Vol. 5, No. 6, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 A1(1) found by Sherman and Zelevinsky (who called it the canonical basis) and those of type A2(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