We study the category Rep(Q, F 1 ) of representations of a quiver Q over "the field with one element", denoted by F 1 , and the Hall algebra of Rep(Q, F 1 ). Representations of Q over F 1 often reflect combinatorics of those over F q , but show some subtleties -for example, we prove that a connected quiver Q is of finite type over F 1 if and only if Q is a tree. Then, to each representation V of Q over F 1 , we associate a combinatorial gadget Γ V , which we call a colored quiver, possessing the same information as V. This allows us to translate representations over F 1 purely in terms of combinatorics of associated colored quivers. We also explore the growth of indecomposable representations of Q over F 1 searching for the tame-wild dichotomy over F 1 -this also shows a similar tame-wild dichotomy 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 F 1 with the Hopf algebra of skew shapes introduced by Szczesny.

A quiver representation assigns a vector space to each vertex, and a linear map to each arrow. When one considers the category Vect(F 1 ) of vector spaces "over F 1 " (the field with one element), one obtains F 1 -representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficients quivers. To be precise, we prove that the category Rep(Q, 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 "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 F 1 -representations. These techniques apply to a large class of F 1 -representations, which we call the F 1 -representations with finite nice length: we prove sufficient conditions for an 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 F 1 -representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent 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.

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