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arXiv · 0912.0657

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

Bo Chen
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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.

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BibTeXRIS

Bo Chen. 2010-04-27. The Gabriel-Roiter measures of the indecomposables in a regular component of the 3-Kronecker quiver. https://arxiv.org/abs/0912.0657

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