Polytropes and Tropical Eigenspaces: Cones of Linearity

TL;DR

This paper studies the piecewise linear structure of the polytrope map for matrices, revealing a polytopal fan structure that refines the eigenvector cones and offers a new combinatorial classification of polytropes and tropical eigenspaces.

Contribution

It establishes that cones of linearity form a polytopal fan with a face lattice linked to connected relations, advancing the understanding of tropical eigenspaces.

Findings

Cones of linearity form a polytopal fan partition.

The fan refines the eigenvector map partition.

Provides a new combinatorial classification of polytropes.

Abstract

The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\R^{n \times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces.