Abstract
Let G = GL n be the general linear group over an algebraically closed field k and let \( \mathfrak{g}=\mathfrak{g}{\mathfrak{l}}_n \) be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[\( \mathfrak{g} \)] be the algebra of polynomial functions on g and let k[\( \mathfrak{g} \)]G be the algebra of invariants under the conjugation action of G. For all weights χ ∈ ℤ n with χ2 ≤ 0 or χ n − 1 ≥ 0 we give explicit bases for the k[\( \mathfrak{g} \)]G-module k \( {\left[\mathfrak{g}\right]}_{\chi}^U \) of highest weight vectors of weight χ. We also give bases for the vector spaces k[C] U χ where C is a nilpotent orbit closure. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Boos, M. Reineke, B-orbits of 2-nilpotent matrices and generalizations, in: Highlights in Lie Algebraic Methods, Progress in Mathematics, Vol. 295, Birkhäuser/Springer, New York, 2012, pp. 147–166.
A. Broer, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants, Indag. Math. 6 (1995), no. 4, 385–396.
R. Brylinski (formerly Gupta), Generalized exponents via Hall-Littlewood symmetric functions, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 2, 287–291.
R. Brylinski (formerly Gupta), Characters and the q-analog of weight multiplicity, J. London Math. Soc. (2) 36 (1987), no. 1, 68–76.
R. Brylinski (formerly Gupta), Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), no. 3, 517–533.
R. Brylinski (formerly Gupta), Stable calculus of the mixed tensor character. I, in: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, Lecture Notes in Mathematics, Vol. 1404, Springer, Berlin, 1989, pp. 35–94.
C. De Concini, D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129–165.
M. Domokos, Hermitian matrices with a bounded number of eigenvalues, Linear Algebra Appl. 439 (2013), 3964–3979.
S. Donkin, The normality of closures of conjugacy classes of matrices, Invent. Math. 101 (1990), no. 3, 717–736.
S. Donkin, Invariants of several matrices, Invent. Math. 110 (1992), no. 2, 389–401.
P. Doubilet, G.-C. Rota, J. Stein, On the foundations of combinatorial theory. IX. Combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216.
W. H. Hesselink, Characters of the nullcone, Math. Ann. 252 (1980), no. 3, 179–182.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Хартсхорн Алгебраическая геометрия, M.,мир, 1981.
A. Isaev, O. Ogievetsky, P. Pyatov, Generalized Cayley-Hamilton-Newton identities, Czechoslovak J. Phys. 48 (1998), no. 11, 1369–1374.
J. C. Jantzen, Nilpotent orbits in representation theory, in: Lie Theory, Progress in Mathematics, Vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211.
S. Kumar, N. Lauritzen, J. Thomsen, Frobenius splitting of cotangent bundles of flag varieties, Invent. Math. 136 (1999), no. 3, 603–621.
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
H. Kraft, C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), no. 3, 227–247.
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, New York, 1995.
C. Moeglin, Factorialité dans les algèbres enveloppantes, C. R. Acad. Sci. Paris Sér. A–B 282 (1976), no. 22, A1269–A1272.
В. Л. Попов, Группы Пикара однородных пространств линейных алгебраических групп и одномерные однородные векторные расслоения, Изв Акад. Наук СССР, Сер. Матем. 38 (1974), no. 2, 294–322. Engl. transl.: V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR-Izv. 8 (1974), no. 2, 301–327.
R. P. Stanley, The stable behavior of some characters of SL(n;ℂ), Linear and Multilinear Algebra 16 (1984), 3–27.
J. R. Stembridge, Rational tableaux and the tensor algebra of gl n , J. Combin. Theory Ser. A 46 (1987), no. 1, 79–120.
R. Tange, Highest weight vectors for the adjoint action of GL n on polynomials, Pacif. J. Math. 258 (2012), no. 2, 497–510.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
TANGE, R. HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II. Transformation Groups 20, 817–830 (2015). https://doi.org/10.1007/s00031-015-9310-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-015-9310-4