Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)

TL;DR

This paper proves Kac's conjecture that the polynomial counting absolutely indecomposable quiver representations has positive coefficients and relates its constant term to root multiplicities in Kac-Moody Lie algebras, for indivisible vectors.

Contribution

It provides a proof of Kac's conjecture for indivisible dimension vectors, establishing a key link between quiver representations and Kac-Moody algebra root multiplicities.

Findings

Confirmed positivity of the polynomial coefficients.

Established the constant term equals root multiplicity.

Validated Kac's conjecture for indivisible vectors.

Abstract

A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.