TITLE: What is the coordinate ring of symmetric product of affine plane?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: The relations might be complicated.  The multisymmetric functions of degree up to n generate the ring, but very redundantly.  In Lemma 2.2 of 
http://annals.princeton.edu/annals/2006/163-2/p11.xhtml
Venkatesh and I show that you can get by with using many fewer of these multisymmetric functions, if you are content to generate a subring of R^{S_n} whose fraction field is finite-index in the fraction field of R^{S_n}.