Structure of spaces of rhombus tilings in the lexicograhic case

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 R 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 { P αivi 0 ≤ αi ≤ mi}. Each prototile used in a tiling of Z is a rhombohedron constructed from a subsequence of d vectors.