Middle terms of AR-sequences of graded Kronecker modules

Abstract

Let $(T(n),\Omega)$ be the covering of the generalized Kronecker quiver $K(n)$, where $\Omega$ is a bipartite orientation. Then there exists a reflection functor $\sigma$ on the category $\mod(T(n),\Omega)$. Suppose that $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an AR-sequence in the regular component $\mathcal{D}$ of $\mod(T(n),\Omega)$, and $b(Z)$ is the number of flow modules in the $\sigma$-orbit of $Z$. Then the middle term $Y$ is a sink (source or flow) module if and only if $\sigma Z$ is a sink (source or flow) module. Moreover, their radii and centers satisfy $r(Y)=r(\sigma Z)+1$ and $C(Y)=C(\sigma Z)$.