Counting Semistable Representations of Quivers over Finite Fields

Abstract

In this paper, we derive a closed formula for the number of isomorphism classes of absolutely indecomposable semistable representations of an arbitrary quiver over a finite field with a fixed dimension vector. This generalises a formula for Kac polynomials given by Hua. A key step in the proof is to show that any representation of a quiver with a nilpotent endomorphism over an arbitrary field admits a structured filtration by subrepresentations compatible with the nilpotent action.