Geometry of quiver Grassmannians of Kronecker type and canonical basis of cluster algebras

TL;DR

This paper investigates the geometric structure of quiver Grassmannians for the Kronecker quiver, providing cellular decompositions and Betti number computations, and applies these results to realize canonical bases in specific cluster algebras.

Contribution

It offers a detailed geometric analysis of quiver Grassmannians for Kronecker quivers and connects this to the canonical bases of related cluster algebras.

Findings

Cellular decomposition of quiver Grassmannians

Computed Betti numbers for these Grassmannians

Realized canonical bases geometrically for specific cluster algebras

Abstract

We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.