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

Abstract

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.

Proto-exact categories of modules over semirings and hyperrings

“…For instance, this is the case of matroids, finite pointed sets, and representations of a quiver over "the field with one element". See [EJS20], [Szc12], [JS20a], [JS21], [Szc18].…”

“…Roughly speaking, a proto-exact category is a pointed category with two distinguished classes of morphisms (admissible monomorphisms and admissible epimorphisms) satisfying certain conditions on pullback and pushout diagrams from which one can obtain a notion of admissible exact sequences. 1 Several interesting "combinatorial" categories are equipped with a proto-exact structure, for instance, the category of matroids [EJS20], the category of representations over a quiver (and more generally any monoid) over "the field with one element" [Szc12], [JS20a], [JS21]. Categories with more algebro-geometric flavors, which are not additive, have been explored in [Szc18], [JS20b], [ELY20].…”

Matroids with coefficients and F1-geometry

“…Namely, Jun and Sistko's work [JS21] provides a nice sequence of gradings of Λ that distinguishes elements. These gradings determine the weights of actions…”