Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians

TL;DR

This paper explores the connection between quiver Grassmannians and totally nonnegative Grassmannians, revealing their shared cellular structures and providing new geometric and combinatorial insights.

Contribution

It establishes that certain quiver Grassmannians have cell posets matching those of totally nonnegative Grassmannians and develops resolutions of singularities for their irreducible components.

Findings

Cell posets of quiver Grassmannians match reversed cell posets of nonnegative Grassmannians

Descriptions of irreducible components and their automorphism group actions

Constructed resolutions of singularities via extended cyclic quivers

Abstract

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.