Are the elementary symmetric polynomials "unique"? The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary symmetrics.
There are other finite sets of polynomials that generate the set of symmetric polynomials in this way, the most obvious one being the family $(X_1, ... X_n)$. Those aren't symmetric polynomials, but even if we look for a finite set of symmetric polynomials that generate the set of all symmetric polynomials, there are other possibilities - I think.


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*Am I correct that the symmetric polynomials aren't the only finite family of symmetric polynomials that generates the total set of symmetric polynomials?

*In that case, is there anything intrinsically special about the elementary symmetrics? Some property of the the way in which they generate the rest of the set which characterizes them? In other words, can they be defined without simply writing down their formulae? Or does their interest simply come from the fact that they happen to be how the roots of a polynomial express the coefficients?

 A: In answer to the first question: you're correct. Consider the power sum symmetric polynomials $\sum_i X_i{}^k$: the Newton-Girard formulae show how to express the elementary symmetric polynomials in terms of the power sum symmetric polynomials.
A: Peter Taylor's answer is very much ok. I just want to add two remarks.


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*The power sums will not always generate all the symmetric polynomials. They do over a field of characteristic zero, but using them over rings of a positive characteristic introduces problems, because Newton-Girard formulas impose the need to divide with an integer ($\le k=$ the degree of the polynomial). For example over the field $\Bbb{F}_2$ we cannot write the product $x_1x_2$ using the power sums $p_1=x_1+x_2$ and $p_2=x_1^2+x_2^2$, because here we have $p_2=p_1^2$ as a consequence of the freshman's dream. The elementary symmetric polynomials generate the ring of symmetric polynomials over any ring of coefficients with any characteristic (Theorem 2.20 in Jacobson's Basic Algebra I).

*The elementary symmetric polynomials are invariant under a finite group generated by reflections: 
$$s_{ij}:(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)\mapsto (x_1,\ldots,x_j,\ldots,x_i,\ldots,x_n).$$ A similar theory of invariants is available for all finite groups generated by reflections. Unlike in the case of the permutation groups the degree of the basic invariants (here $1,2,3,\ldots,n$) are more varied. For example, with the bigger group of signed permutations a polynomial must have even degree w.r.t all the variables to be invariant. Read Humphreys' book to learn more. A classical result (Chevalley?) tells that we always get an algebraically independent set of generators for the algebra of invariants.

