The ring of multisymmetric functions
[L'anneau des fonctions multisymétriques]
Vaccarino, Francesco1
1 Politecnico di Torino, dipartimento di Matematica, Corso Duca degli Abruzzi 24, 10129 Torino (Italy)
Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 717-731.

Si R est un anneau commutatif, on présente par générateurs et relations, l’algèbre des fonctions multisymétriques à coefficients dans R, de façon à répondre à une question classique liée aux travaux de F. Junker [J1, J2, J3] et implicitement à ceux de H. Weyl [W].

We give a presentation (in terms of generators and relations) of the ring of multisymmetric functions that holds for any commutative ring R, thereby answering a classical question coming from works of F. Junker [J1, J2, J3] in the late nineteen century and then implicitly in H. Weyl book “The classical groups” [W].

DOI : 10.5802/aif.2111
Classification : 05E05, 13A50, 20C30
Keywords: invariants theory, symmetric functions, representations of symmetric groups
Mot clés : théorie des invariants, polynômes symétriques, représentations du groupe symétrique
Vaccarino, Francesco 1

1 Politecnico di Torino, dipartimento di Matematica, Corso Duca degli Abruzzi 24, 10129 Torino (Italy) 
@article{AIF_2005__55_3_717_0,
     author = {Vaccarino, Francesco},
     title = {The ring of multisymmetric functions},
     journal = {Annales de l'Institut Fourier},
     pages = {717--731},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     doi = {10.5802/aif.2111},
     mrnumber = {2149400},
     zbl = {1062.05143},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2111/}
}
TY  - JOUR
AU  - Vaccarino, Francesco
TI  - The ring of multisymmetric functions
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 717
EP  - 731
VL  - 55
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2111/
DO  - 10.5802/aif.2111
LA  - en
ID  - AIF_2005__55_3_717_0
ER  - 
%0 Journal Article
%A Vaccarino, Francesco
%T The ring of multisymmetric functions
%J Annales de l'Institut Fourier
%D 2005
%P 717-731
%V 55
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2111/
%R 10.5802/aif.2111
%G en
%F AIF_2005__55_3_717_0
Vaccarino, Francesco. The ring of multisymmetric functions. Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 717-731. doi : 10.5802/aif.2111. http://archive.numdam.org/articles/10.5802/aif.2111/

[A] M. Feschbach The mod 2 cohomology rings of the symmetric groups and invariants, Topology (2002), pp. 57-84 | MR | Zbl

[B] N. Bourbaki Elements of mathematics - Algebra II Chapters 4-7, Springer-Verlag, Berlin, 1988 | MR

[D] J. Dalbec Multisymmetric functions, Beiträge Algebra Geom., Volume 40 (1999) no. 1, pp. 27-51 | MR | Zbl

[F] P. Fleischmann A new degree bound for vector invariants of symmetric groups, Trans. Am. Math. Soc., Volume 350 (1998), pp. 1703-1712 | DOI | MR | Zbl

[G] I. Gelfand; M. Kapranov; A. Zelevinsky Discriminants, resultants and multidimensional determinants, Birkahuser, Boston, 1994 | MR | Zbl

[J1] F. Junker Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen, Math. Ann., Volume 38 (1891), pp. 91-114 | DOI | JFM | MR

[J2] F. Junker Über symmetrische Functionen von mehreren Reihen von Veränderlichen, Math. Ann., Volume 43 (1893), pp. 225-270 | DOI | JFM | MR

[J3] F. Junker Die symmetrische Functionen und die Relationen zwischen den Elementarfunctionen derselben, Math. Ann., Volume 45 (1894), pp. 1-84 | DOI | JFM | MR

[M] I.G. Macdonald Symmetric Functions and Hall Polynomials - second edition, Oxford mathematical monograph (1995) | MR | Zbl

[W] H. Weyl The classical groups, Princeton University Press, Princeton N.J., 1946 | MR | Zbl

Cité par Sources :