dc.contributor.authorMalikiosis, Romanos-Diogenes
dc.contributor.authorRobins, Sinai
dc.contributor.authorZhang, Yichi
dc.date.accessioned2016-05-13T07:05:20Z
dc.date.available2016-05-13T07:05:20Z
dc.date.copyright2016
dc.date.issued2016
dc.identifier.citationMalikiosis, R.-D., Robins, S., & Zhang, Y. (2016). Polyhedral Gauss sums, and polytopes with symmetry. Journal of Computational Geometry, 7(1), 149-170.en_US
dc.identifier.issn1920-180Xen_US
dc.identifier.urihttp://hdl.handle.net/10220/40540
dc.description.abstractWe 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).en_US
dc.format.extent22 p.en_US
dc.language.isoenen_US
dc.relation.ispartofseriesJournal of Computational Geometryen_US
dc.rights© 2016 The Author(s) (Journal of Computational Geometry). This article is distributed under the terms of the Creative Commons Attribution International License.en_US
dc.subjectGauss sumen_US
dc.subjectlatticeen_US
dc.subjectWeyl groupen_US
dc.subjectmulti-tilingen_US
dc.subjectpolyhedronen_US
dc.subjectsolid angleen_US
dc.subjectGram relationsen_US
dc.titlePolyhedral Gauss sums, and polytopes with symmetryen_US
dc.typeJournal Article
dc.contributor.schoolSchool of Physical and Mathematical Sciencesen_US
dc.description.versionPublished versionen_US
dc.identifier.rims191059
dc.identifier.urlhttp://arxiv.org/abs/1508.01876


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