Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations
Alexander Pütz
TL;DR
This work classifies when quiver Grassmannians for string representations admit a GKM-variety structure, showing that straight, equioriented string representations yield a skeletal torus action with finite fixed points and one-dimensional orbits, and that their odd-degree cohomology vanishes via affine paving. It develops constructible gradings, alignments of coefficient quivers, and a BB-decomposition to realize a cellular structure; it also provides a detailed combinatorial framework for the resulting moment graphs through fundamental mutations, and describes a KT-basis for the torus-equivariant cohomology as well as the tangent spaces via mutations. The results give explicit procedures to compute equivariant cohomology and tangent-space data from the coefficient-quiver combinatorics, and connect these to broader questions about degenerations and exceptional collections, with partial extensions suggested for certain forest/tree cases. Overall, the paper delivers a complete classification for string representations and supplies concrete tools (moment graphs, KT-basis, mutation theory) for calculations in the GKM setting of quiver Grassmannians.
Abstract
Quiver Grassmannians of equioriented type $\texttt{A}$ and nilpotent equioriented type $\tilde{\texttt{A}}$ 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.