The symmetric product of a variety $M$ is the quotient of $M^n/S\_n$ where $S\_n$ is the symmetric group permuting components of n-fold product $M^n$. IF $M$ is an affine plane $C^k$ over complex numbers, the coordinate ring of the symmetric product is the invariant polynomials in $R:=C[x^1\_1,...,x^1\_k, x^2\_1,...,x^2\_k,... ,x^n\_1,...,x^n\_k]$ under the action of $S\_n$ where $S\_n$ permutes the variables $x\_i^1,...,x\_i^n$ simultaneously for $i=1,...,k$. I want to know the invariant subring $R^{S\_n}$ in terms of generators and relations. Could anybody help me?



What is the coordinate ring of symmetric product of affine plane?
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>.


