Table of contents for this issue | Next article
David Rydh
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, 57 no. 6 (2007), p. 1741-1769, doi: 10.5802/aif.2312
Article PDF | Reviews MR 2377885 | Zbl 1130.13005
Class. Math.: 13A50, 05E05, 14L30, 14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
[1] Emmanuel Briand, “When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?”, Beiträge Algebra Geom. 45 (2004) no. 2, p. 353-368 MR 2093171 | Zbl 1062.05140
[2] H. E. A. Campbell, I. Hughes & R. D. Pollack, “Vector invariants of symmetric groups”, Canad. Math. Bull. 33 (1990) no. 4, p. 391-397 Article | MR 1091341 | Zbl 0695.14007
[3] Pierre Deligne, Cohomologie à supports propres, exposé XVII of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, 1973, p. 250–480. Lecture Notes in Math., Vol. 305 MR 354654 | Zbl 0255.14011
[4] Daniel Ferrand, “Un foncteur norme”, Bull. Soc. Math. France 126 (1998) no. 1, p. 1-49 Numdam | MR 1651380 | Zbl 1017.13005
[5] P. Fleischmann, “A new degree bound for vector invariants of symmetric groups”, Trans. Amer. Math. Soc. 350 (1998) no. 4, p. 1703-1712 Article | MR 1451600 | Zbl 0891.13002
[6] A. Grothendieck, “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas”, Inst. Hautes Études Sci. Publ. Math. (1964-67) Numdam | Zbl 0135.39701
[7] A. Grothendieck & J. L. Verdier, Prefaisceaux, exposé I of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer-Verlag, 1972, p. 1–217. Lecture Notes in Math., Vol. 269 Zbl 0249.18021
[8] David Hilbert, “Ueber die Theorie der algebraischen Formen”, Math. Ann. 36 (1890) no. 4, p. 473-534 Article | MR 1510634 | JFM 22.0133.01
[9] Fr. Junker, “Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen”, Math. Ann. 38 (1891) no. 1, p. 91-114 Article | MR 1510665 | JFM 23.0156.02
[10] Fr. Junker, “Uber symmetrische Functionen von mehreren Reihen von Veränderlichen”, Math. Ann. 43 (1893) no. 2-3, p. 225-270 Article | MR 1510811 | JFM 25.0230.01
[11] Fr. Junker, “Die symmetrischen Functionen und die Relationen zwischen den Elementarfunctionen derselben”, Math. Ann. 45 (1894) no. 1, p. 1-84 Article | MR 1510854 | JFM 25.0230.02
[12] Christian Lundkvist, “Counterexamples regarding Symmetric Tensors and Divided Powers”, Preprint, Feb 2007, arXiv:math/0702733 arXiv
[13] Masayoshi Nagata, “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. 29 (1955), p. 165-176 Article | MR 96668 | Zbl 0066.14701
[14] Amnon Neeman, “Zero cycles in ${\mathbb{P}^n}$”, Adv. Math. 89 (1991) no. 2, p. 217-227 Article | MR 1128613 | Zbl 0787.14004
[15] Emmy Noether, “Der Endlichkeitssatz der Invarianten endlicher Gruppen”, Math. Ann. 77 (1915) no. 1, p. 89-92 Article | MR 1511848 | JFM 45.0198.01
[16] Emmy Noether, “Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik $p$”, Nachr. Ges. Wiss. Göttingen (1926), p. 28-35 Article | JFM 52.0106.01
[17] David R. Richman, “Explicit generators of the invariants of finite groups”, Adv. Math. 124 (1996) no. 1, p. 49-76 Article | MR 1423198 | Zbl 0879.13003
[18] Norbert Roby, “Lois polynomes et lois formelles en théorie des modules”, Ann. Sci. École Norm. Sup. (3) 80 (1963), p. 213-348 Numdam | MR 161887 | Zbl 0117.02302
[19] Norbert Roby, “Lois polynômes multiplicatives universelles”, C. R. Acad. Sci. Paris Sér. A-B 290 (1980) no. 19, p. A869-A871 MR 580160 | Zbl 0471.13008
[20] David Rydh, “Families of zero cycles and divided powers”, In preparation, 2007
[21] David Rydh, “Hilbert and Chow schemes of points, symmetric products and divided powers”, In preparation, 2007
[22] Ludwig Schläfli, “Über die Resultante eines systemes mehrerer algebraischen Gleichungen”, Denkschr. Kais. Akad. Wiss. Math.-Natur. Kl. 4 (1852), p. 9-112, Reprinted in “Gesammelte matematische Abhandlungen”, Band II, Verlag Birkhäuser, Basel, (1953)
[23] Francesco Vaccarino, “The ring of multisymmetric functions”, Ann. Inst. Fourier (Grenoble) 55 (2005) no. 3, p. 717-731 Cedram | MR 2149400 | Zbl 1062.05143
[24] Heinrich Weber, Lehrbuch der Algebra 2, Braunschweig, Berlin, 1899 JFM 30.0093.01
[25] Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 MR 1488158 | Zbl 1024.20502
[26] Dieter Ziplies, “Generators for the divided powers algebra of an algebra and trace identities”, Beiträge Algebra Geom. (1987) no. 24, p. 9-27 Article | MR 888200 | Zbl 0632.16004
David Rydh
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, 57 no. 6 (2007), p. 1741-1769, doi: 10.5802/aif.2312
Article PDF | Reviews MR 2377885 | Zbl 1130.13005
Class. Math.: 13A50, 05E05, 14L30, 14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
Résumé - Abstract
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 ${\mathrm{T}S}^d_A(A[x_1,\dots ,x_r])= \bigl (A[x_1,\dots ,x_r]^{\otimes _A d}\bigr )^{{\mathfrak{S}}_d}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a ${\mathbb{Q}}$-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 $\Gamma ^d_A(A[x_1,\dots ,x_r])={\mathrm{T}S}^d_A(A[x_1,\dots ,x_r])$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$-algebra with a given surjection $A[x_1,x_2,\dots ,x_r]\rightarrow B$ then using the corresponding surjection $\Gamma ^d_A(A[x_1,\dots ,x_r])\rightarrow \Gamma ^d_A(B)$ we get generators for $\Gamma ^d_A(B)$.
