Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Remarks on classical invariant theory

Author(s): Roger Howe
Journal: Trans. Amer. Math. Soc. 313 (1989), 539-570.
MSC: Primary 22E45; Secondary 11E57, 15A72, 20G05, 22E47
Errata: Trans. Amer. Math. Soc. 318 (1990), null.
MathSciNet review: 986027
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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.


References:

[ABP]
M. Atiyah, R. Bott and V. Patodi, On the heat equation and the Index Theorem, Invent. Math. 19 (1973), 279-330. MR 0650828 (58:31287)

[Ch]
S. S. Chern, On a generalization of Kähler geometry, Algebraic Geometry and Topology, A Symposium in Honor of S. Lefschetz, Princeton Univ. Press, Princeton, N.J., 1975. MR 0087172 (19:314d)

[Ge]
S. Gelbart, Holomorphic discrete series for the real symplectic group, Invent. Math. 19 (1973), 49-58. MR 0320231 (47:8770)

[G-K]
K. Gross and R. Kunze, Bessel functions and representation theory. I, II, J. Funct. Anal. 25 (1977), 1-49. MR 0453928 (56:12181)

[He]
S. Helgason, A duality for symmetric spaces with applications to group representations, Adv. in Math. 5 (1970), 1-54. MR 0263988 (41:8587)

[Hol]
R. Howe, Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, Lectures in Appl. Math., vol. 21, Amer. Math. Soc, Providence, R.I., 1985, pp. 179-206. MR 789290 (86i:22036)

[J]
N. Jacobson, Basic algebra. II, Freeman, New York, 1980. MR 571884 (81g:00001)

[K]
V. Kac, Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR 32 (1968), 1323-1367; English transl., Math. USSR-Izv. 2 (1968), 1271-1311. MR 0259961 (41:4590)

[K-V]
N. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1-47. MR 0463359 (57:3311)

[Ks1]
B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometrical Methods in Mathematical Physics (Proc. Sympos. Univ. Bonn, Bonn, 1975), Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin, 1977, pp. 177-306. MR 0580292 (58:28326)

[Ks2]
-, Lie algebra cohomology and the generalized Borel-Weil Theorem, Ann. of Math. (2) 74 (1961), 324-387. MR 0142696 (26:265)

[Ks3]
-, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. Math. Mech. 7 (1958), 237-264. MR 0092755 (19:1153e)

[L]
D. Littlewood, The theory of group characters and matrix representations of groups, Oxford Univ. Press, Oxford, 1940. MR 0002127 (2:3a)

[Sa]
M. Saito, Représentations unitaires des groupes symplectiques, J. Math. Soc. Japan 24 (1972), 232-251. MR 0299728 (45:8776)

[Se]
J.-P. Serre, Algèbres de Lie semi-simples complexes, Benjamin, Reading, Mass., 1966. MR 0215886 (35:6721)

[Sh]
C. Seshadri, Letter to M. F. Atiyah.

[Sh]
D. Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149-167. MR 0137504 (25:956)

[Th]
R. Thrall, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942), 371-388. MR 0006149 (3:262d)

[H]
T. Tilgner, (a) Graded generalizations of Weyl and Clifford algebras, J. Pure Appl. Algebra 10 (1977), 163-168. (b) A graded generalization of Lie triples, J. Algebra 47 (1977), 190-196. MR 0457515 (56:15720)

[Wi]
A. Weil, Introduction a l'étude des variétés Kähleriennes, Hermann, Paris, 1958. MR 0111056 (22:1921)

[Wi2]
-, Sur certains groupes d'opérateurs unitaires, Acta Math. 11 (1964), 143-211.

[W]
H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N.J., 1946. MR 1488158 (98k:01049)

[Z.]
D. Zhelobenko, Compact Lie groups and their representations, Transl. Math. Mono., no. 40, Amer. Math. Soc., Providence, R.I., 1973. MR 0473098 (57:12776b)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E45, 11E57, 15A72, 20G05, 22E47

Retrieve articles in all Journals with MSC: 22E45, 11E57, 15A72, 20G05, 22E47


Additional Information:

DOI: 10.1090/S0002-9947-1989-0986027-X
PII: S0002-9947-1989-0986027-X
Keywords: Invariant theory, Weyl algebra, spherical harmonics
Copyright of article: Copyright 1989, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia