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A quiver representation assigns a vector space to each vertex, and a linearmap to each arrow. When one considers the category$\textrm{Vect}(\mathbb{F}_1)$ of vector spaces ``over $\mathbb{F}_1$'' (thefield with one element), one obtains $\mathbb{F}_1$-representations of aquiver. In this paper, we study representations of a quiver over the field withone element in connection to coefficient quivers. To be precise, we prove thatthe category $\textrm{Rep}(Q,\mathbb{F}_1)$ is equivalent to the (suitablydefined) category of coefficient quivers over $Q$. This provides a conceptualway to see Euler characteristics of a class of quiver Grassmannians as thenumber of ``$\mathbb{F}_1$-rational points'' of quiver Grassmannians. Wegeneralize techniques originally developed for string and band modules tocompute 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 sufficientconditions for an $\mathbb{F}_1$-representation to have finite nice length, andclassify such representations for certain families of quivers. Finally, weexplore the Hall algebras associated to $\mathbb{F}_1$-representations ofquivers. We answer the question of how a change in orientation affects the Hallalgebra of nilpotent $\mathbb{F}_1$-representations of a quiver with boundedrepresentation type. We also discuss Hall algebras associated torepresentations with finite nice length, and compute them for certain familiesof quivers.

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