Sylvester versus Gundelfinger
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $V_n$ be the $\mathrm{SL}_2$-module of binary forms of degree $n$ and let $V=V_1\oplus V_3\oplus V_4$. We show that the minimum number of generators of the algebra $R = \mathbb C[V]^{\mathrm{SL}_2}$ of polynomial functions on $V$ invariant under the action of $\mathrm{SL}_2$ equals 63. This settles a 143-year old question.
Keywords: invariants; covariants; binary forms.
@article{SIGMA_2012_8_a74,
     author = {Andries E. Brouwer and Mihaela Popoviciu},
     title = {Sylvester versus {Gundelfinger}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a74/}
}
TY  - JOUR
AU  - Andries E. Brouwer
AU  - Mihaela Popoviciu
TI  - Sylvester versus Gundelfinger
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a74/
LA  - en
ID  - SIGMA_2012_8_a74
ER  - 
%0 Journal Article
%A Andries E. Brouwer
%A Mihaela Popoviciu
%T Sylvester versus Gundelfinger
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a74/
%G en
%F SIGMA_2012_8_a74
Andries E. Brouwer; Mihaela Popoviciu. Sylvester versus Gundelfinger. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a74/

[1] Brion M., “Invariants de plusieurs formes binaires”, Bull. Soc. Math. France, 110 (1982), 429–445 | MR | Zbl

[2] Cayley A., “A second memoir upon quantics”, Phil. Trans. Royal Soc. London, 146 (1856), 101–126 | DOI

[3] Derksen H., Kemper G., Computational invariant theory, Encyclopaedia of Mathematical Sciences, 130, Springer-Verlag, Berlin, 2002 | MR | Zbl

[4] Gordan P., “Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist”, J. Reine Angew. Math., 69 (1868), 323–354 | DOI

[5] Gordan P., “Die simultanen Systeme binärer Formen”, Math. Ann., 2 (1870), 227–280 | DOI | MR

[6] Gundelfinger S., Zur Theorie des simultanen Systems einer cubischen und einer biquadratischen binären Form, Habilitationsschrift, J.B. Metzler, Stuttgart, 1869

[7] Hammond J., “Note on an exceptional case in which the fundamental postulate of professor Sylvester's theory of tamisage fails”, Proc. London Math. Soc., 14 (1882), 85–88 | DOI | MR

[8] Hilbert D., “Ueber die vollen Invariantensysteme”, Math. Ann., 42 (1893), 313–373 | DOI | MR

[9] Hochster M., Roberts J.L., “Rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay”, Adv. Math., 13 (1974), 115–175 | DOI | MR | Zbl

[10] Morley R.K., “On the fundamental postulate of tamisage”, Amer. J. Math., 34 (1912), 47–68 | DOI | MR | Zbl

[11] Olver P.J., Classical invariant theory, London Mathematical Society Student Texts, 44, Cambridge University Press, Cambridge, 1999 | DOI | MR

[12] Procesi C., Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007 | MR | Zbl

[13] Springer T.A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, Berlin, 1977 | MR | Zbl

[14] Springer T.A., “On the invariant theory of $\mathrm{SU}_2$”, Indag. Math., 42 (1980), 339–345 | MR | Zbl

[15] Sylvester J.J., “Détermination du nombre exact des covariants irréductibles du système cubo-biquadratique binaire”, C. R. Acad. Sci. Paris, 87 (1878), 477–481

[16] Sylvester J.J., “Proof of the hitherto undemonstrated fundamental theorem of invariants”, Phil. Mag., 5 (1879), 178–188

[17] Sylvester J.J., “Sur le vrai nombre des covariants élémentaires d'un système de deux formes biquadratiques binaires”, C. R. Acad. Sci. Paris, 84 (1877), 1285–1289

[18] Sylvester J.J., “Sur le vrai nombre des formes irréductibles du système cubo-biquadratique”, C. R. Acad. Sci. Paris, 87 (1878), 445–448

[19] Sylvester J.J., “Sur les covariants fundamentaux d'un système cubo-quartique binaire”, C. R. Acad. Sci. Paris, 87 (1878), 287–289

[20] Sylvester J.J., Franklin F., “Tables of the generating functions and groundforms for the binary quantics of the first ten orders”, Amer. J. Math., 2 (1879), 223–251 | DOI | MR