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

TL;DR

This paper establishes a connection between triangulations of product simplices and tropical oriented matroids, advancing the combinatorial understanding of subdivisions of these polytopes and proposing new related structures.

Contribution

It proves that any triangulation of elta_{n-1} imes elta_{d-1} encodes a tropical oriented matroid, confirming a conjecture and introducing new combinatorial objects.

Findings

Triangulations encode tropical oriented matroids.

Supports the conjecture of bijection between subdivisions and tropical oriented matroids.

Proposes new combinatorial objects for broader classes of polytopes.

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.