The Gabriel-Roiter measures admitting no direct predecessors over $n$-Kronecker quivers

TL;DR

This paper demonstrates that for $n$-Kronecker quivers, there exist infinitely many Gabriel-Roiter measures that have no direct predecessors, revealing complex ordering properties in their representation theory.

Contribution

It establishes the existence of infinitely many Gabriel-Roiter measures without direct predecessors specifically for $n$-Kronecker quivers, a new insight in the structure of their measure hierarchy.

Findings

Infinitely many Gabriel-Roiter measures admit no direct predecessors.

The result applies to $n$-Kronecker quivers with two vertices and $n$ arrows.

This reveals intricate ordering properties in the representation theory of these quivers.

Abstract

Let $Q$ be an $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n$ arrows from 2 to 1. We show that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.