Definition

We will consider polynomials in $n$ variables $x_1,\dots,x_n$ and use the shortcut $p(x)$ instead of $p(x_1,\dots,x_n)$. A permutation $w$ is a one to one map of the set $\{1,\dots,n\}$ to itself. There are $n!$ permutations. The product of permutations $w_1w_2$ is just the composition of maps. We will write $w\cdot x$ for $x_{w(1)},\dots,x_{w(n)}$. An inversion in permutation $w$ is a pair of numbers $1\leq i<j\leq n$, such that $w(i)>w(j)$. A permutation $w$ is called even or odd if the number of inversions is even or odd. The sign of a permutation $w$, $sgn(w)$ is $-1$ if $w$ is odd and $sgn(w)=1$ if $w$ is even.

Exercise:Prove that $sgn(w_1w_2)=sgn(w_2w_1)=sgn(w_1)sgn(w_2)$. $\;\Box$

Symmetric polynomials are polynomials which do not change values if some arguments are switched.

Definition:A polynomial $p(x)$ is called symmetric if $p(x)=p(w\cdot x)$ for any permutation $w$.

For example, let $n=3$, then a polynomial $p(x)=x_1+x_2+x_3$ is symmetric, say $p(13,-5,2)=p(-5,2,13)$. The polynomial $q(x)=x_1+x_2+x_3x_1$ is not symmetric, $q(1,2,3)\neq q(2,1,3)$. Note that $p(x)$ is the sum of all variables, no matter how you shuffle the variables, but if you permute the variables in $q$, you can also obtain expressions $x_2+x_1+x_3x_2$, $x_3+x_2+x_1x_2$ and $x_3+x_1+x_1x_2$.

Exercise:Prove that a polynomial $p(x)$ is symmetric if and only if $p(x)$ does not change under the permutations of variables as an expression. $\;\Box$