Abstract
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 v ∈ V 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kostant B., “Lie group representations on polynomial rings,” Amer. J. Math., 85, No. 3, 327–404 (1963).
Kraft H. and Procesi C., “Closures of conjugacy classes of matrices are normal,” Invent. Math., 53, No. 3, 227–247 (1979).
Donkin S., “The normality of closures of conjugacy classes of matrices,” Invent. Math., 101, No. 3, 717–736 (1990).
Kraft H. and Procesi C., “On the geometry of conjugacy classes in classical groups,” Comment. Math. Helv., 57, No. 4, 539–602 (1982).
Sommers E., “Normality of very even nilpotent varieties in D2l,” Bull. London Math. Soc., 37, No. 3, 351–360 (2005).
Broer A., “Normal nilpotent varieties in F 4,” J. Algebra, 207, No. 2, 427–448 (1998).
Kraft H., “Closures of conjugacy classes in G 2,” J. Algebra, 126, No. 2, 454–465 (1989).
Sommers E., “Normality of nilpotent varieties in E 6,” J. Algebra, 270, No. 1, 288–306 (2003).
Fulton W., Introduction to Toric Varieties, Princeton Univ. Press, Princeton (1993).
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).
Sturmfels B., “Equations defining toric varieties,” Proc. Sympos. Pure Math., 62, No. 2, 437–449 (1997).
Sturmfels B., Gröbner Band Convex Polytopes, Amer. Math. Soc., Providence (1996) (Univ. Lect. Ser.; No. 8).
Kuyumzhiyan K., “Simple SL(n)-modules with normal closures of maximal torus orbits,” J. Algebraic Combin., 30, No. 4, 515–538 (2009).
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).
Vinberg È. B. and Onishchik A. L., Lie Groups and Algebraic Groups, Amer. Math. Soc., Providence (1995).
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).
Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, etc. (1978) (Grad. Texts Math.; V. 9).
Fulton W. and Harris J., Representation Theory: a First Course, Springer-Verlag, New York, Berlin, and Heidelberg (1991) (Grad. Texts Math.; V. 129).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2012 Kuyumzhiyan K.G.
The author was supported by the AG Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023, the “EADS Foundation Chair in Mathematics,” the Russian Foundation for Basic Research (Grants 12-01-31342 mol a and 12-01-33101 mol a ved), the Russian-French Poncelet Laboratory (UMI 2615 of CNRS), and the Dmitry Zimin Foundation “Dynasty.”
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 6, pp. 1354–1372, November–December, 2012.
Rights and permissions
About this article
Cite this article
Kuyumzhiyan, K.G. Simple modules of classical linear groups with normal closures of maximal torus orbits. Sib Math J 53, 1089–1104 (2012). https://doi.org/10.1134/S0037446612060122
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446612060122