How can I split a polynomial into parts which are symmetric and antisymmetric under exchange of the variables? 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. In the following image you can see the polynomial as well as further explanations and some ideas I already had.
If $f(x_1, x_2, \dots x_n)$ is a polynomial, then its symmetric part is
$$\frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, x_{\sigma(2)}, \dots x_{(\sigma(n)})$$
and its antisymmetric part is
$$\frac{1}{n!} \sum_{\sigma \in S_n} \text{sgn}(\sigma) f(x_{\sigma(1)}, x_{\sigma(2)}, \dots x_{(\sigma(n)}).$$
Here $S_n$ is the group of permutations of $n$ objects and $\text{sgn}(\sigma)$ is the sign of a permutation.
However, when $n \ge 3$ it is just not true that a polynomial is the sum of its symmetric part and its antisymmetric part. In general there are other parts that behave in more complicated ways under the action of $S_n$ given by the other irreducible representations of $S_n$.