We study quiver Grassmannians associated with indecomposable representations (of finite dimension) of the Kronecker quiver. We find a cellular decomposition for them and we compute their Betti numbers. As an application, we find a geometric realization for the atomic basis of cluster algebras of type A(1)((1)) found by Sherman and Zelevinsky (who called it the canonical basis) and those of type A(2)((1)) found in an earlier paper of the first author.
GEOMETRY OF QUIVER GRASSMANNIANS OF KRONECKER TYPE AND APPLICATIONS TO CLUSTER ALGEBRAS
ESPOSITO, FRANCESCO
2011
Abstract
We study quiver Grassmannians associated with indecomposable representations (of finite dimension) of the Kronecker quiver. We find a cellular decomposition for them and we compute their Betti numbers. As an application, we find a geometric realization for the atomic basis of cluster algebras of type A(1)((1)) found by Sherman and Zelevinsky (who called it the canonical basis) and those of type A(2)((1)) found in an earlier paper of the first author.
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