Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-Theory

TL;DR

This paper explores the permutation actions on the equivariant cohomology of quiver Grassmannians associated with nilpotent representations of the equioriented cycle, using GKM theory, and connects these actions to known results on flag varieties.

Contribution

It introduces a new perspective on permutation actions on quiver Grassmannians via GKM theory, extending Tymoczko's results to a broader class of varieties.

Findings

The equivariant cohomology ring admits a permutation action by a product of symmetric groups.

GKM techniques effectively analyze these permutation actions.

Connections to classical results on flag varieties are established.

Abstract

In previous work we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko's results on permutation representations.