Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Remarks on classical invariant theory

Author: Roger Howe
Journal: Trans. Amer. Math. Soc. 313 (1989), 539-570
MSC: Primary 22E45; Secondary 11E57, 15A72, 20G05, 22E47
Erratum: Trans. Amer. Math. Soc. 318 (1990), null.
MathSciNet review: 986027
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Abstract: A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.

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  • [ABP] M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR 0650828
  • [Ch] Shiing-shen Chern, On a generalization of Kähler geometry, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 103–121. MR 0087172
  • [Ge] Stephen Gelbart, Holomorphic discrete series for the real symplectic group, Invent. Math. 19 (1973), 49–58. MR 0320231
  • [G-K] Kenneth I. Gross and Ray A. Kunze, Bessel functions and representation theory. II. Holomorphic discrete series and metaplectic representations, J. Functional Analysis 25 (1977), no. 1, 1–49. MR 0453928
  • [He] Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 0263988
  • [Hol] Roger Howe, Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons, Applications of group theory in physics and mathematical physics (Chicago, 1982) Lectures in Appl. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1985, pp. 179–207. MR 789290
  • [J] Nathan Jacobson, Basic algebra. II, W. H. Freeman and Co., San Francisco, Calif., 1980. MR 571884
  • [K] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367 (Russian). MR 0259961
  • [K-V] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 0463359
  • [Ks1] Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Springer, Berlin, 1977, pp. 177–306. Lecture Notes in Math., Vol. 570. MR 0580292
  • [Ks2] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 0142696
  • [Ks3] Bertram Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. Math. Mech. 7 (1958), 237–264. MR 0092755
  • [L] Dudley E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940. MR 0002127
  • [Sa] Masahiko Saito, Représentations unitaires des groupes symplectiques, J. Math. Soc. Japan 24 (1972), 232–251 (French). MR 0299728
  • [Se] Jean-Pierre Serre, Algèbres de Lie semi-simples complexes, W. A. Benjamin, inc., New York-Amsterdam, 1966 (French). MR 0215886
  • [Sh] C. Seshadri, Letter to M. F. Atiyah.
  • [Sh] David Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167. MR 0137504, 10.1090/S0002-9947-1962-0137504-6
  • [Th] R. M. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942), 371–388. MR 0006149
  • [H] Hans Tilgner, Graded generalizations of Weyl- and Clifford algebras, J. Pure Appl. Algebra 10 (1977/78), no. 2, 163–168. MR 0457515
  • [Wi] André Weil, Introduction à l’étude des variétés kählériennes, Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056
  • [Wi2] -, Sur certains groupes d'opérateurs unitaires, Acta Math. 11 (1964), 143-211.
  • [W] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
  • [Z.] D. P. Želobenko, Compact Lie groups and their representations, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR 0473098

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Keywords: Invariant theory, Weyl algebra, spherical harmonics
Article copyright: © Copyright 1989 American Mathematical Society