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Subgroups of symplectic groups that contain a subsystem subgroup

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Abstract

Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 ∈ R*; and let ν be a symmetric equivalence relation on the index set {1,…, l, −l,…, 1} all of whose classes contain at least 3 elements. In the present paper, we prove that if a subgroup H of Γ contains the group EΓ(ν) of elementary block diagonal matrices of type ν, then H normalizes the subgroup generated by all elementary symplectic transvections Tij(ξ) ∈ H. Combined with the previous results, this completely describes the overgroups of subsystem subgroups in this case. Similar results for subgroups of GL(n, R) were established by Z. I. Borewicz and the author in the early 1980s, while for GSp(2l, R) and GO(n, R) they have been announced by the author in the late 1980s, but a complete proof for the symplectic case has not been published before. Bibliography: 74 titles.

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References

  1. Z. I. Borewicz, “Description of the subgroups of the general linear group that contain the group of diagonal matrices,” Zap. Nauchn. Semin. LOMI, 64, 12–29 (1976).

    MATH  Google Scholar 

  2. Z. I. Borewicz and N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Tr. Mat. Inst. AN SSSR, 148, 43–57 (1978).

    Google Scholar 

  3. Z. I. Borewicz and N. A. Vavilov, “Arrangement of the subgroups that contain the group of block diagonal matrices in the general linear group over a ring,” Izv. VUZ’ov, 11, 12–16 (1982).

    Google Scholar 

  4. Z. I. Borewicz and N. A. Vavilov, “Subgroups of the general linear group over a commutative ring,” Dokl. Akad Nauk, 267, No. 4, 777–778 (1982).

    MathSciNet  Google Scholar 

  5. Z. I. Borewicz and N. A. Vavilov, “Arrangement of subgroups in the general linear group over a commutative ring,” Tr. Mat. Inst. AN SSSR, 165, 24–42 (1984).

    Google Scholar 

  6. Z. I. Borewicz, N. A. Vavilov, and V. Narkevich, “Subgroups of the general linear group over a Dedekind ring,” Zap. Nauchn. Semin. LOMI, 94, 13–20 (1979).

    Google Scholar 

  7. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a semilocal ring,” Zap. Nauchn. Semin. LOMI, 75, 43–58 (1978).

    MATH  MathSciNet  Google Scholar 

  8. N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring that contain the group of block diagonal matrices,” Vestn. LGU, 1, 10–15 (1981).

    MathSciNet  Google Scholar 

  9. N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring that contain the group of block diagonal matrices,” Vestn. LGU, 1, 16–21 (1983).

    MathSciNet  Google Scholar 

  10. N. A. Vavilov, “Subgroups of split classical groups,” Doctoral Thesis, Leningrad State Univ. (1987).

  11. N. A. Vavilov, “The structure of split classical groups over a commutative ring,” Dokl. Akad. Nauk, 299, No. 6, 1300–1303 (1988).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  14. N. A. Vavilov, “Subgroups of the general symplectic group over a commutative ring,” in: Rings and Modules. Limit Theorems in Probability Theory, Izd. SPbGu (1993), pp. 16–38.

  15. N. A. Vavilov, “Subgroups of split orthogonal groups. II,” Zap. Nauchn. Semin. POMI, 265, 42–63 (1999).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  17. N. A. Vavilov, “Subgroups of the spinor group that contain a split maximal torus. II,” Zap. Nauchn. Semin. POMI, 289, 37–56 (2002).

    Google Scholar 

  18. N. A. Vavilov, “Subgroups of the group SLn over a semilocal ring,” Zap. Nauchn. Semin. POMI, 343, 33–53 (2007).

    MathSciNet  Google Scholar 

  19. N. A. Vavilov and M. R. Gavrilovich, “A 2-proof of structure theorems for Chevalley groups of type E 6 and E 7,” Algebra Analiz, 16, No. 4, 54–87 (2004).

    MathSciNet  Google Scholar 

  20. N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book,” Zap. Nauchn. Semin. POMI, 330, 36–76 (2006).

    MATH  Google Scholar 

  21. N. A. Vavilov and S. I. Nikolenko, “A 2-proof of structure theorems for Chevalley groups of type F 4,” Algebra Analiz, 20, No. 3 (2008).

  22. N. A. Vavilov and E. V. Dybkova, “Subgroups of the general symplectic groups that contain the group of diagonal matrices. I, II,” Zap. Nauchn. Semin. POMI, 103, 31–47 (1980); 132, 44–56 (1983).

    MATH  MathSciNet  Google Scholar 

  23. N. A. Vavilov and V. A. Petrov, “Overgroups of EO(2l, R),” Zap. Nauchn. Semin. POMI, 272, 68–85 (2002).

    Google Scholar 

  24. N. A. Vavilov and V. A. Petrov, “Overgroups of Ep(2l, R),” Algebra Analiz, 15, No. 3, 72–114 (2003).

    MathSciNet  Google Scholar 

  25. N. A. Vavilov and V. A. Petrov, “Overgroups of EO(n, R),” Algebra Analiz, 19, No. 2, 10–51 (2007).

    MathSciNet  Google Scholar 

  26. N. A. Vavilov and A. V. Stepanov, “Subgroups of the general linear group over a ring that satisfy stability conditions,” Izv. VUZ’ov, 10, 19–25 (1989).

    MathSciNet  Google Scholar 

  27. N. A. Vavilov and E. A. Filippova, “Overgroups of A l−1 in the hyperbolic embedding” (to appear).

  28. I. Z. Golubchik, “Subgroups of the general linear groups over an associative ring,” Usp. Mat. Nauk, 39, No. 1, 125–126 (1984).

