A minimal Set of Generators for the Ring of multisymmetric Functions
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 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 ${\mathrm{T}S}_{A}^{d}\left(A\left[{x}_{1},\cdots ,{x}_{r}\right]\right)={\left(A{\left[{x}_{1},\cdots ,{x}_{r}\right]}^{{\otimes }_{A}d}\right)}^{{𝔖}_{d}}$ 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 ${\Gamma }_{A}^{d}\left(A\left[{x}_{1},\cdots ,{x}_{r}\right]\right)={\mathrm{T}S}_{A}^{d}\left(A\left[{x}_{1},\cdots ,{x}_{r}\right]\right)$ we also obtain generators for divided powers algebras: If $B$ is a finitely generated $A$-algebra with a given surjection $A\left[{x}_{1},{x}_{2},\cdots ,{x}_{r}\right]\to B$ then using the corresponding surjection ${\Gamma }_{A}^{d}\left(A\left[{x}_{1},\cdots ,{x}_{r}\right]\right)\to {\Gamma }_{A}^{d}\left(B\right)$ we get generators for ${\Gamma }_{A}^{d}\left(B\right)$.

Soit $A$ un anneau commutatif arbitraire. Nous exhibons un ensemble minimal et explicite de générateurs de l’anneau des fonctions multisymétriques ${\mathrm{T}S}_{A}^{d}\left(A\left[{x}_{1},\cdots ,{x}_{r}\right]\right)$ 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.

DOI : https://doi.org/10.5802/aif.2312
Classification:  13A50,  05E05,  14L30,  14C05
Keywords: Symmetric functions, generators, divided powers, vector invariants
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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}
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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. http://www.numdam.org/item/AIF_2007__57_6_1741_0/

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