GKM-Theory for Torus Actions on Cyclic Quiver Grassmannians

TL;DR

This paper develops GKM-theory for torus actions on cyclic quiver Grassmannians, enabling combinatorial computation of their equivariant cohomology and revealing new structural insights.

Contribution

It introduces a GKM-theoretic framework for cyclic quiver Grassmannians, linking their moment graphs to coefficients quivers and applying this to compute equivariant cohomology bases.

Findings

Quiver Grassmannians are GKM-varieties with combinatorial moment graphs.

Moment graph techniques can construct bases for equivariant cohomology.

The approach generalizes known results to cyclic quiver settings.

Abstract

We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional approximations of the $GL_n$-affine flag variety and affine Grassmannian. We show that these quiver Grassmannians, equipped with our torus action, are GKM-varieties and that their moment graph admits a combinatorial description in terms of coefficients 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 above quiver Grassmannians.