    MathSciNet  Google Scholar 

  29. E. V. Dybkova, “Net subgroups of hyperbolic unitary groups,” Algebra Analiz, 9, No. 4, 87–96 (1997).

    Google Scholar 

  30. E. V. Dybkova, “Overdiagonal subgroups of a hyperbolic unitary group for good form ring over a field,” Zap. Nauchn. Semin. POMI, 236, 87–96 (1997).

    Google Scholar 

  31. E. V. Dybkova, “Form nets and the lattice of overdiagonal subgroups in a symplectic group over a field of characteristic 2,” Algebra Analiz, 10, No. 4, 113–127 (1998).

    MathSciNet  Google Scholar 

  32. E. V. Dybkova, “Overdiagonal subgroups of a hyperbolic unitary group over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 289, 154–206 (2002).

    Google Scholar 

  33. E. V. Dybkova, “Overdiagonal subgroups of a hyperbolic unitary group for a good form ring over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 305, 121–135 (2003).

    Google Scholar 

  34. E. V. Dybkova, “The Borewicz theorem for a hyperbolic unitary group over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 321, 136–167 (2005).

    MATH  Google Scholar 

  35. E. V. Dybkova, “Subgroups of hyperbolic unitary groups,” Doctoral Thesis, St. Petersburg Gos. Univ. (2006).

  36. E. V. Dybkova, “Overgroups of the diagonal group in a hyperbolic group over a simple Artin ring. I,” Zap. Nauchn. Semin. POMI, 338, 155–172 (2006).

    MATH  Google Scholar 

  37. V. A. Koibaev, “Subgroups of the general linear group that contain the group of elementary block diagonal matrices,” Vestn. LGU, 13, 33–40 (1982).

    MathSciNet  Google Scholar 

  38. V. A. Koibaev, “Subgroups of the general linear group over a finite field that contain the group of elementary block diagonal matrices,” in: Structural Properties of Groups, Ordzhonikidze (1982), pp. 6–12.

  39. V. I. Kopeiko, “Stabilization of symplectic groups over the ring of polynomials,” Mat. Sb., 106, No. 1, 94–107 (1978).

    MathSciNet  Google Scholar 

  40. A. Khatib, “Subgroups of a split orthogonal group over a commutative ring,” in: Rings and Modules. Limit Theorems in Probability Theory, Izd. SPbGU (1993), pp. 74–86.

  41. O. T. O’Meara, Lectures in Symplectic Groups [Russian translation], Mir, Moscow (1979).

    Google Scholar 

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

    Google Scholar 

  43. V. A. Petrov, “Overgroups of classical groups,” Ph. D. Thesis, SPbGU (2005).

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

    Google Scholar 

  45. A. V. Stepanov, “Stability conditions in the theory of linear groups over rings,” Ph. D. Thesis, Leningrad Gos. Univ. (1987).

  46. E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 1, 185–198 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  47. A. Bak, The Stable Structure of Quadratic Modules, Thesis, Columbia Univ. (1969).

  48. A. Bak, K-Theory of Forms, Princeton Univ. Press, Princeton (1981).

    Google Scholar 

  49. A. Bak, Lectures on Dimension Theory, Group Valued Functors, and Nonstable K-Theory, Buenos Aires (1995).

  50. A. Bak and M. Morimoto, “Equivariant surgery with middle-dimensional singular sets. I,” Forum Math., 8, No. 3, 267–302 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  51. A. Bak and A. Stepanov, “Dimension theory and nonstable K-theory for net groups,” Rend. Sem. Mat. Univ. Padova, 106, 207–253 (2001).

    MATH  MathSciNet  Google Scholar 

  52. 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 

  53. H. Bass, “Unitary algebraic K-Theory,” Lect. Notes Math., 343, 57–265 (1973).

    Article  MathSciNet  Google Scholar 

  54. D. Costa and G. Keller, “Radix redux: normal subgroups of symplectic group,” J. reine angew. Math., 427, No. 1, 51–105 (1992).

    MATH  MathSciNet  Google Scholar 

  55. F. Grünewald, J. Mennicke, and L. N. Vaserstein, “On symplectic groups over polynomial rings,” Math. Z., 206, No. 1, 35–56 (1991).

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  57. R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  58. R. Hazrat and N. A. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  59. R. Hazrat and N. A. Vavilov, “Bak’s work on K-theory of rings” (to appear).

  60. Li Fuan, “The structure of symplectic group over arbitrary commutative rings,” Acta Math. Sinica, New Series, 3, No. 3, 247–255 (1987).

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  62. 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 

  63. A. Stepanov, “Nonstandard subgroups between E n (R) and GLn(A),” Algebra Colloquium, 10, No. 3, 321–334 (2004).

    Google Scholar 

  64. A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, 109–153 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  65. A. Stepanov, N. Vavilov, and You Hong, “Overgroups of semisimple subgroups via localization-completion” (to appear).

  66. G. Taddei, “Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau,” C. R. Acad. Sci Paris, Sér I, 295, No. 2, 47–50 (1982).

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  69. 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.

  70. 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 

  71. L. N. Vaserstein, “Normal subgroups of symplectic groups over rings,” K-Theory, 2, No. 5, 647–673 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  72. 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 

  73. 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.

  74. You Hong, “Overgroups of symplectic group in linear group over commutative rings,” J. Algebra, 282, No. 1, 23–32 (2004).

    Article  MATH  MathSciNet  Google Scholar 

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

    N. A. Vavilov

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 5–29.

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Vavilov, N.A. Subgroups of symplectic groups that contain a subsystem subgroup. J Math Sci 151, 2937–2948 (2008). https://doi.org/10.1007/s10958-008-9020-8

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