Widths of regular components for n-regular tree $T(n)$

Abstract

Let $(T(n),Ω)$ be the covering of the generalized Kronecker quiver $K(n)$, where $Ω$ is a bipartite orientation. Given a regular Auslander--Reiten component $\cD$ of $\modd(T(n),Ω)$, we introduce two invariants: the width $\cW(\cD)$ and the number of flow modules $b(\cD)$. We show that $\cW(\cD)\geq \frac{b(\cD)+1}{2}$. In particular, we get $\{\cW(\cD)| \cD \text{ is a regular component} \}=\mathbb{N}$.