The Gabriel–Roiter Measure for $\widetilde{\mathbb{A}}_{\boldsymbol n}$ II

Abstract

Let Q be a tame quiver of type \(\widetilde{\mathbb{A}}_n\) and Rep(Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel–Roiter submodules. The quivers Q with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel–Roiter measures admitting no direct predecessors. This fact will be generalized for all tame quivers.