2018/10/19 by Giansiracusa, Noah, Manaker, Jacob
#05B35 #05E10 #12K10 #14T05 #20C20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1810.08674
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.