231–237 (1980).

    Google Scholar 

  64. G. M. Seitz, “On the subgroup structure of the classical groups,” Commun. Algebra,10, No. 8, 875–885 (1982).

    Google Scholar 

  65. G. M. Seitz, “Root subgroups for maximal tori in finite groups of Lie type,” Pacif. J. Math.,106, No. 1, 153–244 (1983).

    Google Scholar 

  66. B. S. Stark, “Some subgroups of Ω(V) generated by groups of root type 1,” Illinois J. Math.,17, No. 4, 584–607 (1973).

    Google Scholar 

  67. B. S. Stark, “Some subgroups of Ω(V) generated by groups of root type,” J. Algebra,29, No. 1, 33–41 (1974).

    Google Scholar 

  68. B. S. Stark, “Irreducible subgroups of orthogonal groups generated by groups of root type 1,” Pacif. J. Math.,53, No. 2, 611–625 (1974).

    Google Scholar 

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

    Google Scholar 

  70. K. Suzuki, “On parabolic subgroups of Chevalley groups over local commutative rings,” Sci. Rep. Tokyo Kyoiku Daigaku,A13, No. 366–382, 225–232 (1977).

    Google Scholar 

  71. N. A. Vavilov, “On subgroups of split orthogonal groups in even dimensions,” Bull. Acad. Polon. Sci. Ser. Sci. Math.,29, No. 9–10, 425–429 (1981).

    Google Scholar 

  72. N. A. Vavilov, “A conjugacy theorem for subgroups of GLn containing the group of diagonal matrices,” Colloq. Math.,54, No. 1, 9–14 (1987).

    Google Scholar 

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  1. N. A. Vavilov
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 49–75, 1991.

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Vavilov, N.A. Subgroups of a spinor group containing a maximal split torus. I. J Math Sci 63, 638–652 (1993). https://doi.org/10.1007/BF01097976

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