Bo Chen
Published 2009-12-03, updated 2010-04-27Version 2
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.
Comments: this paper is combined with a new revised preprint arXiv:1001.4954.
Modules over cluster-tilted algebras determined by their dimension vectors
Regular modules with preprojective Gabriel-Roiter submodules over $n$-Kronecker quivers
C-vectors and dimension vectors for cluster-finite quivers
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