Base change action on quiver representations ($(g\cdot f{)}_{\rho }=g_{t(\rho )}{f}_{\rho }{g}_{s(\rho )}^{-1}$)

The group $\mathrm{GL}(\mathbf{n})={\prod }_i{\mathrm{GL}}_{n_i}(k)$ acts by $(g\cdot f{)}_{\rho }=g_{t(\rho )}{f}_{\rho }{g}_{s(\rho )}^{-1}$. Inverse entries are regular on a general linear group, so this is an algebraic group action. Its stabilizer is the automorphism group of a quiver representation.

The infinitesimal base change action on quiver representations is ${\xi }_x(u{)}_{\rho }=u_{t(\rho )}{x}_{\rho }-x_{\rho }{u}_{s(\rho )}$. Its image is the Zariski tangent space to the orbit, because the stabilizer is a smooth open subset of the endomorphism ring. The extension complex of quiver representations identifies the quotient of the ambient tangent space by this image with ${\mathrm{Ext}}_Q^1(X,X)$. Consequently rigid quiver representations have open orbits.

Choose integer vertex weights increasing along every arrow of a finite acyclic quiver. Acting at vertex $i$ by $t^{w_i}I$ multiplies each arrow by a positive power of $t$. At $t=0$ all arrows vanish. Thus every orbit closure contains the origin of the quiver representation space.

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.