Counting Using Hall Algebras I. Quivers

TL;DR

This paper surveys results on counting rational points of moduli spaces of quiver representations, generalizes to Grassmannians and flags, and explores applications to cluster algebras using Hall algebra structures.

Contribution

It introduces new generalizations of counting methods to Grassmannians and flags of quiver representations, connecting Hall algebras to cluster algebra applications.

Findings

Counting formulas for moduli spaces of quiver representations

Extensions to Grassmannians and flags of quiver representations

Applications to cluster algebra theory

Abstract

We survey some results on counting the rational points of moduli spaces of quiver representations. We then make generalizations to Grassmannians and flags of quiver representations. These results have nice applications to the cluster algebra. Along the way, we use the full Hopf structure of the Hall algebra of a quiver.