The Gabriel-Roiter measures and representation type

TL;DR

This paper establishes that the number of Gabriel-Roiter segments in a hereditary algebra over an algebraically closed field characterizes its representation type, linking finiteness to tameness and wildness.

Contribution

It proves that infinitely many GR segments occur if and only if the algebra is of wild representation type, offering a new criterion for classifying algebras.

Findings

Infinite GR segments correspond to wild type

Finite GR segments correspond to tame or finite type

Potential for generalizing tameness and wildness classification

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.