Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials

Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of extended Dynkin type $D$.