On classical invariant theory and binary cubics
Annales de l'Institut Fourier, Tome 37 (1987) no. 3, p. 191-216
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.
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.
@article{AIF_1987__37_3_191_0,
     author = {Schwarz, Gerald W.},
     title = {On classical invariant theory and binary cubics},
     journal = {Annales de l'Institut Fourier},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {3},
     year = {1987},
     pages = {191-216},
     doi = {10.5802/aif.1104},
     zbl = {0597.14011},
     mrnumber = {89h:14036},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_3_191_0}
}
Schwarz, Gerald W. On classical invariant theory and binary cubics. Annales de l'Institut Fourier, Tome 37 (1987) no. 3, pp. 191-216. doi : 10.5802/aif.1104. http://www.numdam.org/item/AIF_1987__37_3_191_0/

[1] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Inv. Math., 88 (1987), 65-68. | MR 88a:14005 | Zbl 0619.14029

[2] J. Grace and A. Young, The Algebra of Invariants, Cambridge University Press, Cambridge, 1903. | JFM 34.0114.01

[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. | MR 50 #311 | Zbl 0289.14010

[4] F. Knop, Über die Glattheit von Quotientenabbildungen, Manuscripta Math., 56 (1986), 419-427. | MR 88f:14041 | Zbl 0585.14033

[5] H. Kraft, Geometrische Methoden in der Invariantentheorie, Viehweg, Braunschweig, 1984. | MR 86j:14006 | Zbl 0569.14003

[6] M. Krämer, Eine Klassifikation bestimmter Untergruppen kompakter zusammenhängender Liegruppen, Comm. in Alg., 3 (1975), 691-737. | Zbl 0309.22013

[7] S. Lang, Algebra, Addison-Wesley, Reading, 1965. | MR 33 #5416 | Zbl 0193.34701

[8] D. Luna and R. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J., 46 (1979), 487-496. | MR 80k:14049 | Zbl 0444.14010

[9] I.G. Mac Donald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. | MR 84g:05003 | Zbl 0487.20007

[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. | MR 80m:14032 | Zbl 0391.20032

[12] G. Schwarz, Representations of simple Lie groups with a free module of covariants, Inv. Math., 50 (1978), 1-12. | MR 80c:14008 | Zbl 0391.20033

[13] G. Schwarz, Invariant theory of G2, Bull. Amer. Math. Soc., 9 (1983), 335-338. | MR 85c:20031 | Zbl 0531.14007

[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. | MR 81a:20015 | Zbl 0497.20002

[16] R.P. Stanley, Combinatorics and invariant theory, Proc. Symposia Pure Math., Vol. 34, Amer. Math. Soc., Providence, R.I., 1979, 345-355. | MR 80e:15020 | Zbl 0411.22006

[17] F. Von Gall, Das vollständige Formensystem dreier cubischen binären Formen, Math. Ann., 45 (1894), 207-234. | JFM 25.0186.02

[18] Th. Vust, Sur la théorie des invariants des groupes classiques, Ann. Inst. Fourier, 26-1 (1976), 1-31. | Numdam | MR 53 #8082 | Zbl 0314.20035

[19] Th. Vust, Sur la théorie classique des invariants, Comm. Math. Helv., 52 (1977), 259-295. | MR 56 #15683 | Zbl 0364.15022

[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. | MR 82b:15020 | Zbl 0429.14003

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