The ring of multisymmetric functions
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, p. 717-731
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].
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].
DOI : https://doi.org/10.5802/aif.2111
Classification:  05E05,  13A50,  20C30
Keywords: invariants theory, symmetric functions, representations of symmetric groups
@article{AIF_2005__55_3_717_0,
     author = {Vaccarino, Francesco},
     title = {The ring of multisymmetric functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     pages = {717-731},
     doi = {10.5802/aif.2111},
     mrnumber = {2149400},
     zbl = {1062.05143},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_3_717_0}
}
The ring of multisymmetric functions. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 717-731. doi : 10.5802/aif.2111. https://aif.centre-mersenne.org/item/AIF_2005__55_3_717_0/

[A] M. Feschbach The mod 2 cohomology rings of the symmetric groups and invariants, Topology (2002), pp. 57-84 | MR 1871241 | Zbl 1039.20029

[B] N. Bourbaki Elements of mathematics - Algebra II Chapters 4-7, Springer-Verlag, Berlin (1988) | MR 1080964

[D] J. Dalbec Multisymmetric functions, Beiträge Algebra Geom., Tome 40 (1999) no. 1, pp. 27-51 | MR 1678567 | Zbl 0953.05077

[F] P. Fleischmann A new degree bound for vector invariants of symmetric groups, Trans. Am. Math. Soc., Tome 350 (1998), pp. 1703-1712 | Article | MR 1451600 | Zbl 0891.13002

[G] I. Gelfand; M. Kapranov; A. Zelevinsky Discriminants, resultants and multidimensional determinants, Birkahuser, Boston (1994) | MR 1264417 | Zbl 0827.14036

[J1] F. Junker Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen, Math. Ann., Tome 38 (1891), pp. 91-114 | Article | JFM 23.0156.02 | MR 1510665

[J2] F. Junker Über symmetrische Functionen von mehreren Reihen von Veränderlichen, Math. Ann., Tome 43 (1893), pp. 225-270 | Article | JFM 25.0230.01 | MR 1510811

[J3] F. Junker Die symmetrische Functionen und die Relationen zwischen den Elementarfunctionen derselben, Math. Ann., Tome 45 (1894), pp. 1-84 | Article | JFM 25.0230.02 | MR 1510854

[M] I.G. Macdonald Symmetric Functions and Hall Polynomials - second edition, Oxford mathematical monograph (1995) | MR 1354144 | Zbl 0487.20007

[W] H. Weyl The classical groups, Princeton University Press, Princeton N.J. (1946) | MR 1488158 | Zbl 0020.20601