Schubert decompositions for quiver Grassmannians of tree modules

Abstract

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\cB$ and $\ue$ a dimension vector for $Q$. In this note we extend the methods of \cite{L12} to establish Schubert decompositions of quiver Grassmannians $\Gr_\ue(M)$ 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.\ $\cB$ is not necessarily unramified. In particular, we determine the Euler characteristic of $\Gr_\ue(M)$ as the number of \emph{extremal successor closed subsets of $T_0$}, which extends the results of Cerulli Irelli (\cite{Cerulli11}) and Haupt (\cite{Haupt12}) (under certain additional assumptions on $\cB$).

22 pages

References in corpus (3)

• Bases for cluster algebras from surfaces
• Cluster categories for marked surfaces: punctured case
• Tree modules

Cited by in corpus (4)

• Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces
• Representation type via Euler characteristics and singularities of quiver Grassmannians
• Coefficient Quivers, ${\mathbb{F}}_1$-Representations, and Euler Characteristics of Quiver Grassmannians
• Plücker relations for quiver Grassmannians