A minimal Set of Generators for the Ring of multisymmetric Functions
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 1741-1769.

The purpose of this article is to give, for any (commutative) ring A, an explicit minimal set of generators for the ring of multisymmetric functions TS A d (A[x 1 ,,x r ])=A [x 1 ,,x r ] A d 𝔖 d as an A-algebra. In characteristic zero, i.e. when A is a -algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously obtained by Fleischmann.

As Γ A d (A[x 1 ,,x r ])=TS A d (A[x 1 ,,x r ]) we also obtain generators for divided powers algebras: If B is a finitely generated A-algebra with a given surjection A[x 1 ,x 2 ,,x r ]B then using the corresponding surjection Γ A d (A[x 1 ,,x r ])Γ A d (B) we get generators for Γ A d (B).

Soit A un anneau commutatif arbitraire. Nous exhibons un ensemble minimal et explicite de générateurs de l’anneau des fonctions multisymétriques TS A d (A[x 1 ,,x r ]) et obtenons, par conséquent, une borne stricte sur le degré des générateurs. Dans le cas où la caractéristique de A est égale à zéro, un tel ensemble est connu depuis le 19ème siècle. Dans le cas général par contre, il n’existait jusque-là qu’une borne, généralement non stricte, sur le degré des générateurs, et un ensemble, généralement non minimal, de générateurs.

DOI: 10.5802/aif.2312
Classification: 13A50, 05E05, 14L30, 14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
Mot clés : Fonctions Symétriques, générateurs, puissances divisées, théorie des invariants
Rydh, David 1

1 KTH Department of Mathematics 100 44 Stockholm (Sweden)
@article{AIF_2007__57_6_1741_0,
     author = {Rydh, David},
     title = {A minimal {Set} of {Generators} for the {Ring} of multisymmetric {Functions}},
     journal = {Annales de l'Institut Fourier},
     pages = {1741--1769},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     doi = {10.5802/aif.2312},
     zbl = {1130.13005},
     mrnumber = {2377885},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2312/}
}
TY  - JOUR
AU  - Rydh, David
TI  - A minimal Set of Generators for the Ring of multisymmetric Functions
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 1741
EP  - 1769
VL  - 57
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2312/
DO  - 10.5802/aif.2312
LA  - en
ID  - AIF_2007__57_6_1741_0
ER  - 
%0 Journal Article
%A Rydh, David
%T A minimal Set of Generators for the Ring of multisymmetric Functions
%J Annales de l'Institut Fourier
%D 2007
%P 1741-1769
%V 57
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2312/
%R 10.5802/aif.2312
%G en
%F AIF_2007__57_6_1741_0
Rydh, David. A minimal Set of Generators for the Ring of multisymmetric Functions. Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 1741-1769. doi : 10.5802/aif.2312. http://www.numdam.org/articles/10.5802/aif.2312/

[1] Briand, Emmanuel When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?, Beiträge Algebra Geom., Volume 45 (2004) no. 2, pp. 353-368 | EuDML | MR | Zbl

[2] Campbell, H. E. A.; Hughes, I.; Pollack, R. D. Vector invariants of symmetric groups, Canad. Math. Bull., Volume 33 (1990) no. 4, pp. 391-397 | DOI | MR | Zbl

[3] Deligne, Pierre Cohomologie à supports propres, exposé XVII of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, Berlin, 1973, p. 250-480. Lecture Notes in Math., Vol. 305 | MR | Zbl

[4] Ferrand, Daniel Un foncteur norme, Bull. Soc. Math. France, Volume 126 (1998) no. 1, pp. 1-49 | EuDML | Numdam | MR | Zbl

[5] Fleischmann, P. A new degree bound for vector invariants of symmetric groups, Trans. Amer. Math. Soc., Volume 350 (1998) no. 4, pp. 1703-1712 | DOI | MR | Zbl

[6] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. (1964-67), pp. 259, 231, 255, 361 | Numdam | Zbl

[7] Grothendieck, A.; Verdier, J. L. Prefaisceaux, exposé I of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer-Verlag, Berlin, 1972, p. 1-217. Lecture Notes in Math., Vol. 269 | Zbl

[8] Hilbert, David Ueber die Theorie der algebraischen Formen, Math. Ann., Volume 36 (1890) no. 4, pp. 473-534 | DOI | EuDML | JFM | MR

[9] Junker, Fr. Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen, Math. Ann., Volume 38 (1891) no. 1, pp. 91-114 | DOI | EuDML | JFM | MR

[10] Junker, Fr. Uber symmetrische Functionen von mehreren Reihen von Veränderlichen, Math. Ann., Volume 43 (1893) no. 2-3, pp. 225-270 | DOI | MR

[11] Junker, Fr. Die symmetrischen Functionen und die Relationen zwischen den Elementarfunctionen derselben, Math. Ann., Volume 45 (1894) no. 1, pp. 1-84 | DOI | MR

[12] Lundkvist, Christian Counterexamples regarding Symmetric Tensors and Divided Powers, Preprint, 2007 (arXiv:math/0702733)

[13] Nagata, Masayoshi On the normality of the Chow variety of positive 0-cycles of degree m in an algebraic variety, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., Volume 29 (1955), pp. 165-176 | MR | Zbl

[14] Neeman, Amnon Zero cycles in n , Adv. Math., Volume 89 (1991) no. 2, pp. 217-227 | DOI | MR | Zbl

[15] Noether, Emmy Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., Volume 77 (1915) no. 1, pp. 89-92 | DOI | MR

[16] Noether, Emmy Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p, Nachr. Ges. Wiss. Göttingen (1926), pp. 28-35

[17] Richman, David R. Explicit generators of the invariants of finite groups, Adv. Math., Volume 124 (1996) no. 1, pp. 49-76 | DOI | MR | Zbl

[18] Roby, Norbert Lois polynomes et lois formelles en théorie des modules, Ann. Sci. École Norm. Sup. (3), Volume 80 (1963), pp. 213-348 | Numdam | MR | Zbl

[19] Roby, Norbert Lois polynômes multiplicatives universelles, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980) no. 19, p. A869-A871 | MR | Zbl

[20] Rydh, David Families of zero cycles and divided powers (2007) (In preparation)

[21] Rydh, David Hilbert and Chow schemes of points, symmetric products and divided powers (2007) (In preparation)

[22] Schläfli, Ludwig Über die Resultante eines systemes mehrerer algebraischen Gleichungen, Denkschr. Kais. Akad. Wiss. Math.-Natur. Kl., Volume 4 (1852), pp. 9-112 Reprinted in “Gesammelte matematische Abhandlungen”, Band II, Verlag Birkhäuser, Basel, (1953)

[23] Vaccarino, Francesco The ring of multisymmetric functions, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 3, pp. 717-731 | DOI | Numdam | MR | Zbl

[24] Weber, Heinrich Lehrbuch der Algebra, 2, Braunschweig, Berlin, 1899

[25] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 | MR | Zbl

[26] Ziplies, Dieter Generators for the divided powers algebra of an algebra and trace identities, Beiträge Algebra Geom. (1987) no. 24, pp. 9-27 | MR | Zbl

Cited by Sources: