Cellular decompositions for affine Dynkin quiver Grassmannians

Establish that every quiver Grassmannian associated with a representation of a quiver whose underlying graph is a simply laced affine Dynkin diagram admits a cellular decomposition.

Background

The paper explains that cellular decompositions are important for proving the vanishing of odd-degree cohomology of quiver Grassmannians. It cites prior work establishing cellular decompositions for representations of Dynkin quivers and identifies the affine extension as an unresolved conjecture. Proving this statement would extend the known cellular-decomposition theory from finite-type Dynkin quivers to orientations of simply laced affine Dynkin diagrams.

References

They prove that for representations of Dynkin quivers, every quiver Grassmannian admits a cellular decomposition and conjecture the same for quivers which are orientations of the simply laced Dynkin diagrams of affine type.

— Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations  (2503.08233 - Pütz, 11 Mar 2025) in Section 1, Introduction, page 1