Flag arrangements and triangulations of products of simplices

By Federico Ardila, Sara C. Billey. 2006

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Abstract We investigate the line arrangement that results from intersecting d complete flags in C n . We give a combinatorial description of the matroid T n , d that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T n , 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 T n , d , the triangulations of the product of simplices Δ n − 1 × Δ 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 non-vanishing ones.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9e96e53e9189c51695cdd4a134d0bdc2f03d6889