Projective base change group of a quiver ($\mathrm{PGL}(\mathbf{n})=\mathrm{GL}(\mathbf{n})/\Delta k^{\times }$)

For a nonzero dimension vector, quotient ${\prod }_i{\mathrm{GL}}_{n_i}(k)$ by the common scalar subgroup $(\lambda I_{n_i}{)}_i$. It acts trivially, so the quotient acts algebraically on the quiver representation space. This is one scalar quotient, rather than a product of the individual projective general linear groups. At dimension vector zero the action is trivial.