The Gabriel-Roiter measure for $\widetilde{\mathbb{A}}_n$ II

TL;DR

This paper investigates the Gabriel-Roiter measure for tame quivers of type  ilde{A}_n, characterizing certain modules and measures, and establishing finiteness results that generalize to all tame quivers.

Contribution

It provides a detailed analysis of Gabriel-Roiter measures for  ilde{A}_n quivers, including characterizations and finiteness results, extending previous work.

Findings

Regular string modules have at most two Gabriel-Roiter submodules.

Quivers with sink-source orientations are characterized by their central parts not containing preinjective modules.

Finitely many central Gabriel-Roiter measures admit no direct predecessors.

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.