On the cardinalities of Kronecker quiver Grassmannians

TL;DR

This paper provides explicit formulas and a recursive algorithm for calculating the number of points over finite fields of Grassmannians associated with the Kronecker quiver, extending known Euler characteristic formulas.

Contribution

It introduces a new recursive method and explicit formulas for counting points on Kronecker quiver Grassmannians over finite fields, linking to previous Euler characteristic results.

Findings

Explicit formulas for Grassmannian cardinalities over finite fields

A recursive algorithm for computing these cardinalities

Extension of Euler characteristic formulas to finite field counts

Abstract

We deduce using the Ringel-Hall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. We also present a recursive algorithm for computing the cardinality of every Kronecker quiver Grassmannian over a finite field.