Flag arrangements and triangulations of products of simplices

We investigate the line arrangement that results from intersecting d complete flags in Cn. We give a combinatorial description of the matroid Tn,d that keeps track of the linear dependence relations among these lines.We prove that the bases of the matroid Tn,3 characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid Tn,d, the triangulations of the product of simplices Δn−1×Δd−1, and the arrangements of d tropical hyperplanes in tropical (n−1)-space.Our work provides a simple and effective criterion to ensure the vanishing of many Schubert structure constants in the flag manifold, and a new perspective on Billey and Vakil's method for computing the non-vanishing ones.