The Fundamental Theorem of Symmetric Polynomials

Question

This theorem stays that any symmetric polynomial can be expressed as a polynomial of elementary polynomials. So let's suppose I have a symmetric polynomial $f(x_1,x_2,...,x_n)$ in $R[\mathbb{X}]$. I can find a unique polynomial $g(s_1,s_2,...,s_n)$ in $R[\mathbb{X}]$ so I can express f in term of g, where $s_1,s_2,...,s_n$ are elementary polynomials. My question to you is: Is there a algorithm/set of stepts to determine g? Thanks.

Answer 1

Let $c x_1^{a_1} x_2^{a_2} \dots x_n^{a_n}$ be the lexicographically largest monomial of $f$, that is there are no monomials with strictly larger exponent of $x_1$, and no monomials with $x_1$ exponent ${a_1}$ that have a higher power of $x_2$, and so on. We'll think of this as being the leading term of the polynomial.

Now the key thing to notice is that $e_n^{a_n}e_{n-1}^{(a_{n-1}-{a_n})} \dots e_1^{(a_1-a_2)}$ contains the monomial $x_1^{a_1} x_2^{a_2} \dots x_n^{a_n}$ with coefficient $1$ and all other monomials it contains are smaller lexicographically.

Now the point is you can consider the leading term of $f - c e_n^{a_n}e_{n-1}^{(a_{n-1}-{a_n})} \dots e_1^{(a_1-a_2)}$, and so on. Each step reduces the leading term (in lexicographic order), so this process must eventually terminate, at which point you have written $f$ in terms of the elementary symmetric functions.

Answer 2

You can use divided difference operators to express the polynomial in terms of Schur polynomials, then you can apply the second Jacobi-Trudi formula, which expresses a Schur polynomial as the determinant of a matrix whose entries are elementary symmetric polynomials. Computationally this is rather inefficient, but it is nonrecursive, which is sometimes advantageous.