Counting Representations of Quivers Respecting Nilpotent Relations over Finite Fields

TL;DR

This paper extends Hua's work by providing formulas for counting quiver representations over finite fields that respect nilpotent relations, including a $q$-deformation of a key algebraic identity.

Contribution

It introduces a closed formula for counting isomorphism classes of absolutely indecomposable representations with specific dimension vectors and establishes a $q$-deformation of the Weyl-Kac denominator identity.

Findings

Derived a closed formula for counting representations respecting nilpotent relations.

Established a $q$-deformation of the Weyl-Kac denominator identity.

Provided a method to determine the number of indecomposable representations from representation counts.

Abstract

This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely indecomposable representations with given dimension vectors is given and a $q$-deformation of Weyl-Kac denominator identity is established. In principle, if the numbers of representations are known, then the numbers of isomorphism classes of absolutely indecomposable representations are known.