Asymptotic Behavior of Kac Polynomials

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 behavior of the coefficients of Kac polynomials. All computations were made using SageMath.

Keywords:

• quiver
• Kac polynomial
• Nakajima quiver varieties
• Hall algebra of a quiver

Acknowledgments

The author warmly thanks to Olivier Schiffmann for useful comments and corrections concerning this work and the anonymous referees whose comments have allowed considerable improvements and corrections of inacurracies in a previous version.