Elementary modules of the generalized Kronecker quivers K(n)

Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. We use $\text{modd}{\text{cK}}_n$ to denote the category of the finite-dimensional modules of the path algebra ${\text{cK}}_n=kK(n)$. There exist a function called dimension vector $\text{dimu}:\text{modd}{\text{cK}}_n\to {\mathbb{Z}}^2;M\mapsto ({\dim }_k{M}_1,{\dim }_k{M}_2)$, and two functors $\sigma :{\mathbb{Z}}^2\to {\mathbb{Z}}^2;(x,y)\mapsto (nx-y,x)$, and $\delta :{\mathbb{Z}}^2\to {\mathbb{Z}}^2;(x,y)\mapsto (y,x)$. Then the set $\mathbf{F}=\{(x,y)\mid \frac{2}{n}x\le y\le x\}$ is the fundamental domain of the dimension vectors of regular modules under the action of $\delta$ and $\sigma$ in $\text{modd}{\text{cK}}_n$. We call a regular module $M\in \text{modd}{\text{cK}}_n$ $\mathit{\text{elementary}}$ if there does not exist a short exact sequence $(0)\to L\to M\to N\to (0)$ with $L,N$ being non-zero regular. In this note, we focus on the set $\mathbf{F}$, and when $x<n$, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.