Representations of regular trees and invariants of AR-components for generalized Kronecker quivers

Abstract

We investigate the generalized Kronecker algebra ${\mathcal{K}}_r=k{\Gamma }_r$ with $r\ge 3$ arrows. Given a regular component $\mathcal{C}$ of the Auslander-Reiten quiver of ${\mathcal{K}}_r$, we show that the quasi-rank $rk(\mathcal{C})\in {\mathbb{Z}}_{\le 1}$ can be described almost exactly as the distance $\mathcal{W}(\mathcal{C})\in {\mathbb{N}}_0$ between two non-intersecting cones in $\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{W}(\mathcal{C}) \leq rk(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.\] Utilizing covering theory, we construct for each $n\in {\mathbb{N}}_0$ a bijection ${\phi }_n$ between the field $k$ and $\{\mathcal{C}\mid \mathcal{C}\text{regular component},\mathcal{W}(\mathcal{C})=n\}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.