DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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Structure of spaces of rhombus tilings in the lexicograhic case

Éric Rémila


Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of lexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence (v1, v2,…, vD) of vectors of ℝd and a sequence (m1, m2,…, mD) of positive integers. We assume (lexicographic hypothesis) that for each subsequence (vi1, vi2,…, vid) of length d, we have det(vi1, vi2,…, vid) > 0. The zonotope Z is the set { Σαivi  0 ≤αi ≤mi }. Each prototile used in a tiling of Z is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of Z is a graded poset, with minimal and maximal element.

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