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Tohoku Mathematical Journal
SECOND SERIES VOL. 65, NO. 1
|Tohoku Math. J.|
65 (2013), 1-30
VORONOI TILINGS HIDDEN IN CRYSTALS ---THE CASE OF MAXIMAL ABELIAN COVERINGS---
(Received July 14, 2011)
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related to the tropical Abel-Jacobi map of the associated compact tropical curve.
2000 Mathematics Subject Classification.
Primary 52C22; Secondary 05C40, 14T05, 74E15, 82B20, 82D25, 14M25.
Key words and phrases.
Graph, strongly connected, bridgeless, crystal, discrete geometric analysis, geometric invariant theory, standard realization, Voronoi cell, Wigner-Seitz cell, Voronoi tiling, tropical Abel-Jacobi map.
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