For a given finite-type quiver Γ \varGamma , we will consider scalar-removed representations ( S d , R d ( Γ ) ) (S_{d}, R_{d}(\varGamma )) , where S d S_{d} is a direct product of special linear algebraic groups and R d ( Γ ) R_{d}(\varGamma ) is the representation defined naturally by Γ \varGamma and a dimension vector d d . In this paper, we give a necessary and sufficient condition on d d that R d ( Γ ) R_{d}(\varGamma ) has only finitely many S d S_{d} -orbits. This condition can be paraphrased as a condition concerning lattices of small rank spanned by positive roots of Γ \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.