Bibliography
[2] H. E. A. Campbell, I. Hughes & R. D. Pollack, “Vector invariants of symmetric groups”, Canad. Math. Bull. 33 (1990) no. 4, p. 391-397 Article | MR 1091341 | Zbl 0695.14007
[3] Pierre Deligne, Cohomologie à supports propres, exposé XVII of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, 1973, p. 250–480. Lecture Notes in Math., Vol. 305 MR 354654 | Zbl 0255.14011
[4] Daniel Ferrand, “Un foncteur norme”, Bull. Soc. Math. France 126 (1998) no. 1, p. 1-49 Numdam | MR 1651380 | Zbl 1017.13005
[5] P. Fleischmann, “A new degree bound for vector invariants of symmetric groups”, Trans. Amer. Math. Soc. 350 (1998) no. 4, p. 1703-1712 Article | MR 1451600 | Zbl 0891.13002
[6] A. Grothendieck, “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas”, Inst. Hautes Études Sci. Publ. Math. (1964-67) Numdam | Zbl 0135.39701
[7] A. Grothendieck & J. L. Verdier, Prefaisceaux, exposé I of SGA 4, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Springer-Verlag, 1972, p. 1–217. Lecture Notes in Math., Vol. 269 Zbl 0249.18021
[8] David Hilbert, “Ueber die Theorie der algebraischen Formen”, Math. Ann. 36 (1890) no. 4, p. 473-534 Article | MR 1510634 | JFM 22.0133.01
[9] Fr. Junker, “Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen”, Math. Ann. 38 (1891) no. 1, p. 91-114 Article | MR 1510665 | JFM 23.0156.02
[10] Fr. Junker, “Uber symmetrische Functionen von mehreren Reihen von Veränderlichen”, Math. Ann. 43 (1893) no. 2-3, p. 225-270 Article | MR 1510811 | JFM 25.0230.01
[11] Fr. Junker, “Die symmetrischen Functionen und die Relationen zwischen den Elementarfunctionen derselben”, Math. Ann. 45 (1894) no. 1, p. 1-84 Article | MR 1510854 | JFM 25.0230.02
[12] Christian Lundkvist, “Counterexamples regarding Symmetric Tensors and Divided Powers”, Preprint, Feb 2007, arXiv:math/0702733 arXiv
[13] Masayoshi Nagata, “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. 29 (1955), p. 165-176 Article | MR 96668 | Zbl 0066.14701
[14] Amnon Neeman, “Zero cycles in ${\mathbb{P}^n}$”, Adv. Math. 89 (1991) no. 2, p. 217-227 Article | MR 1128613 | Zbl 0787.14004
[15] Emmy Noether, “Der Endlichkeitssatz der Invarianten endlicher Gruppen”, Math. Ann. 77 (1915) no. 1, p. 89-92 Article | MR 1511848 | JFM 45.0198.01
[16] Emmy Noether, “Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik $p$”, Nachr. Ges. Wiss. Göttingen (1926), p. 28-35 Article | JFM 52.0106.01
[17] David R. Richman, “Explicit generators of the invariants of finite groups”, Adv. Math. 124 (1996) no. 1, p. 49-76 Article | MR 1423198 | Zbl 0879.13003
[18] Norbert Roby, “Lois polynomes et lois formelles en théorie des modules”, Ann. Sci. École Norm. Sup. (3) 80 (1963), p. 213-348 Numdam | MR 161887 | Zbl 0117.02302
[19] Norbert Roby, “Lois polynômes multiplicatives universelles”, C. R. Acad. Sci. Paris Sér. A-B 290 (1980) no. 19, p. A869-A871 MR 580160 | Zbl 0471.13008
[20] David Rydh, “Families of zero cycles and divided powers”, In preparation, 2007
[21] David Rydh, “Hilbert and Chow schemes of points, symmetric products and divided powers”, In preparation, 2007
[22] Ludwig Schläfli, “Über die Resultante eines systemes mehrerer algebraischen Gleichungen”, Denkschr. Kais. Akad. Wiss. Math.-Natur. Kl. 4 (1852), p. 9-112, Reprinted in “Gesammelte matematische Abhandlungen”, Band II, Verlag Birkhäuser, Basel, (1953)
[23] Francesco Vaccarino, “The ring of multisymmetric functions”, Ann. Inst. Fourier (Grenoble) 55 (2005) no. 3, p. 717-731 Cedram | MR 2149400 | Zbl 1062.05143
[24] Heinrich Weber, Lehrbuch der Algebra 2, Braunschweig, Berlin, 1899 JFM 30.0093.01
[25] Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 MR 1488158 | Zbl 1024.20502
[26] Dieter Ziplies, “Generators for the divided powers algebra of an algebra and trace identities”, Beiträge Algebra Geom. (1987) no. 24, p. 9-27 Article | MR 888200 | Zbl 0632.16004
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310