Subdivisions of root polytopes and generalized tropical oriented matroids (Extended abstract)

Abstract

We study a generalization of tropical oriented matroids by Ardila and Develin, and show that they are in bijection with subdivisions of root polytopes, which are sub-polytopes of a product of two simplices.

Figures (3)

• Figure 1: A graph $G$ (left) and the corresponding $P_G$ (right), formed by taking the Minkowski sums of $4$ faces of $\Delta^2$ (middle). Here $n = 4$ and $d = 3$.
• Figure 2: A subdivision of $\Delta^{1}\times \Delta^{2}$ and the corresponding subdivision of $2\Delta^{2}$.
• Figure 3: An arrangement of $4$ tropical pseudo-hyperplanes in $\mathbb{TP}^2$ in dashed lines, obtained from a mixed subdivision of $4\Delta^2$.

Theorems & Definitions (26)

• Definition 2.1: santos2003cayley, Section 1.1
• Definition 2.2: santos2003cayley, Section 1.2
• Definition 2.3: postnikov2009permutohedra, Sections 9 and 12
• Example 2.4
• Theorem 2.5: postnikov2009permutohedra, Corollary 14.6
• Proposition 2.6: ardila2007tropical, Lemma 6.1 and galashin2018trianguloids, Proposition 7.5
• Definition 2.7
• Theorem 2.8: ardila2007tropical, Theorem 6.2, generalized
• Remark 2.9
• Definition 3.1: ardila2007tropical, Definition 3.1, 3.4