Stabilization of Kac polynomials along root strings

TL;DR

This paper investigates how the cohomology groups of moduli spaces of quiver representations stabilize when scaled, providing explicit formulas and demonstrating stabilization of Kac polynomial coefficients.

Contribution

It establishes stabilization results for cohomology groups and Kac polynomial coefficients, with explicit formulas under mild conditions on the dimension vector.

Findings

Cohomology group dimensions stabilize with large multiples of the dimension vector.

Explicit formulas for stabilized cohomology dimensions are derived.

Coefficients of Kac polynomials also stabilize under the studied conditions.

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 $\delta$, we show that the dimensions of these cohomology groups stabilize when a sufficiently large multiple of $\delta$ is added. We derive explicit formulas for the stabilized dimensions and, in particular, obtain stabilization of the coefficients of Kac polynomials.