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

Question

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?

Answer 1

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.

Answer 2

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}.

Answer 3

I may be making a very trivial mistake [Edit: yes, indeed], but isn't it just that:

$(\mathbb{A}^k)^{(n)}=(\mathbb{A}^k)^n/S_n\cong(\mathbb{A}^n)^k/S_n\cong(\mathbb{A}^n/S_n)^k\cong(\mathbb{A}^n)^k$

with affine coordinate ring $\mathbb{C}[\sigma_1,\cdots,\sigma_n]^{\otimes k}$ (where $\sigma_d=\sigma_d(x_1,\cdots,x_n)$ is the degree-$d$ symmetric function in the $n$ variables $x_1,\cdots, x_n$) ?