Polyhedral Gauss Sums, and polytopes with symmetry

Abstract

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 $(0,0,0)$, $(1,0,0)$, $(1,1,0)$, $(1,1,1)$.

18 pages, 2 figures