Siberian Mathematical Journal

, Volume 53, Issue 6, pp 1089–1104 | Cite as

Simple modules of classical linear groups with normal closures of maximal torus orbits

  • K. G. KuyumzhiyanEmail author


Let T be a maximal torus in a classical linear group G. In this paper we find all simple rational G-modules V such that for each vector vV the closure of the T-orbit of v is a normal affine variety. For every G-module without this property we present a T-orbit with nonnormal closure. To solve this problem, we use a combinatorial criterion of normality which is formulated in terms of the set of weights of a simple G-module. The same problem for G = SL(n) was solved by the author earlier.


toric variety normality irreducible representation classical root system weight decomposition 


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  1. 1.
    Kostant B., “Lie group representations on polynomial rings,” Amer. J. Math., 85, No. 3, 327–404 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Kraft H. and Procesi C., “Closures of conjugacy classes of matrices are normal,” Invent. Math., 53, No. 3, 227–247 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Donkin S., “The normality of closures of conjugacy classes of matrices,” Invent. Math., 101, No. 3, 717–736 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kraft H. and Procesi C., “On the geometry of conjugacy classes in classical groups,” Comment. Math. Helv., 57, No. 4, 539–602 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Sommers E., “Normality of very even nilpotent varieties in D2l,” Bull. London Math. Soc., 37, No. 3, 351–360 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Broer A., “Normal nilpotent varieties in F 4,” J. Algebra, 207, No. 2, 427–448 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kraft H., “Closures of conjugacy classes in G 2,” J. Algebra, 126, No. 2, 454–465 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Sommers E., “Normality of nilpotent varieties in E 6,” J. Algebra, 270, No. 1, 288–306 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fulton W., Introduction to Toric Varieties, Princeton Univ. Press, Princeton (1993).zbMATHGoogle Scholar
  10. 10.
    Morand J., “Closures of torus orbits in adjoint representations of semisimple groups,” C. R. Acad. Sci. Paris Sér. I Math., 320, No. 3, 197–202 (1999).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sturmfels B., “Equations defining toric varieties,” Proc. Sympos. Pure Math., 62, No. 2, 437–449 (1997).MathSciNetGoogle Scholar
  12. 12.
    Sturmfels B., Gröbner Band Convex Polytopes, Amer. Math. Soc., Providence (1996) (Univ. Lect. Ser.; No. 8).Google Scholar
  13. 13.
    Kuyumzhiyan K., “Simple SL(n)-modules with normal closures of maximal torus orbits,” J. Algebraic Combin., 30, No. 4, 515–538 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bogdanov I. and Kuyumzhiyan K., “Simple modules of exceptional linear groups with normal closures of maximal torus orbits,” Math. Notes, 92, No. 4, 445–457 (2012).CrossRefGoogle Scholar
  15. 15.
    Vinberg È. B. and Onishchik A. L., Lie Groups and Algebraic Groups, Amer. Math. Soc., Providence (1995).Google Scholar
  16. 16.
    Kempf G., Knudsen F., Mumford D., and Saint-Donat B., Toroidal Embeddings. Vol. 1, Springer-Verlag, Berlin, Heidelberg, and New York (1973) (Lecture Notes Math.; V. 339).zbMATHGoogle Scholar
  17. 17.
    Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, etc. (1978) (Grad. Texts Math.; V. 9).zbMATHGoogle Scholar
  18. 18.
    Fulton W. and Harris J., Representation Theory: a First Course, Springer-Verlag, New York, Berlin, and Heidelberg (1991) (Grad. Texts Math.; V. 129).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsIndependent University of Moscow and Poncelet Laboratory (UMI 2615 of CNRS)MoscowRussia

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