The Gabriel-Roiter measures and representation type

Abstract

Let $\Lambda$ be an Artin algebra. A GR segment of $\Lambda$ is a sequence of GR measures which is closed under direct successors and direct predecessors. The number of the GR segments was conjectured to relate to the representation type of $\Lambda$. In this paper, let $k$ be an algebraically closed field and $\Lambda$ be a finite-dimensional hereditary $k$-algebra. We show that $\Lambda$ admits infinitely many GR segments if and only if $\Lambda$ is of wild representation type. Thus the finiteness of the number of the GR segments might be an alternative characterization of the tameness of finite dimensional algebras over algebraically closed fields. Therefore, this might give a possibility to generalize Drozd's tameness and wildness to arbitrary Artin algebras.