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

TL;DR

This paper explores the combinatorial structure of quiver representations over the field with one element, focusing on pseudotree quivers, classifying their indecomposable representations, and linking these to Lie algebras and Grassmannian Euler characteristics.

Contribution

It provides a complete classification of indecomposable representations over _1 for pseudotree quivers and establishes structural results connecting these to Lie algebras and combinatorial Grassmannian computations.

Findings

Classified asymptotic behaviors of indecomposable _1-representations for pseudotrees

Proved structural results about Lie algebras associated to pseudotrees

Constructed _1-representations with combinatorial Euler characteristic computations

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 $\textrm{Gr}^Q_{\underline{d}}(M)$ can be computed in a purely combinatorial way.