Coefficient Quivers, $\mathbb{F}_1$-Representations, and Euler Characteristics of Quiver Grassmannians

TL;DR

This paper explores representations of quivers over the field with one element, establishing their connection to coefficient quivers, and uses this framework to compute Euler characteristics of quiver Grassmannians and analyze associated Hall algebras.

Contribution

It introduces a new categorical equivalence between $ extrm{Rep}(Q, ext{F}_1)$ and coefficient quivers, and extends techniques to compute Euler characteristics and Hall algebras for $ ext{F}_1$-representations.

Findings

Category $ extrm{Rep}(Q, ext{F}_1)$ is equivalent to coefficient quivers.

Euler characteristics of quiver Grassmannians correspond to $ ext{F}_1$-rational points.

Hall algebra structures are affected by quiver orientation changes.

Abstract

A quiver representation assigns a vector space to each vertex, and a linear map to each arrow. When one considers the category $\textrm{Vect}(\mathbb{F}_1)$ of vector spaces ``over $\mathbb{F}_1$'' (the field with one element), one obtains $\mathbb{F}_1$-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\textrm{Rep}(Q,\mathbb{F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over $Q$. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of ``$\mathbb{F}_1$-rational points'' of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated to $\mathbb{F}_1$-representations. These techniques apply to a large class of $\mathbb{F}_1$-representations, which we call the $\mathbb{F}_1$-representations with finite nice length: we prove sufficient conditions for an $\mathbb{F}_1$-representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated to $\mathbb{F}_1$-representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb{F}_1$-representations of a quiver with bounded representation type. We also discuss Hall algebras associated to representations with finite nice length, and compute them for certain families of quivers.