Closer look at
This is, essentially, the space of linear functions on , with values in , invariant under the action of .Let us introduce the coordinates on in the following way.
In other words, every element of can be decomposed as a sum where , and . Therefore elements of can be thought of as linear functions from monomials to elements of :
Invariance under implies that is only nonzero when there are no , and moreover the value of on monomials involving and can be calculated from the value on the monomials only involving .
We conclude that can be realized as the space of functions of , taking values in .