A minimal Set of Generators for the Ring of multisymmetric Functions  [ Un ensemble minimal de générateurs de l’anneau des fonctions multisymétriques ]
Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1741-1769.

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.

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).

DOI : https://doi.org/10.5802/aif.2312
Classification : 13A50,  05E05,  14L30,  14C05
Mots clés : Fonctions Symétriques, générateurs, puissances divisées, théorie des invariants
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     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'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     doi = {10.5802/aif.2312},
     mrnumber = {2377885},
     zbl = {1130.13005},
     language = {en},
     url = {archive.numdam.org/item/AIF_2007__57_6_1741_0/}
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Rydh, David. A minimal Set of Generators for the Ring of multisymmetric Functions. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1741-1769. doi : 10.5802/aif.2312. http://archive.numdam.org/item/AIF_2007__57_6_1741_0/

[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 124600 | MR 2093171 | Zbl 1062.05140

[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 | Article | MR 1091341 | Zbl 0695.14007

[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 354654 | Zbl 0255.14011

[4] Ferrand, Daniel Un foncteur norme, Bull. Soc. Math. France, Volume 126 (1998) no. 1, pp. 1-49 | EuDML 87779 | Numdam | MR 1651380 | Zbl 1017.13005

[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 | Article | MR 1451600 | Zbl 0891.13002

[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 0135.39701

[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 0249.18021

[8] Hilbert, David Ueber die Theorie der algebraischen Formen, Math. Ann., Volume 36 (1890) no. 4, pp. 473-534 | Article | EuDML 157506 | JFM 22.0133.01 | MR 1510634

[9] Junker, Fr. Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen, Math. Ann., Volume 38 (1891) no. 1, pp. 91-114 | Article | EuDML 157537 | JFM 23.0156.02 | MR 1510665

[10] Junker, Fr. Uber symmetrische Functionen von mehreren Reihen von Veränderlichen, Math. Ann., Volume 43 (1893) no. 2-3, pp. 225-270 | Article | MR 1510811

[11] Junker, Fr. Die symmetrischen Functionen und die Relationen zwischen den Elementarfunctionen derselben, Math. Ann., Volume 45 (1894) no. 1, pp. 1-84 | Article | MR 1510854

[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 96668 | Zbl 0066.14701

[14] Neeman, Amnon Zero cycles in n , Adv. Math., Volume 89 (1991) no. 2, pp. 217-227 | Article | MR 1128613 | Zbl 0787.14004

[15] Noether, Emmy Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., Volume 77 (1915) no. 1, pp. 89-92 | Article | MR 1511848

[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 | Article | MR 1423198 | Zbl 0879.13003

[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 161887 | Zbl 0117.02302

[19] Roby, Norbert Lois polynômes multiplicatives universelles, C. R. Acad. Sci. Paris Sér. A-B, Volume 290 (1980) no. 19, p. A869-A871 | MR 580160 | Zbl 0471.13008

[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 | Article | Numdam | MR 2149400 | Zbl 1062.05143

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

[25] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 | MR 1488158 | Zbl 1024.20502

[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 888200 | Zbl 0632.16004