A minimal Set of Generators for the Ring of multisymmetric Functions
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 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 : https://doi.org/10.5802/aif.2312
Classification:  13A50,  05E05,  14L30,  14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     pages = {1741-1769},
     doi = {10.5802/aif.2312},
     zbl = {1130.13005},
     mrnumber = {2377885},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_6_1741_0}
}
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. https://aif.centre-mersenne.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., Tome 45 (2004) no. 2, pp. 353-368 | MR 2093171 | Zbl 1062.05140

[2] Campbell, H. E. A.; Hughes, I.; Pollack, R. D. Vector invariants of symmetric groups, Canad. Math. Bull., Tome 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, Tome 126 (1998) no. 1, pp. 1-49 | Numdam | MR 1651380 | Zbl 1017.13005

[5] Fleischmann, P. A new degree bound for vector invariants of symmetric groups, Trans. Amer. Math. Soc., Tome 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., Tome 36 (1890) no. 4, pp. 473-534 | Article | JFM 22.0133.01 | MR 1510634

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

[10] Junker, Fr. Uber symmetrische Functionen von mehreren Reihen von Veränderlichen, Math. Ann., Tome 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., Tome 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., Tome 29 (1955), pp. 165-176 | MR 96668 | Zbl 0066.14701

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

[15] Noether, Emmy Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., Tome 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., Tome 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), Tome 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, Tome 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., Tome 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), Tome 55 (2005) no. 3, pp. 717-731 | Article | Numdam | MR 2149400 | Zbl 1062.05143

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