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Title: Polyhedral Gauss sums, and polytopes with symmetry
Authors: Malikiosis, Romanos-Diogenes
Robins, Sinai
Zhang, Yichi
Keywords: Gauss sum
Weyl group
solid angle
Gram relations
Issue Date: 2016
Source: Malikiosis, R.-D., Robins, S., & Zhang, Y. (2016). Polyhedral Gauss sums, and polytopes with symmetry. Journal of Computational Geometry, 7(1), 149-170.
Series/Report no.: Journal of Computational Geometry
Abstract: We define certain natural finite sums of nn'th roots of unity, called GP(n)GP(n), that are associated to each convex integer polytope PP, and which generalize the classical 11-dimensional Gauss sum G(n)G(n) defined over Z/nZZ/nZ, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group WW, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let GG be the group generated by WW as well as all integer translations in ZdZd. We prove that if PP multi-tiles RdRd under the action of GG, then we have the closed form GP(n)=vol(P)G(n)dGP(n)=vol(P)G(n)d. Conversely, we also prove that if PP is a lattice tetrahedron in R3R3, of volume 1/61/6, such that GP(n)=vol(P)G(n)dGP(n)=vol(P)G(n)d, for n∈{1,2,3,4}n∈{1,2,3,4}, then there is an element gg in GG such that g(P)g(P) is the fundamental tetrahedron with vertices (0,0,0)(0,0,0), (1,0,0)(1,0,0), (1,1,0)(1,1,0), (1,1,1)(1,1,1).
ISSN: 1920-180X
Schools: School of Physical and Mathematical Sciences 
Rights: © 2016 The Author(s) (Journal of Computational Geometry). This article is distributed under the terms of the Creative Commons Attribution International License.
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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