Tropical representations and valuated matroids

TL;DR

This paper investigates tropical subrepresentations of linear group representations over the tropical semifield, establishing their equivalence to valuated matroidal representations and analyzing related modules and automorphisms.

Contribution

It introduces a novel intrinsic description of tropical subrepresentations using quasi-free modules and connects them to valuated matroids, advancing tropical linear algebra theory.

Findings

Tropical subrepresentations are characterized via quasi-free modules.

Tropical subrepresentations are equivalent to valuated matroidal representations.

Analysis of automorphisms of weakly free modules and tropical prevarieties.

Abstract

We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a semiring: weakly free and quasi-free modules. We also investigate subgroups of $\text{GL}_n(K)$ for $K=\mathbb{T}$, $ \mathbb{R}_{\geq 0}$, and automorphisms of weakly free modules and tropical prevarieties defined by tropical linear equations. As an application of our results, we provide an intrinsic description of tropical subrepresentation via certain quasi-free modules, and prove that a tropical subrepresentation is equivalent to a valuated matroidal representation.