Schubert decompositions for quiver Grassmannians of tree modules

Abstract

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\mathcal{ℬ}$ and $\underset{¯}{e}$ a dimension vector for $Q$. In this note we extend the methods of Lorscheid (2014) to establish Schubert decompositions of quiver Grassmannians Gr${}_{\underset{¯}{e}}\left(M\right)$ into affine spaces to the ramified case, i.e., the canonical morphism $F:T\to Q$ from the coefficient quiver $T$ of $M$ w.r.t. $\mathcal{ℬ}$ is not necessarily unramified.

In particular, we determine the Euler characteristic of Gr${}_{\underset{¯}{e}}\left(M\right)$ as the number of extremal successor closed subsets of $T_0$, which extends the results of Cerulli Irelli (2011) and Haupt (2012) (under certain additional assumptions on $\mathcal{ℬ}$).