On Schubert decompositions of quiver Grassmannians

TL;DR

This paper introduces Schubert decompositions for quiver Grassmannians, linking their cell structures to simpler cases, and extends known affine space decompositions to broader classes, impacting understanding of their topology.

Contribution

It develops a new framework for Schubert decompositions in quiver Grassmannians and extends affine cell decompositions to more complex representations.

Findings

Cells relate to simpler quiver Grassmannians

Extended affine decompositions to new classes

Derived implications for Euler characteristics and cohomology

Abstract

In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert decomposition into relation to the cells of a certain simpler quiver Grassmannian. This allows us to extend known examples of Schubert decompositions into affine spaces to a larger class of quiver Grassmannians. This includes exceptional representations of the Kronecker quiver as well as representations of forests with block matrices of the form $\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$. Finally, we draw conclusions on the Euler characteristics and the cohomology of quiver Grassmannians.