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

TL;DR

This paper explores the category of quiver representations over the hypothetical field with one element, revealing combinatorial structures, finite representation types, and connections to Hall algebras and skew shapes.

Contribution

It introduces a combinatorial approach to quiver representations over _1, characterizes finite representation types, and links Hall algebras to skew shapes.

Findings

Connected quivers are of finite type iff they are trees.

Representations over _1 can be described via coefficient quivers.

Hall algebra of nilpotent representations relates to skew shapes.

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.