Dimension vectors with the equal kernels property

Abstract

Let $r \in \mathbb N$, $\Gamma_r$ be the generalized Kronecker quiver with $r$ arrows $\gamma_1,\ldots,\gamma_r \colon 1 \to 2$ 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 $M^\alpha := \sum^r_{i=1} \alpha_i M(\gamma_i) \colon M_1 \to M_2$ is injective. We show that $\delta$ has the equal kernels property if and only if $q_{\Gamma_r}(\delta) + \delta_2 - \delta_1 \geq 1$, where $q_{\Gamma_r} \colon \mathbb Z^2 \to \mathbb Z, (x,y) \mapsto x^2 + y^2 - rxy$ denotes the Tits quadratic form of $\Gamma_r$.