Shift orbits for elementary representations of Kronecker quivers

Abstract

Let $r \in \mathbb{N}_{\geq 3}$. We denote by $K_r$ the wild $r$-Kronecker quiver with $r$ arrows $\gamma_i \colon 1 \to 2$ and consider the action of the group $G_r \subseteq \mathrm{Aut}(\mathbb{Z}^2)$ generated by $\delta \colon \mathbb{Z}\to \mathbb{Z}, (x,y) \mapsto (y,x)$ and $\sigma_{r} \colon \mathbb{Z} \to \mathbb{Z}, (x,y) \mapsto (rx-y,x)$ on the set of regular dimension vectors \[\mathcal{R} = \{ (x,y) \in \mathbb{N}^2 \mid x^2 + y^2 - rxy < 1\}.\] A fundamental domain of this action is given by $\mathcal{F}_r := \{ (x,y) \in \mathbb{N}^2 \mid \frac{2}{r} x \leq y \leq x \}$. We show that $(x,y) \in \mathcal{F}_r$ is the dimension vector of an elementary representation if and only if \[y \leq \min \{ \lfloor \frac{x}{r} \rfloor+\frac{x}{\lfloor \frac{x}{r} \rfloor} - r, \lceil \frac{x}{r} \rceil -\frac{x}{\lceil \frac{x}{r} \rceil} +r,r-1\},\] where we interpret $\lfloor \frac{x}{r} \rfloor+\frac{x}{\lfloor \frac{x}{r} \rfloor} - r$ as $\infty$ for $1 \leq x < r$. 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 $r \geq 4$, altough the representation theory of $K_3$ is as difficult as the representation theory of $K_r$ for $r \geq 4$.