Triangulations of $Δ_{n-1} \times Δ_{d-1}$ and Tropical Oriented Matroids

Details Triangulations of $Δ_{n-1} \times Δ_{d-1}$ and Tropical Oriented Matroids Triangulations of $Δ_{n-1} \times Δ_{d-1}$ and Tropical Oriented Matroids

2010-09-24

arxiv.org/abs/1009.4750v3

Mathematics

Combinatorics

11 pages and 3 figures. Any comment or feedback would be welcomed v2. Our result is that triangulations of product of simplice

Scientific paper

Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.

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