Matroidal representations of groups

We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic one modular representation theory, or a matroidal representation theory—and we draw from all three perspectives. After some general properties and constructions, including a weak tropical analogue of Maschke’s theorem, we turn to a study of the regular representation of a finite group and its tropicalization. For abelian groups we find an interesting interplay between elementary number theory and matroid theory—even cyclic groups are surprisingly rich—and we conclude with some possible first steps toward a tropical character theory.