Matroidal representations of groups

TL;DR

This paper develops a tropical and matroidal representation theory of finite groups over idempotent semifields, exploring properties, regular representations, and connections to number theory and matroids.

Contribution

It introduces a novel tropical and matroidal framework for group representations, including a Maschke-like theorem and initial steps toward a tropical character theory.

Findings

Weak tropical Maschke's theorem established

Rich structure found in cyclic groups via number theory and matroids

Initial concepts for tropical character theory proposed

Abstract

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.