A minimal Set of Generators for the Ring of multisymmetric Functions

Rydh, David

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 ${\mathrm{T}S}_A^d\left(A[x_1,\cdots ,x_r]\right)$ 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 ${\mathrm{T}S}_A^d\left(A[x_1,\cdots ,x_r]\right)={\left(A{[x_1,\cdots ,x_r]}^{{\otimes }_{A}d}\right)}^{{𝔖}_d}$ 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 ${\Gamma }_A^d\left(A[x_1,\cdots ,x_r]\right)={\mathrm{T}S}_A^d\left(A[x_1,\cdots ,x_r]\right)$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$-algebra with a given surjection $A[x_1,x_2,\cdots ,x_r]\to B$ then using the corresponding surjection ${\Gamma }_A^d\left(A[x_1,\cdots ,x_r]\right)\to {\Gamma }_A^d\left(B\right)$ we get generators for ${\Gamma }_A^d\left(B\right)$.

Classification : 13A50, 05E05, 14L30, 14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
Mot clés : Fonctions Symétriques, générateurs, puissances divisées, théorie des invariants

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