A refinement of the Kac polynomials for quivers with enough loops

TL;DR

This paper refines the understanding of Kac polynomials for quivers with enough loops by expressing them as sums of refined polynomials with non-negative coefficients and providing a closed formula, along with a new representation class.

Contribution

It introduces refined Kac polynomials parametrized by partitions, provides a closed formula, and proposes a conjectural interpretation involving block representations.

Findings

Refined Kac polynomials have non-negative coefficients.

A closed formula for the refined Kac polynomials is established.

A new class of representations called blocks is introduced.

Abstract

A conjecture of Kac now a theorem asserts that the polynomial now known as the Kac polynomial, which counts the isomorphism classes of absolutely indecomposable representations of a quiver over a finite field with a given dimension vector, has non-negative integer coefficients only. In this paper, we show that, for quivers with enough loops, every Kac polynomial can be expressed as a sum of the refined Kac polynomials which are parametrized by tuples of partitions and have non-negative integer coefficients only. A closed formula for the refined Kac polynomials is given. We further introduce a new class of representations called blocks and make a conjectural interpretation of the refined Kac polynomials for quivers with enough loops in terms of the numbers of block representations.