COEFFICIENT QUIVERS, ${\mathbb {F}}_1$ -REPRESENTATIONS, AND EULER CHARACTERISTICS OF QUIVER GRASSMANNIANS

Abstract

AbstractA quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ 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 $\mathrm {Rep}(Q,\mathbb {F}_1)$ 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 with $\mathbb {F}_1$ -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 with $\mathbb {F}_1$ -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 with representations with finite nice length, and compute them for certain families of quivers.