The GR-segments for tame quivers

Abstract

A GR-segment for an artin algebra is a sequence of Gabriel-Roiter measures, which is closed under direct predecessors and successors. The number of the GR-segments indexed by natural numbers $\mathbb{N}$ and integers $\mathbb{Z}$ probably relates to the representation types of artin algebras. Let $k$ be an algebraically closed field and $Q$ be a tame quiver (of type $\widetilde{\mathbb{A}}_n$, $\widetilde{\mathbb{D}}_n$, $\widetilde{\mathbb{E}}_6$, $\widetilde{\mathbb{E}}_7$, or $\widetilde{\mathbb{E}}_8$). Let $b$ be the number of the isomorphism classes of the exceptional quasi-simple modules over the path algebra $\Lambda=kQ$. We show that the number of the $\mathbb{N}$- and $\mathbb{Z}$-indexed GR-segments in the central part for $Q$ is bounded by $b+1$. Therefore, there are at most $b+3$ GR segments.