%0 Generic %1 ArdilaBilley2006 %A Ardila, Federico %A Billey, Sara %D 2006 %K Ardila Billey Schubert Vakil arrangement base combinatorics flag hyperplane intersection line manifold matroid product simplex space triangulation tropics %T Flag arrangements and triangulations of products of simplices %U http://arxiv.org/abs/math/0605598 %X 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 Delta_n-1 x \Delta_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.

@misc{ArdilaBilley2006, 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 Delta_{n-1} x \Delta_{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. }, added-at = {2008-12-23T17:15:35.000+0100}, author = {Ardila, Federico and Billey, Sara}, biburl = {https://www.bibsonomy.org/bibtex/2cf2300f9ca5cf491c389946c63739716/fwd13}, description = {Flag arrangements and triangulations of products of simplices}, interhash = {9c6338afa0286d480e67178049c1d50b}, intrahash = {cf2300f9ca5cf491c389946c63739716}, keywords = {Ardila Billey Schubert Vakil arrangement base combinatorics flag hyperplane intersection line manifold matroid product simplex space triangulation tropics}, note = {cite arxiv:math/0605598 Comment: 39 pages, 12 figures, best viewed in color}, timestamp = {2008-12-23T17:28:13.000+0100}, title = {Flag arrangements and triangulations of products of simplices}, url = {http://arxiv.org/abs/math/0605598}, year = 2006 }