Shift orbits for elementary representations of Kronecker quivers

Let r N ₃. We denote by Kᵣ the wild r-Kronecker quiver with r arrows ᵢ 1 2 and consider the action of the group Gᵣ Aut (Z²) generated by Z Z, (x, y) (y, x) and ₑ Z Z, (x, y) (rx-y, x) on the set of regular dimension vectors = \ (x, y) N² x² + y² - rxy < 1\. \ A fundamental domain of this action is given by Fᵣ: = \ (x, y) N² 2{r x y x \}. We show that (x, y) Fᵣ is the dimension vector of an elementary representation if and only if \ x{r +x x{r } - r, xr -x x{r } +r, r-1\}, \ where we interpret xr +x x{r } - r as for 1 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 4, altough the representation theory of K₃ is as difficult as the representation theory of Kᵣ for r 4.