Accumulation points of real Schur roots

TL;DR

This paper investigates the accumulation points of real Schur roots in the root system of acyclic quivers, providing new insights into their properties and a complete description for weakly hyperbolic types.

Contribution

It introduces new properties of canonical decompositions and uses them to describe both rational and irrational accumulation points of real Schur roots.

Findings

Rational accumulation points are characterized via canonical decompositions.

Complete description of irrational accumulation points for weakly hyperbolic quivers.

New properties of canonical decompositions of dimension vectors.

Abstract

Let $k$ be an algebraically closed field and $Q$ be an acyclic quiver with $n$ vertices. Consider the category ${\rm rep}(Q)$ of finite dimensional representations of $Q$ over $k$. The exceptional representations of $Q$, that is, the indecomposable objects of ${\rm rep}(Q)$ without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to $Q$. These roots are special elements of the Grothendieck group $\mathbb{Z}^n$ of ${\rm rep}(Q)$. When we identify the dimension vectors of the representations (that is, the non-negative vectors of $\mathbb{Z}^n$) up to positive multiple, we see that the real Schur roots can accumulate in some directions of $\mathbb{R}^n \supset \mathbb{Z}^n$. This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case $Q$ is of weakly hyperbolic type.