Tropical subrepresentations of the boolean regular representation in low dimension

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.