“And every science, when we understand it not as an instrument of power and domination but as an adventure in knowledge pursued by our species across the ages, is nothing but this harmony, more or less vast, more or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the delicate counterpoint of all the themes appearing in turn, as if summoned from the void.”
Alexander Grothendieck
This post is about symmetric polynomials. A symmetric polynomial is \(f(x_1,\dots,x_n)\in \mathbb Z[x_1,\dots,x_n]\) such that for any permutation \(\sigma\in S_n\), we have \(\sigma f=f\) where \(\sigma\) acts by permuting the variables. The symmetric polynomials form a subring of \(\mathbb Z[x_1,\dots,x_n]\) which we denote as \(\Lambda_n\). One example of symmetric polynomials is the Newton power sums \(p_k=x_1^k+\cdots+x_n^k\) for \(k\ge 1\). Another example is elementary symmetric polynomials \(e_k=\sum_{1\le i_1<\cdots< i_k\le n}x_{i_1}\cdots x_{i_k}\) for \(1\le k\le n\). They arise in Vieta’s relations for a polynomial equation. Obviously \(\Lambda_n\) is a graded ring in degree. For a monomial \(u=ax^{k_1}_1\cdots x^{k_n}_n\), we define its weight as \(\mathrm{wt}(u)=k_1+2k_2+\cdots+nk_n\), and for a polynomial \(f\) define \(\mathrm{wt}(f)\) as the largest weight occuring among its monomials. We note that the weight of a polynomial is the same as the degree of \(f(e_1,\dots,e_n)\). One of the first results in symmetric polynomials is that the elementary symmetric polynomials forms a generating set of \(\Lambda_n\) as a (graded) algebra over \(\mathbb Z\). This is our first theorem.