Geometry of quiver Grassmannians of Kronecker type and canonical basis of 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 $A_2^{(1)}$.

21 pages

References in corpus (4)

• Quiver varieties and cluster algebras
• Generic bases for cluster algebras and the Chamber Ansatz
• Quiver Grassmannians associated with string modules
• A homological interpretation of the transverse quiver Grassmannians

Cited by in corpus (7)

• Quiver Grassmannians associated with string modules
• Bases for cluster algebras from orbifolds
• Cluster algebras of type $A_2^{(1)}$
• Categorified canonical bases and framed BPS states
• Representation type via Euler characteristics and singularities of quiver Grassmannians
• A homological interpretation of the transverse quiver Grassmannians
• Affine cluster monomials are generalized minors