Quiver Grassmannians and degenerate flag varieties

TL;DR

This paper explores the geometric properties of quiver Grassmannians, revealing their connections to degenerate flag varieties, and provides explicit formulas for topological invariants in type A cases.

Contribution

It establishes isomorphisms between certain quiver Grassmannians and degenerate flag varieties, and analyzes their geometric and topological structure, including explicit formulas for invariants.

Findings

Quiver Grassmannians for type A are isomorphic to degenerate flag varieties.

These varieties are irreducible, normal, local complete intersections with finitely many orbits.

Explicit formulas for Euler characteristic and Poincare polynomials are derived for type A.

Abstract

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.