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Structure of spaces of rhombus tilings in the lexicograhic case

Abstract : 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 $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. 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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Éric Rémila. Structure of spaces of rhombus tilings in the lexicograhic case. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.145-150, ⟨10.46298/dmtcs.3400⟩. ⟨hal-01184356⟩

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