  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.
