symmetric polynomial

We are going to be dealing with polynomials in n variables x₁, x₂, ..., x_n. Such a polynomial is called symmetric if any rearrangement of the variables leaves the polynomial unchanged.

Formally, a polynomial f(x₁,...,x_n) is symmetric iff f(x₁,...,x_n)=f(x_s(1),...,x_s(n)) for any permutation s in the nth Symmetric group S_n

For example,

• If n=1, so there is just one variable, then every polynomial is symmetric.
• If n=2, then
  • x₁ + x₂ is symmetric.
  • x₁x₂ is symmetric.
  • x₁ + (x₂)² is not. (Because when we swap over x₁ and x₂ for this polynomial we get x₂ + (x₁)².)
• If n=3, then
  • x₁ + x₂ is not symmetric because if we consider the permutation that sends 1 -> 2 -> 3 and apply it to our polynomial we get x₂ + x₃.
  • x₁ + x₂ + x₃ is symmetric though and so is (x₁)² + (x₂)² + (x₃)².

The elementary symmetric polynomials are defined as follows:

• First we take the sum of all the variables
  s₁=Sum(1<= i <=n) x_i. Thus, s₁=x₁+x₂ + ... + x_n.
• Next we take the sum of all products of pairs of distinct variables
  s₂=Sum(1<= i < j <=n) x_ix_j.
• Then we take the sum of all products of three distinct variables
  s₃=Sum(1<= i < j < k <=n) x_ix_jx_k.
• ...
• Finally we take the product of all of the variables s_n=x₁...x_n

It is not too hard to see that s₁, ..., s_n are symmetric polynomials.

In the case n=2 we get

• s₁=x₁ + x₂.
• s₂=x₁x₂.

For n=3 we get

• s₁=x₁ + x₂ + x₃.
• s₂=x₁x₂ + x₁x₃ + x₂x₃.
• s₃=x₁x₂x₃.

It turns out that any symmetric polynomial can be constructed out of these elementary ones.

Theorem Let f(x₁,...,x_n) be a symmetric polynomial over a field then there exists a polynomial g(x₁,...,x_n) such that f(x₁,...,x_n)=g(s₁,...,s_n).

Here is a proof of the fundamental theorem on symmetric polynomials.