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Stabilization of Kac polynomials along root strings

Vladyslav Zveryk

TL;DR

The paper investigates stabilization phenomena for Kac polynomials and the cohomology of quiver moduli when a large multiple of a fixed δ is added to the dimension vector. By combining geometric invariant theory, Harder–Narasimhan stratification, and Kirwan surjectivity, it derives explicit formulas for stabilized cohomology dimensions and the stabilized coefficients of Kac polynomials, and shows these stabilize for quiver and Nakajima quiver varieties as n grows large. The stabilized generating function is (1−q) p^{|supp(δ)|}(q) divided by a product over coordinates not in supp(δ), tying root multiplicities in Kac–Moody algebras to the topology of quiver moduli. The results extend to Nakajima varieties via the Crawley–Boevey construction and suggest conjectural exact limits and Kirwan surjectivity extensions, with connections to Hilbert schemes and Frenkel-type bounds.

Abstract

We study the stabilization behavior of cohomology groups associated with moduli spaces of quiver representations for a fixed quiver $Q$. Under mild conditions on a dimension vector $δ$, we show that the dimensions of these cohomology groups stabilize when a sufficiently large multiple of $δ$ is added. We derive explicit formulas for the stabilized dimensions and, in particular, obtain stabilization of the coefficients of Kac polynomials.

Stabilization of Kac polynomials along root strings

TL;DR

The paper investigates stabilization phenomena for Kac polynomials and the cohomology of quiver moduli when a large multiple of a fixed δ is added to the dimension vector. By combining geometric invariant theory, Harder–Narasimhan stratification, and Kirwan surjectivity, it derives explicit formulas for stabilized cohomology dimensions and the stabilized coefficients of Kac polynomials, and shows these stabilize for quiver and Nakajima quiver varieties as n grows large. The stabilized generating function is (1−q) p^{|supp(δ)|}(q) divided by a product over coordinates not in supp(δ), tying root multiplicities in Kac–Moody algebras to the topology of quiver moduli. The results extend to Nakajima varieties via the Crawley–Boevey construction and suggest conjectural exact limits and Kirwan surjectivity extensions, with connections to Hilbert schemes and Frenkel-type bounds.

Abstract

We study the stabilization behavior of cohomology groups associated with moduli spaces of quiver representations for a fixed quiver . Under mild conditions on a dimension vector , we show that the dimensions of these cohomology groups stabilize when a sufficiently large multiple of is added. We derive explicit formulas for the stabilized dimensions and, in particular, obtain stabilization of the coefficients of Kac polynomials.
Paper Structure (12 sections, 19 theorems, 101 equations)

This paper contains 12 sections, 19 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Kac Polynomials and Quiver Varieties
  3. Representation Varieties and Stability
  4. Representation varieties for the double quiver
  5. Nakajima quiver varieties
  6. Harder-Narasimhan Stratification
  7. Kirwan surjectivity for quiver varieties
  8. Kac Polynomials
  9. Stabilization for Quiver Representations
  10. Stabilization for Kac polynomials
  11. Relaxing the (\ref{['eq:condition']}) condition
  12. Stabilization for Nakajima quiver varieties

Key Result

Theorem 1.1

Write the Kac polynomial corresponding to a dimension vector $d+n\delta$ as Then for every $i$ the coefficients $a_i^{(d+n\delta)}$ stabilize for sufficiently large $n$. Moreover, the generating function of the stabilized coefficients is given by where $p^l(q)$ denotes the generating function of the partition number into $l$ colors.

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: Rei08, 3.5
  • Theorem 2.3: Kir, Theorem 10.12
  • Example 2.4
  • Example 2.5
  • Proposition 2.6: pp. 261-262, CB01
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 36 more