Splitting a polynomial into a symmetric and an antisymmetric part.

Question

How can I split a polynomial into a symmetric and an antisymmetric part? I have an explicit polynomial, which is a function of three variables (and some further constants). The symmetry properties should be with respect to all three variables.

Answer 1

Suppose you have a polynomial $f(X,Y,Z) = X^aY^bZ^c$ of your polynomial. Then $f(-X,-Y,-Z) = (-1)^{a+b+c}X^aY^bZ^c$, which equals $f(X,Y,Z)$ if and only if $a+b+c$ is divisible by two. Therefore the symmetric parts are those terms which have an even sum of powers, and the antiymmetric terms are two whcih have an odd sum of powers.