no. 6
A minimal Set of Generators for the Ring of multisymmetric Functions
[Un ensemble minimal de générateurs de l’anneau des fonctions multisymétriques]
Rydh, David1
1 KTH Department of Mathematics 100 44 Stockholm (Sweden)
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 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 ,⋯,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 : 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) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2312/

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