The Gabriel-Roiter measures of the indecomposables in a regular component of the 3-Kronecker quiver

TL;DR

This paper demonstrates that in a specific regular component of the 3-Kronecker quiver, the Gabriel-Roiter measure uniquely identifies indecomposable modules based on their dimension vectors, leveraging Fibonacci number properties.

Contribution

It establishes a unique correspondence between Gabriel-Roiter measures and indecomposable modules in a regular component of the 3-Kronecker quiver, using Fibonacci numbers.

Findings

Gabriel-Roiter measures are uniquely determined by dimension vectors.

Two indecomposable modules are isomorphic iff their measures differ.

Fibonacci numbers play a key role in the measure determination.

Abstract

Let $Q$ be the 3-Kronecker quiver, i.e., $Q$ has two vertices, labeled by 1 and 2, and three arrows from 2 to 1. Fix an algebraically closed field $k$. Let $\mathcal{C}$ be a regular component of the Auslander-Reiten quiver containing an indecomposable module $X$ with dimension $(1,1)$ or $(2,1)$. Using the properties of the Fibonacci numbers, we show that the Gabriel-Roiter measures of the indecomposable modules in $\mathcal{C}$ are uniquely determined by the dimension vectors. In other words, two indecomposable modules in $\mathcal{C}$ are not isomorphic if and only if their Gabriel-Roiter measures are different.