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Equivariant geometry of symmetric quiver orbit closures

Ryan Kinser, Martina Lanini, Jenna Rajchgot

TL;DR

This work establishes a precise bridge between the equivariant geometry of symmetric quiver representations of type $A$ and the symmetric quotients $GL(n)/K$ (with $K=O(n)$ or $Sp(n)$). By embedding symmetric quiver representation spaces into symmetric matrix spaces via Mars–Springer slices and a symmetric Zelevinsky map, it transfers orbit-closure questions, singularity information, and equivariant (co)homology data to the well-studied framework of parabolic orbits and Bruhat order. The authors prove an main theorem giving an injective, order-preserving correspondence between orbit closures and a parallel orbit-closure poset, along with smooth families and homomorphisms on $K$-theory and Chow groups; they also derive normality and Cohen–Macaulayness results in characteristic $0$ for the $\\epsilon=-1$ case and provide counterexamples in the $\\epsilon=+1$ case. Furthermore, they introduce symmetric Zelevinsky permutations to give a combinatorial model for the orbit-closure poset, and show how to reduce arbitrary orientations to bipartite ones via homogeneous fiber bundles. The paper thus unifies representation theory, algebraic geometry of symmetric varieties, and combinatorial Bruhat–order machinery, enabling explicit calculations of orbit closures and their singularities in a broad class of problems.

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.

Equivariant geometry of symmetric quiver orbit closures

TL;DR

This work establishes a precise bridge between the equivariant geometry of symmetric quiver representations of type and the symmetric quotients (with or ). By embedding symmetric quiver representation spaces into symmetric matrix spaces via Mars–Springer slices and a symmetric Zelevinsky map, it transfers orbit-closure questions, singularity information, and equivariant (co)homology data to the well-studied framework of parabolic orbits and Bruhat order. The authors prove an main theorem giving an injective, order-preserving correspondence between orbit closures and a parallel orbit-closure poset, along with smooth families and homomorphisms on -theory and Chow groups; they also derive normality and Cohen–Macaulayness results in characteristic for the case and provide counterexamples in the case. Furthermore, they introduce symmetric Zelevinsky permutations to give a combinatorial model for the orbit-closure poset, and show how to reduce arbitrary orientations to bipartite ones via homogeneous fiber bundles. The paper thus unifies representation theory, algebraic geometry of symmetric varieties, and combinatorial Bruhat–order machinery, enabling explicit calculations of orbit closures and their singularities in a broad class of problems.

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 where 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 symmetric quiver representation varieties into symmetric varieties.

Paper Structure

This paper contains 32 sections, 42 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Summary of results
  4. Symmetric quivers and their representations
  5. Symmetric quivers
  6. Symmetric quiver representations
  7. Symmetric representation varieties
  8. Symmetric varieties, Borel orbit closures, and Mars-Springer slices
  9. Conventions
  10. Mars-Springer slices
  11. The Mars-Springer Lemma
  12. Symmetric quotients as matrix spaces
  13. A specific Mars-Springer slice
  14. Borel orbit closures in symmetric varieties via rank conditions
  15. Parametrization of Borel orbit closures via involutions
  16. ...and 17 more sections

Key Result

Theorem 1.1

Given a symmetric type $A$ quiver $Q^\epsilon$ with dimension vector $\mathbf{d}$, there is an associated $G = GL(n)$, a symmetric subgroup and parabolic subgroup $P\subseteq G$ such that all the following hold.

Theorems & Definitions (82)

  • Theorem 1.1
  • Corollary 1.5
  • Theorem 1.6
  • Example 2.14
  • Lemma 2.15
  • Theorem 2.16
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.15
  • proof
  • ...and 72 more