Tropical subrepresentations of the boolean regular representation in low dimension

TL;DR

This paper classifies tropical subrepresentations of the boolean regular representation for finite groups in low dimensions, revealing a complete correspondence with subgroups in dimension two but not in dimension three.

Contribution

It provides a complete classification of two-dimensional tropical subrepresentations and explores the complexity of three-dimensional cases, including subgroup correspondences.

Findings

Complete classification of 2D tropical subrepresentations

Equivalence with subgroups in dimension two

Existence of 3D subrepresentations not corresponding to subgroups

Abstract

We study two dimensional and three dimensional tropical subrepresentations of the regular representation $\mathbb{B}[G]$ of a finite group over the tropical booleans, utilizing the theory of group representations over a fixed idempotent semifield as developed by Giansiracusa--Manaker. In dimension two we completely classify all two dimensional tropical subrepresentations of $\mathbb{B}[G]$, provide an explicit characterization for the set of bases of the corresponding matroids, and show an equivalence with the subgroups of $G$. In dimension three we show such an equivalence no longer holds. Towards a classification in dimension three we give a collection of tropical subrepresentations corresponding to subgroups of index 2, and we show that in the special case of finite cyclic groups, one can find three dimensional tropical subrepresentations that do not correspond to subgroups in a similar way.