no. 6
A minimal Set of Generators for the Ring of multisymmetric Functions
Rydh, David1
1 KTH Department of Mathematics 100 44 Stockholm (Sweden)
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 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 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).

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: 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, Volume 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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