Polyhedral Gauss sums, and polytopes with symmetry

We define certain natural finite sums of n’th roots of unity, called GP (n), that are associated to each convex integer polytope P , and which generalize the classical 1-dimensional Gauss sum G(n) defined over Z/nZ, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group W, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let G be the group generated by W as well as all integer translations in Zd. We prove that if P multi-tiles Rd under the action of G, then we have the closed form GP (n) = vol(P )G(n) d. Conversely, we also prove that if P is a lattice tetrahedron in R3, of volume 1/6, such that GP (n) = vol(P )G(n)d, for n ∈ {1, 2, 3, 4}, then there is an element g in G such that g(P ) is the fundamental tetrahedron with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1). 2010 Mathematics Subject Classification. Primary: 11L05, 52C22, 05B45. Secondary: 52B10, 52B15, 51M20.