The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is simply the ring of symmetric functions. The complementary summand--that is, the direct sum of all the other isotypic components--is a free module of rank $n!-1$ over the ring of symmetric functions. I would like to have an explicit basis, or at least an explicit set of generators, for this module.

For example, if $n=2$, we have $\mathbb{C}[x_1,x_2] = \mathbb{C}[x_1,x_2]^{S_2} \oplus \mathbb{C}[x_1,x_2]^{S_2}\cdot (x_1-x_2)$, so we have a basis consisting of the single element $x_1-x_2$. More generally, the isotypic component of the sign representation is equal to symmetric functions times the Vandermonde determinant, so the Vandermonde determinant should probably be one of the elements of my generating set.