Tropical matrix groups

TL;DR

This paper investigates the structure of maximal subgroups within the semigroup of tropical matrices, revealing their isomorphism to automorphism groups of tropical polytopes and their composition as products of real numbers and finite groups.

Contribution

It characterizes the maximal subgroups of tropical matrix semigroups, showing their isomorphism to automorphism groups of tropical polytopes and their embedding into units of tropical matrices.

Findings

Maximal subgroups are isomorphic to automorphism groups of tropical polytopes.

Each maximal subgroup is a direct product of real numbers and a finite group.

Every automorphism of a full rank tropical polytope extends to the ambient space.

Abstract

We study the subgroup structure of the semigroup of finitary tropical matrices under multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of the real numbers with a finite group. We also show that there is a natural and canonical embedding of each full rank maximal subgroup into the group of units of the semigroup of matrices over the tropical semiring with minus infinity. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space, and that every full rank subgroup has a common eigenvector.