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

Abstract

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.

11 pages and 3 figures. Any comment or feedback would be welcomed v2. Our result is that triangulations of product of simplices is a tropical oriented matroid. We are trying to extend this to all subdivisions. v3 Replaces the proof of Lemma 2.6 with a reference.. Proof of the matrix being totally unimodular is now more detailed. Extended abstract will be submitted to FPSAC '11