Representation theory over semifields

TL;DR

This work establishes a foundational framework for representation theory over idempotent semifields, linking algebraic and combinatorial perspectives motivated by tropical geometry and matroid theory. It shows that representations of a torsion group $G$ over a semifield $K$ can be classified via basis lines and $G$-sets, with a bijection between indecomposables and conjugacy classes of subgroups $H\subseteq G$ and explicit descriptions of morphisms through double cosets $H_V\backslash G / H_W$. Over the Boolean semifield $\mathbb{B}$ these results simplify, yielding a transitive-basis-lines picture and a quasi-freeness property for $\mathbb{B}[G]$-modules, while for algebraic groups the representations correspond to equivariant vector bundles on $\mathrm{Spec}\,K$. The paper thus bridges tropical representation theory and semiring/module theory, setting the stage for applications to tropical/matroidal representations and subsequent work on matroids of low rank.

Abstract

We study and classify representations of a torsion group $G$ over an idempotent semifield with special attention on the case over the Boolean semifield $\mathbb{B}$. In subsequent work we extend this theory to studying representations of matroids of low rank.

Theorems & Definitions (59)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3