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Harder-Narasimhan Filtration on Moment Map for Quiver Representations

Ching Yan Timothy Yau

TL;DR

This work connects the geometry of quiver varieties to the arithmetic of Kac polynomials by analyzing Harder-Narasimhan strata on the zero level set of the moment map. It extends Crawley-Boevey–Reineke techniques to quivers with many edges via enlargements $Q_{ n}$, deriving a dimension formula for HN strata on $μ^{-1}(0)$ and relating those dimensions to linear forms that appear in the Kac polynomial decomposition. The results tie equivariant cohomology and stratification theory (Hesselink/Kirwan-type frameworks) to stabilization phenomena in Kac polynomials (Hennecart’s linear forms and Hua’s counting formula). Overall, the paper provides a unified view linking geometric invariant theory, cohomology, and representation counting for quivers, with explicit conditions under which the dimension computations stabilize as edge multiplicities grow.

Abstract

For a quiver without loops with many edges, we generalize the methods of Kac, Crawley-Boevey and Reineke and compute the dimension of Harder-Narasimhan strata of the zero set of the moment map. We notice a link between this dimension and the terms in the Kac's polynomial, which is given by the equivariant cohomology of this zero set.

Harder-Narasimhan Filtration on Moment Map for Quiver Representations

TL;DR

This work connects the geometry of quiver varieties to the arithmetic of Kac polynomials by analyzing Harder-Narasimhan strata on the zero level set of the moment map. It extends Crawley-Boevey–Reineke techniques to quivers with many edges via enlargements , deriving a dimension formula for HN strata on and relating those dimensions to linear forms that appear in the Kac polynomial decomposition. The results tie equivariant cohomology and stratification theory (Hesselink/Kirwan-type frameworks) to stabilization phenomena in Kac polynomials (Hennecart’s linear forms and Hua’s counting formula). Overall, the paper provides a unified view linking geometric invariant theory, cohomology, and representation counting for quivers, with explicit conditions under which the dimension computations stabilize as edge multiplicities grow.

Abstract

For a quiver without loops with many edges, we generalize the methods of Kac, Crawley-Boevey and Reineke and compute the dimension of Harder-Narasimhan strata of the zero set of the moment map. We notice a link between this dimension and the terms in the Kac's polynomial, which is given by the equivariant cohomology of this zero set.

Paper Structure

This paper contains 13 sections, 27 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quiver Varieties
  4. Harder-Narasimhan Filtration
  5. Kac's Polynomial
  6. Exact Sequence and Lifting
  7. Dimension Computing
  8. Dimension of HN Strata
  9. One-parameter Subgroup and Improving the Bound
  10. Codimension and Transverse Intersection
  11. Hua's Formula
  12. Strata and Cohomology
  13. Hennecart's Linear Form

Key Result

Theorem 2.2

king1994moduli Under the action of $GL_{\alpha}(\mathbb C)$, points of $\mathcal{R}_0(\alpha)$ corresponds to orbits of semisimple representations and points of $\mathcal{R}_{\theta}(\alpha)$ corresponds to orbits of $\theta$-semistable representations.

Theorems & Definitions (58)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 48 more