Polyhedral Gauss sums, and polytopes with symmetry
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...
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| Format: | Journal Article |
| Language: | English |
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2016
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| Online Access: | https://hdl.handle.net/10356/80394 http://hdl.handle.net/10220/40540 http://arxiv.org/abs/1508.01876 |
| _version_ | 1811692220976201728 |
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| author | Malikiosis, Romanos-Diogenes Robins, Sinai Zhang, Yichi |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Malikiosis, Romanos-Diogenes Robins, Sinai Zhang, Yichi |
| author_sort | Malikiosis, Romanos-Diogenes |
| collection | NTU |
| description | 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). |
| first_indexed | 2024-10-01T06:32:20Z |
| format | Journal Article |
| id | ntu-10356/80394 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:32:20Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | ntu-10356/803942023-02-28T19:30:03Z Polyhedral Gauss sums, and polytopes with symmetry Malikiosis, Romanos-Diogenes Robins, Sinai Zhang, Yichi School of Physical and Mathematical Sciences Gauss sum lattice Weyl group multi-tiling polyhedron solid angle Gram relations 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). Published version 2016-05-13T07:05:20Z 2019-12-06T13:48:29Z 2016-05-13T07:05:20Z 2019-12-06T13:48:29Z 2016 2016 Journal Article Malikiosis, R.-D., Robins, S., & Zhang, Y. (2016). Polyhedral Gauss sums, and polytopes with symmetry. Journal of Computational Geometry, 7(1), 149-170. 1920-180X https://hdl.handle.net/10356/80394 http://hdl.handle.net/10220/40540 http://arxiv.org/abs/1508.01876 191059 en Journal of Computational Geometry © 2016 The Author(s) (Journal of Computational Geometry). This article is distributed under the terms of the Creative Commons Attribution International License. 22 p. application/pdf |
| spellingShingle | Gauss sum lattice Weyl group multi-tiling polyhedron solid angle Gram relations Malikiosis, Romanos-Diogenes Robins, Sinai Zhang, Yichi Polyhedral Gauss sums, and polytopes with symmetry |
| title | Polyhedral Gauss sums, and polytopes with symmetry |
| title_full | Polyhedral Gauss sums, and polytopes with symmetry |
| title_fullStr | Polyhedral Gauss sums, and polytopes with symmetry |
| title_full_unstemmed | Polyhedral Gauss sums, and polytopes with symmetry |
| title_short | Polyhedral Gauss sums, and polytopes with symmetry |
| title_sort | polyhedral gauss sums and polytopes with symmetry |
| topic | Gauss sum lattice Weyl group multi-tiling polyhedron solid angle Gram relations |
| url | https://hdl.handle.net/10356/80394 http://hdl.handle.net/10220/40540 http://arxiv.org/abs/1508.01876 |
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