Equivariant geometry of symmetric quiver orbit closures

TL;DR

This paper establishes a unified framework connecting the equivariant geometry of symmetric quiver orbit closures with symmetric varieties like $GL(n)/K$, enabling transfer of results on singularities, orbit containment, and cohomology.

Contribution

It introduces explicit embeddings of symmetric quiver varieties into symmetric varieties, bridging their geometric and combinatorial properties for the first time.

Findings

Results on singularities of orbit closures transferred between classes

Combinatorial descriptions of orbit closure containment established

Explicit embeddings facilitate analysis of equivariant cohomology and K-theory

Abstract

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In particular, we translate results about singularities of orbit closures; combinatorics of orbit closure containment; and torus equivariant cohomology and K-theory between these classes of varieties. We obtain these results by constructing explicit embeddings with nice properties of homogeneous fiber bundles over type $A$ symmetric quiver representation varieties into symmetric varieties.