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

Develin and Sturmfels showed that regular triangulations of $Δ_{n-1} \times Δ_{d-1}$ 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 $Δ_{n-1} \times Δ_{d-1}$. In this paper, we show that any triangulation of $Δ_{n-1} \times Δ_{d-1}$ 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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