# Generalizing the Fundamental Theorem of Symmetric Polynomials

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.

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I think this is what arxiv.org/pdf/math/0205233.pdf is about. –  darij grinberg Feb 23 '12 at 22:20
Post that as an answer and I'll accept it, thanks! –  Owen Biesel Feb 24 '12 at 4:46

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!

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Even better - thank you so much! –  Owen Biesel Feb 24 '12 at 13:13
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