ScalingStacks

Proof. In the case ($A$) there are only finitely many isomorphism classes of quiver representation with a fixed dimension vector by Gabriel’s theorem. Hence there are only finitely many $\operatorname{GL}(\mathbf{d})$ orbits in $\operatorname{Rep}(\mathbf{d})$ and condition (FO) is fulfilled. The works of Cerulli-Irelli–Esposito–Franzen–Reineke [CIEFR21] and Maksimau [Mak19] show that type $A$ partial quiver flag varieties admit an affine pavings. This implies condition (PT) using Proposition 3.4 and Proposition 3.21.

In the case ($\widetilde{A}$) there can be infinitely many isomorphism classes of quiver representations with fixed dimension vector. However, any representation $\rho\in\operatorname{Rep}(\mathbf{d})$ in the image of $\mu_{\underline{\mathbf{d}}}$ fulfills (6.1) for some flag $\underline{V}\in\operatorname{Fl}(\underline{\mathbf{d}}).$ This implies that $\rho$ is nilpotent, that is, there is some $n\in\mathbb{Z}_{\geq 0}$ such that $\rho_{a_{1}}\rho_{a_{2}}\dots\rho_{a_{n}}=0$ for any sequence of composable arrows $a_{1},\dots,a_{n}\in Q_{1}.$ There are only finitely many isomorphism classes of nilpotent quiver representations with fixed dimension vector in this case, see Section 6.3. This implies condition (FO). Moreover, we will show in Section 6.3 that the partial quiver flag varieties admit affine pavings which implies condition (PT). ∎