On the Combinatorics of ${\mathbb{F}}_1$-Representations of Pseudotree Quivers

Abstract

We investigate quiver representations over ${\mathbb{F}}_1$. Coefficient quivers are combinatorial gadgets equivalent to ${\mathbb{F}}_1$-representations of quivers. We focus on the case when the quiver $Q$ is a pseudotree. For such quivers, we first use the notion of coefficient quivers to provide a complete classification of asymptotic behaviors of indecomposable representations over ${\mathbb{F}}_1$. Then, we prove some fundamental structural results about the Lie algebras associated to pseudotrees. Finally, we construct examples of ${\mathbb{F}}_1$-representations $M$ of a quiver $Q$ by using coverings, under which the Euler characteristics of the quiver Grassmannians ${\text{Gr}}_{\underline{d}}^Q(M)$ can be computed in a purely combinatorial way.

26 pages