#### Vol. 5, No. 6, 2011

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Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras

### Giovanni Cerulli Irelli and Francesco Esposito

Vol. 5 (2011), No. 6, 777–801
##### 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 ${A}_{1}^{\left(1\right)}$ found by Sherman and Zelevinsky (who called it the canonical basis) and those of type ${A}_{2}^{\left(1\right)}$ found in an earlier paper of the first author.

##### Keywords
complex algebraic geometry, quiver Grassmannians, cluster algebras, quiver representations
##### Mathematical Subject Classification 2000
Primary: 06B15
Secondary: 16G20, 14N05, 13F99, 16G99, 05E10