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

Cerulli Irelli, Giovanni; Esposito, Francesco

doi:10.2140/ant.2011.5.777

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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

DOI: 10.2140/ant.2011.5.777

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.

Keywords

complex algebraic geometry, quiver Grassmannians, cluster algebras, quiver representations

Mathematical Subject Classification 2000

Primary: 06B15

Secondary: 16G20, 14N05, 13F99, 16G99, 05E10

Milestones

Received: 22 March 2010

Revised: 14 September 2010

Accepted: 30 October 2010

Published: 2 April 2012

Proposed: Andrei Zelevinsky

Authors

Giovanni Cerulli Irelli
Università degli studi di Padova Dipartimento di Matematica Pura ed Applicata Via Trieste 63 I-35121 Padova Italy	Università di Roma “La Sapienza” Dipartimento di Matematica Piazzale Aldo Moro 5 I-00185 Roma Italy

Francesco Esposito
Università degli studi di Padova Dipartimento di Matematica Pura ed Applicata Via Trieste 63 I-35121 Padova Italy

Vol. 5, No. 6, 2011