no. 3
On classical invariant theory and binary cubics
Annales de l'Institut Fourier, Volume 37 (1987) no. 3, pp. 191-216

Let G be a reductive complex algebraic group, and let C[mV] G denote the algebra of invariant polynomial functions on the direct sum of m copies of the representations space V of G. There is a smallest integer n=n(V) such that generators and relations of C[mV] G can be obtained from those of C[nV] G by polarization and restitution for all m>n. We bound and the degrees of generators and relations of C[nV] G , extending results of Vust. We apply our techniques to compute the invariant theory of binary cubics.

Soit G un groupe algébrique complexe réductif et C[mV] G l’algèbre des polynômes G-invariants sur la somme directe de m copies de l’espace de représentation V de G. Il existe un nombre entier n=n(V) minimal tel que les générateurs et relations de C[mv] G puissent s’obtenir à partir de ceux de C[nv] G par polarisation et restitution pour chaque m>n. On borne n et les degrés des générateurs et relations de C[nV] G , en étendant des résultats de Vust. Ces techniques sont alors appliquées au calcul des invariants de plusieurs formes binaires cubiques.

Schwarz, Gerald W. On classical invariant theory and binary cubics. Annales de l'Institut Fourier, Volume 37 (1987) no. 3, pp. 191-216. doi: 10.5802/aif.1104
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[1] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Inv. Math., 88 (1987), 65-68. | Zbl | MR

[2] J. Grace and A. Young, The Algebra of Invariants, Cambridge University Press, Cambridge, 1903. | JFM

[3] M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. in Math., 13 (1974), 115-175. | Zbl | MR

[4] F. Knop, Über die Glattheit von Quotientenabbildungen, Manuscripta Math., 56 (1986), 419-427. | Zbl | MR

[5] H. Kraft, Geometrische Methoden in der Invariantentheorie, Viehweg, Braunschweig, 1984. | Zbl | MR

[6] M. Krämer, Eine Klassifikation bestimmter Untergruppen kompakter zusammenhängender Liegruppen, Comm. in Alg., 3 (1975), 691-737. | Zbl

[7] S. Lang, Algebra, Addison-Wesley, Reading, 1965. | Zbl | MR

[8] D. Luna and R. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J., 46 (1979), 487-496. | Zbl | MR

[9] I.G. Mac Donald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. | Zbl | MR

[10] C. Procesi, A Primer of Invariant Theory, Brandeis Lecture Notes 1, Department of Mathematics, Brandeis University, 1982.

[11] G. Schwarz, Representations of simple Lie groups with regular rings of invariants, Inv. Math., 49 (1978), 167-191. | Zbl | MR

[12] G. Schwarz, Representations of simple Lie groups with a free module of covariants, Inv. Math., 50 (1978), 1-12. | Zbl | MR

[13] G. Schwarz, Invariant theory of G2, Bull. Amer. Math. Soc., 9 (1983), 335-338. | Zbl | MR

[14] G. Schwarz, Invariant theory of G2 and Spin7, to appear.

[15] R.P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc., 1 (1979), 475-511. | Zbl | MR

[16] R.P. Stanley, Combinatorics and invariant theory, Proc. Symposia Pure Math., Vol. 34, Amer. Math. Soc., Providence, R.I., 1979, 345-355. | Zbl | MR

[17] F. Von Gall, Das vollständige Formensystem dreier cubischen binären Formen, Math. Ann., 45 (1894), 207-234. | JFM

[18] Th. Vust, Sur la théorie des invariants des groupes classiques, Ann. Inst. Fourier, 26-1 (1976), 1-31. | Zbl | MR | Numdam

[19] Th. Vust, Sur la théorie classique des invariants, Comm. Math. Helv., 52 (1977), 259-295. | Zbl | MR

[20] Th. Vust, Foncteurs polynomiaux et théorie des invariants, in Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin, Springer Lecture Notes, No. 725, Springer Verlag, New York, 1980, pp. 330-340. | Zbl | MR

[21] H. Weyl, The Classical Groups, 2nd edn., Princeton Univ. Press, Princeton, N.J., 1946.

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