Quiver Grassmannians of Type $\widetilde {D}_{n}$, Part 2: Schubert Decompositions and F-polynomials

Oliver Lorscheid, Thorsten Weist
2021 Algebras and Representation Theory  
AbstractExtending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type $\tilde D_{n}$ D ~ n 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
more » ... 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 type $\tilde D_{n}$ D ~ n .
doi:10.1007/s10468-021-10097-z fatcat:6qcezcttnzayjo6ijcmk4l5e4e