= Projective base change group of a quiver
{title2=$\operatorname{PGL}(\mathbf n)=\operatorname{GL}(\mathbf n)/\Delta k^\times$}
For a nonzero dimension vector, quotient $\prod_i\operatorname{GL}_{n_i}(k)$ 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.
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