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 .