Abstract
In [7] the author has given an exposition of the theory of invariants of binary forms in terms of a particular version of Classical Invariant Theory. Reflection shows that many aspects of the development apply also ton-ary forms. The purpose of this paper is to make explicit this more general application. The plethysms S’(Sp(ℂn)) are computed quite explicitly forl = 2, 3 and 4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen Y, Garsia A and Remmel J, Algorithms for plethysm in combinatorics and algebra,Contemp. Math. AMS Providence RI 34 (1984) 109–153
Curtis C and Reiner I,Representation theory of finite groups and associative algebras (New York: Wiley-Interscience) (1962)
Duncan D, On D E Littlewood’s algebra of S-functions,Can. J. Math. 4 (1952) 504–512
Foulkes H, Concomitants of the quintic and sextic up to degree four in coefficients of the ground form.J. London Math. Soc. 25 (1950) 205–209
Gross K and Kunze R, Bessel functions and representation theory II. Holomorphic discrete series and metaplectic representations,J. Funct. Anal. 25 (1977) 1–49
Guillemin V. and Sternberg S,The Frobenius reciprocity theorems from a symplectic point of view in Nonlinear partial differential operators and quantization procedures, Lecture notes in math. No. 1037 (New York: Springer-Verlag) 1983, pp. 212–256
Howe R, The classical groups and invariants of binary forms in:Proceedings of the Hermann Weyl Centennial Symposium (Durham: AMS Publication) (1987) (to appear)
Howe R, Remarks on classical invariant theory (preprint)
Howe R, Transcending classical invariant theory (preprint)
Howe R, Reciprocity laws in the theory of dual pairs, in:Representation theory of reductive groups, Progress in mathematics (ed.) P Trombi (Boston: Birkhauser) vol 40 (1983) pp. 159–175
Howe R, Asymptotic dimensions of spaces of invariants (in preparation)
Jacobson N,Lie algebras (New York: Interscience Publishers) 1962
James G D, The representation theory of the symmetric groups,Lecture notes in mathematics, No. 682 (New York: Springer-Verlag) (1978)
Kashiwara M and Vergne M, On the Segal-Shale-Weil representations and harmonic polynomials,Invent. Math. 44 (1978) 1–47
Kempe A, On regular difference terms,Proc. London Math. Soc. 25 (1894) 465–470
Klink W and Ton-That T, Calculation of Clebsch-Gordan and Racah coefficients using symbolic manipulation programs (preprint)
Kung J and Rota G-C, The invariant theory of binary forms,BAMS (new series) 10 (1984) 27–85
Lang S,Algebra (Reading: Addison-Wesley) (1984)
Littelman P, A decomposition formula for Levi sub-groups (preprint)
Littlewood D,The theory of group characters and matrix representations of groups (Oxford: University Press) (1940)
MacDonald I,Symmetric functions and Hall polynomials (Oxford: University Press) (1979)
Stanley R, GLn(ℂ) for combinatorialists, in:Surveys in combinatorics (ed.) E K Lloyd (Cambridge: University Press) London Math. Soc. Lecture Notes No. 82 (1983)
Thrall R, On symmetrized Kronecker powers and the structure of the free Lie ring,Am. J. Math. 64 (1942) 371–388
Weyl H,The classical groups (Princeton: University Press) (1946)
Zhelobenko D, Compact Lie groups and their representations, Trans. Math. Mono. No. 40, AMS Providence (1973)
Author information
Authors and Affiliations
Additional information
Partially supported by NSF Grant No. DMS 8506130.
Rights and permissions
About this article
Cite this article
Howe, R. (GLn, GLm)-duality and symmetric plethysm. Proc. Indian Acad. Sci. (Math. Sci.) 97, 85–109 (1987). https://doi.org/10.1007/BF02837817
Issue Date:
DOI: https://doi.org/10.1007/BF02837817