The number of the Gabriel-Roiter measures admitting no direct predecessors over a wild quiver

TL;DR

This paper investigates the Gabriel-Roiter measures for a specific wild quiver, demonstrating the existence of infinitely many measures with no direct predecessors, which relates to the wildness of the algebra.

Contribution

It provides a detailed analysis of Gabriel-Roiter measures for a particular wild quiver, showing the existence of infinitely many measures without direct predecessors, advancing understanding of wild algebra classification.

Findings

Infinitely many Gabriel-Roiter segments exist for the studied wild quiver.

There are infinitely many Gabriel-Roiter measures with no direct predecessors.

The results relate the structure of measures to the wildness of the algebra.

Abstract

A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel-Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel-Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of artin algebras. As the first step, we are going to study the wild quiver with three vertices, labeled by $1$,$2$ and $3$, and one arrow from $1$ to $2$ and two arrows from $2$ to $3$. The Gabriel-Roiter submodules of the indecomposable preprojective modules and quasi-simple modules $\tau^{-i}M$, $i\geq 0$ are described, where $M$ is a Kronecker module and $\tau=DTr$ is the Auslander-Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.