Shift orbits for elementary representations of Kronecker quivers

Let $r\in {\mathbb{N}}_{\ge 3}$. We denote by $K_r$ the wild $r$-Kronecker quiver with $r$ arrows ${\gamma }_i\text{\hspace{0.17em}⁣}:1\to 2$ and consider the action of the group $G_r\subseteq \mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{Z}}^2)$ generated by $\delta \text{\hspace{0.17em}⁣}:\mathbb{Z}\to \mathbb{Z},(x,y)\mapsto (y,x)$ and ${\sigma }_r\text{\hspace{0.17em}⁣}:\mathbb{Z}\to \mathbb{Z},(x,y)\mapsto (rx-y,x)$ on the set of regular dimension vectors $$\mathcal{R}=\{(x,y)\in {\mathbb{N}}^2\mid x^2+y^2-rxy<1\}\mathrm{.}$$ A fundamental domain of this action is given by ${\mathcal{F}}_r:=\{(x,y)\in {\mathbb{N}}^2\mid \frac{2}{r}x\le y\le x\}$. We show that $(x,y)\in {\mathcal{F}}_r$ is the dimension vector of an elementary representation if and only if $$y\le \min \{\lfloor \frac{x}{r}\rfloor +\frac{x}{\lfloor \frac{x}{r}\rfloor }-r,\lceil \frac{x}{r}\rceil -\frac{x}{\lceil \frac{x}{r}\rceil }+r,r-1\},$$ where we interpret $\lfloor \frac{x}{r}\rfloor +\frac{x}{\lfloor \frac{x}{r}\rfloor }-r$ as $\mathrm{\infty }$ for $1\le x<r$. In this case we also identify the set of elementary representations as a dense open subset of the irreducible variety of representations with dimension vector $(x,y)$. A complete combinatorial description of elementary representations for $r=3$ has been given by Ringel. We show that such a compact description is out of reach when we consider $r\ge 4$, altough the representation theory of $K_3$ is as difficult as the representation theory of $K_r$ for $r\ge 4$.