Face monoid actions and tropical hyperplane arrangements

TL;DR

This paper explores the combinatorial structure of tropical hyperplane arrangements and their connection to classical hyperplane face monoids, revealing new actions and characterizations in tropical geometry.

Contribution

It introduces a novel action of the classical braid arrangement's face monoid on tropical hyperplane arrangements and provides a new characterization of point types in tropical geometry.

Findings

Hyperplane face monoid acts on tropical arrangements

New characterization of tropical point types via partial bijections

Establishment of links between tropical and classical hyperplane structures

Abstract

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropical hyperplane arrangement, in terms of partial bijections which attain permanents of submatrices of a matrix which naturally encodes the arrangement.