Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T06:19:50.393Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial Invariant Theory and Taylor Series

Published online by Cambridge University Press:  20 November 2018

John E. Gilbert
John E. Gilbert*
Affiliation:
The University of Texas at Austin, Austin, Texas, U.S.A. 78712
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1243 - 1262
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Capelli, A., Über die Zurückführung der Cayley'schen Operation Q. auf gewohnliche Polar-Operationen, Math. Ann. 29(1887), 331–338.Google Scholar
2. Capelli, A., Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37(1890), 1–37.Google Scholar
3. Capelli, A., Lezioni sulla Teoria délie forme algebriche. (Pellerano, éd.) Napoli. Libr. Se. 1902.Google Scholar
4. Capelli, A., Quelques formules relatives aux opérations de polaire, Proc. Int. Congress. Math. (1894), 35–39.Google Scholar
5. Carter, R.W. and Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136(1974), 193–242.Google Scholar
6. Davis, K.M., Gilbert, J.E. and Kunze, R.A., Elliptic differential operators in harmonic analysis, I. Generalized Cauchy-Riemann systems. Amer. J. Math. 113(1991), 75–116.Google Scholar
7. Davis, K.M., Invariant differential operators in harmonic analysis on real hyperbolic space. Proc. Int. Conf. Harmonic Analysis and Operator Algebras, Aust. Nat. Univ., Canberra, 1987, 58–78.Google Scholar
8. Davis, K.M., Elliptic differential operators and harmonic analysis on hyperbolic space. In preparation.Google Scholar
9. DeConcini, C., Eisenbud, D. and Procesi, C., Young diagrams anddeterminantal varieties, InventionesMath. 56(1980), 129–165.Google Scholar
10. DeConcini, C. and Procesi, C., A characteristic-free approach to invariant theory, Adv. in Math. 21(1976), 330–354.Google Scholar
11. Dixmier, J., Algèbres enveloppantes. Gauthier-Villars, Paris, 1973.Google Scholar
12. Gilkey, P.B., Invariance Theory, The Heat Equation, and the Atiyah-Singer Index Theorem. Math. Lect. Series, Vol.11, Publish or Perish Inc., Wilmington, Delaware, 1984.Google Scholar
13. Helgason, S., Groups and geometric analysis. Academic Press, New York, 1984.Google Scholar
14. Hodge, W.V.D., Some enumerative results in the theory of forms, Proc. Camb. Phil. Soc. 39(1943), 22–30.Google Scholar
15. Howe, R., Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313(1989), 539–570.Google Scholar
16. Kashiwara, M. and Vergne, M., On the Segal-Shale-Weil representations and harmonic polynomials, Inventiones Mathematicae 44(1978), 1–47.Google Scholar
17. Littlewood, D.E., The theory of group characters and matrix representations of groups, 2nd edition, Oxford Univ. Press, Oxford 1950.Google Scholar
18. MacDonald, I.G., Symmetric functions and Hall polynomials. Clarendon Press, Oxford, 1979.Google Scholar
19. Procesi, C., A primer of invariant theory. Brandeis Lecture Notes, 1(1982).Google Scholar
20. Thrall, R.M., On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64(1942), 371–388.Google Scholar
21. Turnbull, H.W., The Theory of Determinants, Matrices and Invariants. Third Edition, Dover Publications, 1960.Google Scholar
22. Verma, D.-N., On a classical stability result on invariants of isotypical modules, J. of Algebra 63(1980), 15–40.Google Scholar
23. Wallach, N.R., Harmonic analysis on homogeneous spaces. Marcel Dekker Inc., New York, 1973.Google Scholar
24. Weyl, H., Classical Groups. Second Edition, Princeton University Press, Princeton, N.J., 1953.Google Scholar
25. Zelobenko, D.P., Compact Lie Groups and their Representations, .Transi. Math. Monographs, Amer. Math. Soc, Providence, Rhode Island, 1973.Google Scholar