Asymptotic behaviour of Kac polynomials

TL;DR

This work analyzes the asymptotic behavior of Kac polynomials $A_{Q_{\underline{n}},\mathbf{d}}(q)$ as arrow multiplicities grow, proving that renormalized polynomials converge to a universal power series and that $A_{Q_{\underline{n}},\mathbf{d}}(q)$ admits a canonical quotient representation with a denominator independent of $\underline{n}$. It provides a conjectural valuation formula depending on loops, and demonstrates linear-rate convergence along directions in arrow space, with explicit closed forms in key examples such as multi-loop and tennis-racket quivers. The results bridge combinatorial, Lie-theoretic, and geometric viewpoints, linking to Nakajima quiver varieties and Lusztig nilpotent varieties, and offer computational tools via SageMath to study asymptotics and coefficient distributions. Together, these findings shed light on the global structure of Kac polynomials and their stability under increasing quiver complexity, with potential implications for DT invariants and graded Lie algebra representations.

Abstract

We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behaviour of the coefficients of Kac polynomials. All computations were made using SageMath.

Theorems & Definitions (27)

• Theorem 2.1: Kac,MR718127
• proof : A word on the proof
• Theorem 2.2: Hausel,MR2651380
• Theorem 2.3: Hausel--Letellier--Rodriguez-Villegas,MR3034296
• Remark 2.4
• Theorem 2.5: Hua, MR1752774
• Conjecture 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4