Dimension vectors with the equal kernels property

Abstract

Let $r\in \mathbb{N}$, ${\Gamma }_r$ be the generalized Kronecker quiver with $r$ arrows ${\gamma }_1,\dots ,{\gamma }_r:1\to 2$ and $\delta \in {\Delta }_{+}({\Gamma }_r)$ be a positive root of ${\Gamma }_r$. We say that $\delta$ has the equal kernels property if for all $\alpha \in k^r\setminus \{0\}$ and every indecomposable representation $M$ with dimension vector $\underline{dim}M=\delta$ the $k$-linear map $M^{\alpha }:={\sum }_{i=1}^r{\alpha }_{i}M({\gamma }_i):M_1\to M_2$ is injective. We show that $\delta$ has the equal kernels property if and only if $q_{{\Gamma }_r}(\delta )+{\delta }_2-{\delta }_1\ge 1$, where $q_{{\Gamma }_r}:{\mathbb{Z}}^2\to \mathbb{Z},(x,y)\mapsto x^2+y^2-rxy$ denotes the Tits quadratic form of ${\Gamma }_r$.

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