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Young diagrammatic methods in non-commutative invariant theory

Published online by Cambridge University Press:  22 January 2016

Yasuo Teranishi*
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Chikusa-ku, Nagoya, 464-01, Japan
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In this paper we will study some aspects of non-commutative invariant theory. Let V be a finite-dimensional vector space over a field K of characteristic zero and let

K[V] = KVS2(V)⊕…, and

KV› = KV⊕⊕2(V)⊕⊕3V⊕&

be respectively the symmetric algebra and the tensor algebra over V. Let G be a subgroup of GL(V). Then G acts on K[V] and KV›. Much of this paper is devoted to the study of the (non-commutative) invariant ring KVG of G acting on KV›.

In the first part of this paper, we shall study the invariant ring in the following situation.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[D-P] Contini, C. De and Procesi, C., A characteristic free approach to invariant theory, Adv. in Math., 21 (1976), 330354.Google Scholar
[D-R-S] Doubilet, P., Rota, G.-C., and Stein, J., On the foundation of combinatorial theory: IX. Combinatorial method in invariant theory, Stud. Appl. Math., 53 (1974), 185216.CrossRefGoogle Scholar
[K] Kempf, G. R., Computing invariants, pp. 81–94 in “Invariant Theory”, Lecture Notes in Math., No. 1278, Springer, 1987.Google Scholar
[Kh] Kharchenko, V. K., Algebras of invariants of free algebras, Algebra i Logika, 17 (1978), 478487 (Russian); English translation: Algebra and Logic 17 (1978), 316321.Google Scholar
[Ko] Koryukin, A. N., Noncommutative invariants of reductive groups, Algebra i Logika, 23 (1984, 419429 (Russian); English translation: Algebra and Logic, 23 (1984), 290296.Google Scholar
[K-R] Kung, J. P. S. and Rota, G. -C., The invariant theory of binary forms, Bull. Amer. Math. Soc. (New Series), 10 (1985), 2785.Google Scholar
[L] Lane, D. R., Free algebras of rank two and their automorphisms, Ph. D. Thesis, Betford College, London, 1976.Google Scholar
[Le] Bruyn, L. Le, Trace rings of generic 2 by 2 matrices, Mem. Ams Amer. Math. Soc. 363 (1987).Google Scholar
[M] MacDonald, I. G., “Symmetric Functions and Hall Polynomials”, Oxford University Press, Oxford, 1979.Google Scholar
[N] Noether, E., Der Endlichkeitessatz der Invarianten endlichen Gruppen, Math. Ann., 77 (1916), 8992.Google Scholar
[P1] Popov, V., Modern development in invariant theory, pp. 394406 in “Proc. of I. C. M. Barkeley 1986”.Google Scholar
[P2] Popov, V., Constructive invariant theory, Asterisque, 87–88 (1981), 303334.Google Scholar
[Pr] Procesi, C., The invariant theory of nxn matrices, Adv. of Math., 19 (1976), 306381.Google Scholar
[T] Tambour, T., Examples of S-algebras and generating functions, preprint.Google Scholar
[Te1] Teranishi, Y., Noncommutative invariant theory, 321–332, in “Perspectives in Ring Theory”, Nato ASI Series 233, Kluwer Academic Publishers, 1988.Google Scholar
[Te2] Teranishi, Y., Noncommutative classical invariant theory, Nagoya Math. J., 112 (1988), 153169.Google Scholar
[Te3] Teranishi, Y., Universal induced characters and restriction rules for the classical groups, Nagoya Math. J., 117 (1990), 173205.Google Scholar
[W] Weyl, H., “The Classical Groups”, Princeton University Press, Princeton, 1946.Google Scholar