Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations

Quiver Grassmannians of equioriented type $\mathtt{\text{A}}$ and nilpotent equioriented type $\widetilde{\mathtt{\text{A}}}$ quiver representations are GKM-varieties. In particular, they have a cellular decomposition and admit a torus action with finitely many fixed points and one-dimensional orbits (i.e. skeletal action). We examine the case of string representations and provide a classification of all corresponding quiver Grassmannians with a GKM-variety structure.