A minimal Set of Generators for the Ring of multisymmetric Functions

Rydh, David

doi:10.5802/aif.2312

Rydh, David

Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 1741-1769

Abstract
Résumé

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 ${T S}_{A}^{d} (A [x_{1}, \dots, x_{r}]) = {(A {[x_{1}, \dots, x_{r}]}^{\otimes_{A} d})}^{𝔖_{d}}$ as an $A$ -algebra. In characteristic zero, i.e. when $A$ is a $ℚ$ -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 $Γ_{A}^{d} (A [x_{1}, \dots, x_{r}]) = {T S}_{A}^{d} (A [x_{1}, \dots, x_{r}])$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$ -algebra with a given surjection $A [x_{1}, x_{2}, \dots, x_{r}] \to B$ then using the corresponding surjection $Γ_{A}^{d} (A [x_{1}, \dots, x_{r}]) \to Γ_{A}^{d} (B)$ 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 ${T S}_{A}^{d} (A [x_{1}, \dots, x_{r}])$ 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.

MR 2377885 | Zbl 1130.13005

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},
     mrnumber = {2377885},
     zbl = {1130.13005},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIF_2007__57_6_1741_0/

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