Widths of regular components for an $n$-regular tree $T(n)$
Colloquium Mathematicum
MSC: Primary 16G20; Secondary 16G70
DOI: 10.4064/cm9392-6-2026
Published online: 29 July 2026
Abstract
Let $(T(n),\varOmega )$ be the covering of the generalized Kronecker quiver $K(n)$, where $\varOmega $ is a bipartite orientation. Given a regular Auslander–Reiten component $\mathcal {D}$ of ${\rm mod}(T(n),\varOmega )$, we introduce two invariants: the width $\mathcal {W}(\mathcal {D})$ and the number $b(\mathcal {D})$ of flow modules. We show that $$\mathcal {W}(\mathcal {D})\geq \frac{b(\mathcal {D})+1}{2}.$$ In particular, we get $\{\mathcal {W}(\mathcal {D}) \mid \mathcal {D} \text{ is a regular component} \}=\mathbb {N}$.
