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Two Orbits: When is one in the closure of the other?

  • V. L. Popov
Article

Abstract

Let G be a connected linear algebraic group, let V be a finite dimensional algebraic G-module, and let \( \mathcal{O}_1 \) and \( \mathcal{O}_2 \) be two G-orbits in V. We describe a constructive way to find out whether or not \( \mathcal{O}_1 \) lies in the closure of \( \mathcal{O}_2 \).

Keywords

STEKLOV Institute Algebraic Group Nilpotent Orbit Linear Algebraic Group Hasse Diagram 
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References

  1. 1.
    N. Bourbaki, Éléments de mathématique, Fasc. XXXIV: Groupes et algèbres de Lie, Chs. IV–VI (Hermann, Paris, 1968; Mir, Moscow, 1972).Google Scholar
  2. 2.
    È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie Groups and Lie Algebras (VINITI, Moscow, 1990), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 41; Engl. transl. in Lie Groups and Lie Algebras III (Springer, Berlin, 1994), Encycl. Math. Sci. 41.Google Scholar
  3. 3.
    È. B. Vinberg and V. L. Popov, “On a Class of Quasihomogeneous Affine Varieties,” Izv. Akad. Nauk SSSR, Ser. Mat. 36(4), 749–764 (1972) [Math. USSR, Izv. 6, 743–758 (1972)].zbMATHMathSciNetGoogle Scholar
  4. 4.
    È. B. Vinberg and V. L. Popov, “Invariant Theory,” in Algebraic Geometry-4 (VINITI, Moscow, 1989), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 55, pp. 137–309; Engl. transl.: V. L. Popov and É. B. Vinberg, “Invariant Theory,” in Algebraic Geometry IV (Springer, Berlin, 1994), Encycl. Math. Sci. 55, pp. 123–278.Google Scholar
  5. 5.
    B. Ya. Kazarnovskiĭ, “Newton Polyhedra and the Bezout Formula for Matrix-Valued Functions of Finite-Dimensional Representations,” Funkts. Anal. Prilozh. 21(4), 73–74 (1987) [Funct. Anal. Appl. 21, 319–321 (1987)].Google Scholar
  6. 6.
    V. V. Kashin, “Orbits of Adjoint and Coadjoint Actions of Borel Subgroups of Semisimple Algebraic Groups,” in Problems in Group Theory and Homological Algebra (Izd. Yaroslav. Gos. Univ., Yaroslavl’, 1990), pp. 141–158 [in Russian].Google Scholar
  7. 7.
    D. Cox, J. Little, and D. O’shea, Ideals, Varieties, and Algorithms (Springer, New York, 1992; Mir, Moscow, 2000).zbMATHGoogle Scholar
  8. 8.
    D. Mumford, Algebraic Geometry, Vol. 1: Complex Projective Varieties (Springer, Berlin, 1976; Mir, Moscow, 1979).Google Scholar
  9. 9.
    V. L. Popov, “Structure of the Closure of Orbits in Spaces of Finite-Dimensional Linear SL(2) Representations,” Mat. Zametki 16(6), 943–950 (1974) [Math. Notes 16, 1159–1162 (1974)].MathSciNetGoogle Scholar
  10. 10.
    V. L. Popov, “The Cone of Hilbert Nullforms,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 241, 192–209 (2003) [Proc. Steklov Inst. Math. 241, 177–194 (2003)].Google Scholar
  11. 11.
    K. Bongartz, “Degenerations for Representations of Tame Quivers,” Ann. Sci. Éc. Norm. Supér. 28(5), 647–668 (1995).zbMATHMathSciNetGoogle Scholar
  12. 12.
    T. Brüstle, L. Hille, G. Röhrle, and G. Zwara, “The Bruhat-Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules,” Adv. Math. 148(2), 203–242 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D. Burde, “Degenerations of 7-Dimensional Nilpotent Lie Algebras,” Commun. Algebra 33(4), 1259–1277 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Burde and C. Steinhoff, “Classification of Orbit Closures of 4-Dimensional Complex Lie Algebras,” J. Algebra 214, 729–739 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (J. Wiley & Sons, New York, 1985).zbMATHGoogle Scholar
  16. 16.
    D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras (Van Nostrand Reinhold, New York, 1993).zbMATHGoogle Scholar
  17. 17.
    H. Derksen and G. Kemper, Computational Invariant Theory (Springer, Berlin, 2002), Encycl. Math. Sci. 130, Invariant Theory and Algebraic Transformation Groups 1.zbMATHGoogle Scholar
  18. 18.
    S. M. Goodwin, L. Hille, and G. Röhrle, “Orbits of Parabolic Subgroups on Metabelian Ideals,” arXiv: 0711.3711.Google Scholar
  19. 19.
    A. Grothendieck, “Torsion homologique et sections rationnelles,” in Anneaux de Chow et applications (Paris, 1958), Sém. C. Chevalley ENS, pp. 5-01–5-29.Google Scholar
  20. 20.
    Z. Jelonek, “On the Effective Nullstellensatz,” Invent. Math. 162, 1–17 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Magyar, J. Weyman, and A. Zelevinsky, “Multiple Flag Varieties of Finite Type,” Adv. Math. 141(1), 97–118 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    H. Matsumura and P. Monsky, “On the Automorphisms of Hypersurfaces,” J. Math. Kyoto Univ. 3, 347–361 (1964).zbMATHMathSciNetGoogle Scholar
  23. 23.
    L. Moser-Jauslin, “The Chow Rings of Smooth Complete SL(2)-Embeddings,” Compos. Math. 82, 67–106 (1992).zbMATHMathSciNetGoogle Scholar
  24. 24.
    K. D. Mulmuley and M. Sohoni, “Geometric Complexity Theory. I: An Approach to the P vs. NP and Related Problems,” SIAM J. Comput. 31(2), 496–526 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    D. Mumford, Geometric Invariant Theory (Springer, Berlin, 1965), Ergebn. Math. 34.zbMATHGoogle Scholar
  26. 26.
    D. D. Pervouchine, “Hierarchy of Closures of Matrix Pencils,” J. Lie Theory 14, 443–479 (2004).zbMATHMathSciNetGoogle Scholar
  27. 27.
    V. L. Popov, “Constructive Invariant Theory,” Astérisque 87–88, 303–334 (1981).Google Scholar
  28. 28.
    M. Rosenlicht, “Some Basic Theorems on Algebraic Groups,” Am. J. Math. 78, 401–443 (1956).zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    M. Rosenlicht, “On Quotient Varieties and the Affine Embedding of Certain Homogeneous Spaces,” Trans. Am. Math. Soc. 101, 211–223 (1961).zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    C. Seeley, “Degenerations of 6-Dimensional Nilpotent Lie Algebras over C,” Commun. Algebra 18, 3493–3505 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    N. Spaltenstein, Classes unipotentes et sous-groupes de Borel (Springer, Berlin, 1982), Lect. Notes Math. 946.zbMATHGoogle Scholar
  32. 32.
    T. A. Springer, Linear Algebraic Groups, 2nd ed. (Birkhäser, Boston, 1998), Prog. Math. 9.zbMATHGoogle Scholar
  33. 33.
    P. Tauvel and R. W. T. Yu, Lie Algebras and Algebraic Groups (Springer, Berlin, 2005), Springer Monogr. Math.zbMATHGoogle Scholar
  34. 34.
    J. Weyman, “The Equations of Conjugacy Classes of Nilpotent Matrices,” Invent. Math. 98, 229–245 (1989).zbMATHCrossRefMathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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