Dimension vectors with the equal kernels property

Let $r\in \mathbb{N}$, ${\mathrm{\Gamma }}_r$ be the generalized Kronecker quiver with $r$ arrows ${\gamma }_1,\dots ,{\gamma }_r\text{\hspace{0.17em}⁣}:1\to 2$ and $\delta \in \Delta_+(\Gamma_r)$beapositiverootof$\Gamma_r$\mathrm{.}Wesaythat$\delta$ 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$k$-linearmap$M^\alpha := \sum^r_{i=1} \alpha_i M(\gamma_i) \colon M_1 \to M_2$isinjective\mathrm{.}Weshowthat$\delta$ has the equal kernels property if and only if $q_{{\mathrm{\Gamma }}_r}(\delta )+{\delta }_2-{\delta }_1\ge 1$, where $q_{\Gamma_r} \colon \mathbb Z^2 \to \mathbb Z, (x,y) \mapsto x^2 + y^2 - rxy$denotestheTitsquadraticformof$\Gamma_r$.