I'm working on a problem where I have an embedding of the form: $$ K:\mathbb R^n\rightarrow \mathbb R^{2^n}:(x_1,x_2,\cdots,x_n)\mapsto(1,x_1,x_2,\cdots,x_1x_2,x_1x_3,\cdots,x_1\cdots x_n) $$ In case that's hard to parse, the new vector can be broken up into pieces of size $_nC_m$, each piece consisting of all the possible products of $m$ elements of the original vector.
Now let $\pi\in S_n$ be a permutation. I'm interested in constructing the representation of $\pi$, $\mathcal D^K_\pi$, so that $$ K(\pi{\mathbf x}) =\mathcal D^K_\pi K({\mathbf x}) $$
For small $n$, I can construct these matrices tediously by hand, and I hardcode the results. I'd like to know a bit more about the general theory of how to do this for any finite $n$, so that my code could generate these matrices on the fly.
Any comments or reading suggestions would be greatly appreciated.