Counting using Hall Algebras II. Extensions from Quivers

TL;DR

This paper provides explicit formulas for counting rational points of GIT quotients of quiver representations with relations, focusing on specific algebra types and exploring conditions for polynomial-count, using geometric Hall algebra methods.

Contribution

It extends previous work by deriving explicit formulas for new algebra classes and analyzing polynomial-count conditions through geometric Hall algebra techniques.

Findings

Explicit formulas for quiver representation counts

Identification of conditions for polynomial-count

Application of geometric Hall algebra methods

Abstract

We count the $\mathbb{F}_q$-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras -- one is one-point extended from a quiver $Q$, and the other is the Dynkin $A_2$ tensored with $Q$. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions.