TITLE: Generalizing the Fundamental Theorem of Symmetric Polynomials
QUESTION [10 upvotes]: The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary symmetric polynomials $e_i(\bar{x})$, for $i$ between $1$ and $n$.  I'm looking for a reference in the literature for a similar theorem in more variables, which should look something like this:
Consider the action of $S_n$ on $\mathbb{Z}[x_1,\ldots,x_n,y_1,\ldots,y_n]=\mathbb{Z}[\bar{x},\bar{y}]$ given by permuting the $x_i$ and the $y_i$ simultaneously.  The fixed subring $\mathbb{Z}[\bar{x},\bar{y}]^{S_n}$ is generated by the elementary symmetric polynomials $e_i(\bar{m})$, where $m=m(x,y)$ is a monomial.  (For example, if $m(x,y)=x^2y$, then $e_1(\bar{m}) = x_1^2y_1 + x_2^2y_2 + \ldots$.)
As an example, consider the $S_2$-invariant polynomial $(x_1+y_1)(x_2+y_2)$. It can be written as $(x_1+x_2)(y_1+y_2) - (x_1 y_1 + x_2 y_2) + (x_1x_2) + (y_1y_2)$, i.e. $e_1(\bar x)e_1(\bar y) - e_1(\overline{xy}) + e_2(\bar{x}) + e_2(\bar y)$.
I'd also be interested to know what the relations are in such a presentation of $\mathbb{Z}[\bar{x},\bar{y}]^{S_n}$.  Certainly we can do better by excluding the monomials $x^m$ and $y^m$ for $m\geq 2$, as each such $e_i(\bar x^m)$ is already covered by the ordinary fundamental theorem.  There also seem to be a handful of other relations around $i=n$, such as the observation that $e_n(\overline{xy}) = e_n(\bar x)e_n(\bar y)$, and possibly others.

REPLY [3 votes]: Another systematic study of multisymmetric polynomials is in the thesis of Emmanuel Briand
available in .ps format at:
http://emmanuel.jean.briand.free.fr/publications/polms/
see also his other publications at:
http://emmanuel.jean.briand.free.fr/publications/
Although not well-known the study of multisymmetric functions is very old and related
to the search for explicit formulas for multidimensional resultants.
Among the classics who worked on this: Poisson, Schlafli, Brill, Gordan, Junker.
A good account of some of this old work is in the book by Faa di Bruno
"Theorie general de l'elimination", p. 109--114 and 129--131. 
Another little-known reference is the beginning of Lecture Notes in Math no. 896 by Bernard Angeniol
"Familles de Cycles Algebriques - Schema de Chow".