A characterization of finite prehomogeneous vector spaces associated with products of special linear groups and Dynkin quivers

For a given finite-type quiver $\varGamma$, we will consider scalar-removed representations $(S_{d}, R_{d}(\varGamma))$, where $S_{d}$ is a direct product of special linear algebraic groups and $R_{d}(\varGamma)$ is the representation defined naturally by $\varGamma$ and a dimension vector $d$. In this paper, we give a necessary and sufficient condition on $d$ that $R_{d}(\varGamma)$ has only finitely many $S_{d}$-orbits. This condition can be paraphrased as a condition concerning lattices of small rank spanned by positive roots of $\varGamma$. To determine such scalar-removed representations having only finitely many orbits is very fundamental to the open problem of classification of the so-called semisimple finite prehomogeneous vector spaces. We consider everything over an algebraically closed field of characteristic zero.