GKM-theory for torus actions on cyclic quiver Grassmannians

Abstract

We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $A$ flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$. We show that these quiver Grassmannians equipped with our specific torus action are GKM-varieties and that their moment graph admits a combinatorial description in terms of the coefficient quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able to prove that moment graph techniques can be applied to construct module bases for the equivariant cohomology of the quiver Grassmannians listed above.