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Affine Pavings of Quiver Flag Varieties

Xiaoxiang Zhou

TL;DR

The paper develops a unified framework to construct affine pavings for quiver partial flag varieties by encoding flags as quiver Grassmannians via extended quivers and the canonical functor $\Phi$. It proves an inductive mechanism (via short exact sequences and affine-bundle descriptions) that reduces complex flag varieties to simpler pieces, enabling affine pavings for Dynkin quivers and identifying the remaining challenges in affine types. The results extend prior work of Cerulli-Irelli, Esposito, Franzen, Reineke, and Maksimau, and hinge on Ext-vanishing properties and a careful combinatorial analysis of sectional monomorphisms, particularly in the exceptional type $E$. The methods unify Grassmannian and flag-variety perspectives, providing a concrete path to affine pavings for a broad class of quiver representations and clarifying the obstacles in the affine setting.

Abstract

In this article, we construct affine pavings for quiver partial flag varieties when the quiver is of Dynkin type. To achieve our results, we extend methods from Cerulli-Irelli-Esposito-Franzen-Reineke and Maksimau as well as techniques from Auslander-Reiten theory.

Affine Pavings of Quiver Flag Varieties

TL;DR

The paper develops a unified framework to construct affine pavings for quiver partial flag varieties by encoding flags as quiver Grassmannians via extended quivers and the canonical functor . It proves an inductive mechanism (via short exact sequences and affine-bundle descriptions) that reduces complex flag varieties to simpler pieces, enabling affine pavings for Dynkin quivers and identifying the remaining challenges in affine types. The results extend prior work of Cerulli-Irelli, Esposito, Franzen, Reineke, and Maksimau, and hinge on Ext-vanishing properties and a careful combinatorial analysis of sectional monomorphisms, particularly in the exceptional type . The methods unify Grassmannian and flag-variety perspectives, providing a concrete path to affine pavings for a broad class of quiver representations and clarifying the obstacles in the affine setting.

Abstract

In this article, we construct affine pavings for quiver partial flag varieties when the quiver is of Dynkin type. To achieve our results, we extend methods from Cerulli-Irelli-Esposito-Franzen-Reineke and Maksimau as well as techniques from Auslander-Reiten theory.
Paper Structure (10 sections, 30 theorems, 83 equations, 10 figures, 1 table)

This paper contains 10 sections, 30 theorems, 83 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extended quiver
  4. Canonical functor $\Phi$
  5. Dimension vector
  6. Ext-vanishing properties
  7. Main Theorem
  8. Application: Dynkin Case
  9. Affine paving for big order representations
  10. Application: Affine Case

Key Result

Theorem 1.1

Let $Q$ be a quiver, and let $M$ be a representation of $Q$.

Figures (10)

  • Figure 1: Quiver flag variety realized as quiver Grassmannian.
  • Figure 2: The starting functions $s_{P(e)}$.
  • Figure 3: Image in the product space.
  • Figure 4: Minimal sectional monos.
  • Figure 5: Labeling for $E_8$ exception cases.
  • ...and 5 more figures

Theorems & Definitions (77)

  • Theorem 1.1
  • Definition 2.1: Extended quiver
  • Definition 2.2: Strict extended quiver
  • Example 2.3
  • Definition 2.4: Algebra of an extended quiver
  • Definition 2.5: Partial flag
  • Definition 2.6: Strict partial flag
  • Definition 2.7: Grassmannian
  • Definition 2.8: Canonical functor $\Phi$
  • Proposition 2.9
  • ...and 67 more