#### Vol. 9, No. 6, 2015

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Schubert decompositions for quiver Grassmannians of tree modules

### Appendix: Thorsten Weist

Vol. 9 (2015), No. 6, 1337–1362
##### Abstract

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\mathsc{ℬ}$ 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. $\mathsc{ℬ}$ 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 $\mathsc{ℬ}$).

##### Keywords
quiver Grassmannian, Schubert decompositions, Euler characteristics, tree modules
##### Mathematical Subject Classification 2010
Primary: 14M15
Secondary: 16G20, 05C05