The Gabriel–Roiter measure and representation types of quivers

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 KQKQ of a finite connected acyclic quiver QQ over an algebraically closed field KK is of wild representation type if and only if KQKQ admits infinitely many GR segments.