Summary: 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
We prove that the bases of the matroid Tn,3 characterize the trian-
gles 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 n-1 × d-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