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

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.