On Schubert decompositions of quiver Grassmannians

In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate certain 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 (0100). Finally, we draw conclusions on the Euler characteristics and the cohomology of quiver Grassmannians.