Those invariant polynomials are called multisymmetric functions. There are several papers on them; you could start with J. Dalbec, Multisymmetric functions, Beiträge Algebra Geom. 40(1) (1999), 27-51 http://www.emis.de/journals/BAG/vol.40/no.1/b40h1dal.ps.gz.
Angelo
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