Dimension vectors with the equal kernels property

Abstract

Let $r \in \mathbb N$, $Γ_r$ be the generalized Kronecker quiver with $r$ arrows $γ_1,\ldots,γ_r \colon 1 \to 2$ and $δ\in Δ_+(Γ_r)$ be a positive root of $Γ_r$. We say that $δ$ has the equal kernels property if for all $α\in k^r \setminus \{0\}$ and every indecomposable representation $M$ with dimension vector $\underline{dim} M = δ$ the $k$-linear map $M^α:= \sum^r_{i=1} α_i M(γ_i) \colon M_1 \to M_2$ is injective. We show that $δ$ has the equal kernels property if and only if $q_{Γ_r}(δ) + δ_2 - δ_1 \geq 1$, where $q_{Γ_r} \colon \mathbb Z^2 \to \mathbb Z, (x,y) \mapsto x^2 + y^2 - rxy$ denotes the Tits quadratic form of $Γ_r$.