On quiver representations over $\mathbb{F}_1$

Abstract

We study the category $\textrm{Rep}(Q,\mathbb{F}_1)$ of representations of a quiver $Q$ over "the field with one element", denoted by $\mathbb{F}_1$, and the Hall algebra of $\textrm{Rep}(Q,\mathbb{F}_1)$. Representations of $Q$ over $\mathbb{F}_1$ often reflect combinatorics of those over $\mathbb{F}_q$, but show some subtleties - for example, we prove that a connected quiver $Q$ is of finite representation type over $\mathbb{F}_1$ if and only if $Q$ is a tree. Then, to each representation $\mathbb{V}$ of $Q$ over $\mathbb{F}_1$ we associate a coefficient quiver $\Gamma_\mathbb{V}$ possessing the same information as $\mathbb{V}$. This allows us to translate representations over $\mathbb{F}_1$ purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of $Q$ over $\mathbb{F}_1$ - there are also similarities to representations over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an $n$-loop quiver over $\mathbb{F}_1$ with the Hopf algebra of skew shapes introduced by Szczesny.