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Symmetries of Quiver Schemes

Ryo Terada, Daisuke Yamakawa

TL;DR

This work extends Nakajima-style reflection operations to quiver schemes, introducing isomorphisms between schemes $ c{ m S}_{ c{ m Q}, c{f d}}(m{ lambda},m{v})$ and their reflected partners when a designated parameter component is a unit. It formulates a Coxeter-branching framework by defining reflection functors $ c{ m F}_j$ and establishing a detailed regularization theory that connects schemes for $( m Q,d)$ to those for a regularized pair $( ilde{ m Q}, ilde{f d})$ via shifting and irregular legs. The core machinery rests on GLS preprojective algebras, the moment-map/Hamiltonian-reduction viewpoint, and the construction of coadjoint-orbit realizations (Hausel–Wong–Wyss), which together show that these quiver schemes realize all symmetrizable Cartan-type structures. The results generalize Lusztig–Maffei–Nakajima reflection theory to multiplicity settings and provide tools to interpret certain quiver schemes as (affine) algebraic varieties, with Weyl-group actions linking different parameter regimes. Overall, the paper broadens the geometric repertoire of quiver varieties to quiver schemes and elucidates how Weyl-group symmetries and regularization interact in this richer context, with potential implications for representations of symmetrizable Kac–Moody algebras and associated quantum groups.

Abstract

We introduce reflection functors on quiver schemes in the sense of Hausel--Wong--Wyss, generalizing those on quiver varieties. Also we construct some isomorphisms between quiver schemes whose underlying quivers are different.

Symmetries of Quiver Schemes

TL;DR

This work extends Nakajima-style reflection operations to quiver schemes, introducing isomorphisms between schemes and their reflected partners when a designated parameter component is a unit. It formulates a Coxeter-branching framework by defining reflection functors and establishing a detailed regularization theory that connects schemes for to those for a regularized pair via shifting and irregular legs. The core machinery rests on GLS preprojective algebras, the moment-map/Hamiltonian-reduction viewpoint, and the construction of coadjoint-orbit realizations (Hausel–Wong–Wyss), which together show that these quiver schemes realize all symmetrizable Cartan-type structures. The results generalize Lusztig–Maffei–Nakajima reflection theory to multiplicity settings and provide tools to interpret certain quiver schemes as (affine) algebraic varieties, with Weyl-group actions linking different parameter regimes. Overall, the paper broadens the geometric repertoire of quiver varieties to quiver schemes and elucidates how Weyl-group symmetries and regularization interact in this richer context, with potential implications for representations of symmetrizable Kac–Moody algebras and associated quantum groups.

Abstract

We introduce reflection functors on quiver schemes in the sense of Hausel--Wong--Wyss, generalizing those on quiver varieties. Also we construct some isomorphisms between quiver schemes whose underlying quivers are different.
Paper Structure (12 sections, 23 theorems, 187 equations, 1 table)

This paper contains 12 sections, 23 theorems, 187 equations, 1 table.

Table of Contents

  1. Introduction
  2. Quiver schemes
  3. Preliminaries
  4. GLS preprojective algebras and quiver schemes
  5. Some $G_d(V)$-coadjoint orbits
  6. Reflection functors for quiver schemes
  7. Reflection functors
  8. Proof of Theorem \ref{['thm:reflection']}
  9. Regularization
  10. Shifting trick
  11. Irregular legs and regularization
  12. Weyl groups and regularization

Key Result

Theorem 1.1

Take $j \in I$, $\bm{\lambda} =(\lambda_i) \in \bigoplus_{i \in I} \mathbb{C}[\epsilon_i]/(\epsilon_i^{d_i})$, $\mathbf{v} \in \mathbb{Z}_{\geq 0}^I$ so that $\lambda_j$ is a unit of $\mathbb{C}[\epsilon_j]/(\epsilon_j^{d_j})$. Then there exists an isomorphism of schemes

Theorems & Definitions (58)

  • Theorem 1.1: see Section \ref{['section:reflection']}
  • Theorem 1.2: see Section \ref{['section:normalization']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 48 more