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.

中文翻译：

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群的拟阵表示

我们通过考虑热带线性空间上的线性作用，开发了幂等半场上群的有限维表示理论的基本原理。这可以被认为是一种热带表示理论，一种典型的模表示理论，或者一种拟阵表示理论——我们从所有三个角度进行了借鉴。在一些一般性质和构造之后，包括 Maschke 定理的弱热带类似物，我们转向研究有限群的正则表示及其热带化。对于阿贝尔群，我们发现初等数论和拟阵理论之间存在有趣的相互作用——甚至循环群也非常丰富——我们总结了一些可能的热带特征理论的第一步。