Generalizing the Fundamental Theorem of Symmetric Polynomials

Question

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.

Answer 1

I heard of three relatively recent works in that direction, taking different routes and arriving to interesting information about diagonal invariants.

First, there is the paper of Vaccarino that Darij mentioned in his comment (http://arxiv.org/abs/math/0205233).

Second, there are results of Domokos (http://arxiv.org/abs/0706.2154).

Third, there is a by-product of works of Buchshtaber and Rees on Frobenius n-characters, which also leads to new insight in that direction (http://www.ams.org/mathscinet-getitem?mr=2069166).

Hope that helps!

Answer 2

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".