Degenerate affine Grassmannians and loop quivers

TL;DR

This paper explores the structure of degenerate affine Grassmannians of type A, linking them with quiver Grassmannians and affine Schubert varieties, providing explicit descriptions and finite-dimensional approximations.

Contribution

It offers explicit descriptions of degenerate affine Grassmannians, connects them with quiver Grassmannians, and proposes conjectures for symplectic cases and generalizations.

Findings

Explicit description of degenerate affine Grassmannian of type GL_n

Identification with semi-infinite orbit closure of type A_{2n-1}

Finite-dimensional approximations via principal quiver Grassmannians

Abstract

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_1^{(1)}$, propose a conjectural description in the symplectic case and discuss the generalization to the case of the affine degenerate flag varieties.