[hal-01184356] Structure of spaces of rhombus tilings in the lexicograhic case

Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence (v_1, v_2,\ldots, v_D) of vectors of ℝ^d and a sequence (m_1, m_2,\ldots, m_D) of positive integers. We assume (lexicographic hypothesis) that for each subsequence (v_i_1, v_i_2,\ldots, v_i_d) of length d, we have det(v_i_1, v_i_2,\ldots, v_i_d) > 0. The zonotope Z is the set \ Σα _iv_i 0 ≤α _i ≤m_i \. 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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