Type D quiver representation varieties, double Grassmannians, and symmetric varieties

TL;DR

This paper unifies the equivariant geometry of type D quiver varieties, double Grassmannians, and symmetric varieties by generalizing Zelevinsky's construction from type A to type D, revealing deep connections and translating results among these areas.

Contribution

We extend Zelevinsky's construction from type A to type D quivers, providing explicit embeddings into symmetric varieties and unifying their geometric and combinatorial properties.

Findings

Explicit embeddings of type D quiver varieties into symmetric varieties.

Unified description of singularities and orbit closures across three geometric families.

Translation of combinatorial and K-theoretic results among these varieties.

Abstract

We unify aspects of the equivariant geometry of type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties $GL(a+b)/GL(a)\times GL(b)$; in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant $K$-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type $A$ quivers to the type $D$ setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type $D$ quiver representation varieties into these symmetric varieties.