Here is an example of a full-rank vertex of $\hom(\Box_3, \Diamond_4)$, where $\Box_3$ is the unit cube in the positive orthant and $\Diamond_4$ is the standard 4D cross polytope (ie., with vertices at $\pm e_i$); the vertex is expressed in two ways:

v0*F11,v0*F15,

v1*F4,v1*F7,v1*F9,v1*F11,

v2*F4,v2*F5,v2*F6,v2*F7,

v3*F6,v3*F10,

v4*F12,v4*F13,v4*F14,v4*F15,

v5*F9,v5*F13,

v6*F5,v6*F8,

v7*F8,v7*F10,v7*F12,v7*F14:

1 1/3 -1/3 -1/3 1/3 0 -1/3 -1/3 1/3 1/3 -1/3 0 -1/3 -1/3 -1/3 2/3 1/3

Can I have Polymake express this vertex as an explicit affine map, i.e., a linear map and a shift vector? Or is there some way that I can find that expression in terms of what is shown here? None of the vertices of $p$ are sent to vertices of $q$, but rather to faces of dimension 1 or 2, so the vertex-facet pairing doesn't give precise coordinates for the images of the vertices of $p$.

The motivation for this question: We were not expecting to find such full-rank vertex embeddings, and we want to figure out how to count them.

Thank you in advance for any help.