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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.

Thomas
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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.

Marc
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    Sorry, I think I was not explicit enough. I was talking about the symmetry under exchange of variables. So I search for splitting the polynomial into parts which are symmetric or antisymmetric under exchange of any of the three variables. So I think it would be the best to ask the question again, just more specifically. Thanks for your help! – Thomas Mar 25 '15 at 18:23