Polyhedral Gauss Sums, and polytopes with symmetry

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $\mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $\mathcal G$ be the group generated by $\mathcal{W}$ as well as all integer translations in $\mathbb Z^d$. We prove that if $P$ multi-tiles $\mathbb R^d$ under the action of $\mathcal G$, then we have the closed form $G_P(n) = \text{vol}(P) G(n)^d$. Conversely, we also prove that if $P$ is a lattice tetrahedron in $\mathbb R^3$, of volume $1/6$, such that $G_P(n) = \text{vol}(P) G(n)^d$, for $n \in \{ 1,2,3,4 \}$, then there is an element $g$ in $\mathcal G$ such that $g(P)$ is the fundamental tetrahedron with vertices $(0,0,0)$, $(1, 0, 0)$, $(1,1,0)$, $(1,1,1)$.