Polyhedral Gauss Sums, and polytopes with symmetry

TL;DR

This paper introduces a generalization of Gauss sums associated with convex integer polytopes, exploring their properties under symmetry groups and characterizing polytopes that satisfy specific sum formulas, especially in relation to tiling and lattice tetrahedra.

Contribution

It defines polyhedral Gauss sums $G_P(n)$, proves their closed form for multi-tiling polytopes under symmetry groups, and characterizes certain lattice tetrahedra satisfying sum identities.

Findings

If $P$ multi-tiles $ ^d$, then $G_P(n) = ext{vol}(P) G(n)^d$.

For lattice tetrahedra of volume 1/6, the sum formula holds for $n=1,2,3,4$ implies $P$ is equivalent to a fundamental tetrahedron.

The work connects geometric tiling properties with algebraic sum identities.

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