Representations of Regular Trees and Invariants of AR-Components for Generalized Kronecker Quivers

We investigate the generalized Kronecker algebra ?? _r = kΓ _r with r ≥ 3 arrows. Given a regular component ?? of the Auslander-Reiten quiver of ?? _r , we show that the quasi-rank rk(??) ∈ ?_≤1 can be described almost exactly as the distance ??(??) ∈ ?₀ between two non-intersecting cones in ??, 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

$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$

Utilizing covering theory, we construct for each n ∈ ?₀ a bijection φ _n between the field k and {??∣?? regular component, ??(??) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.