The Gabriel–Roiter measure and representation types of quivers

Abstract

A GR segment of an Artin algebra is a sequence of Gabriel–Roiter measures that is closed under direct successors and direct predecessors. The number of GR segments was conjectured to relate to the representation types of finite-dimensional hereditary algebras. We prove in the paper that a path algebra KQ of a finite connected acyclic quiver Q over an algebraically closed field K is of wild representation type if and only if KQ admits infinitely many GR segments.