Vol. 4, No. 5, 2010

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Transverse quiver Grassmannians and bases in affine cluster algebras

Grégoire Dupont

Vol. 4 (2010), No. 5, 599–624

Sherman, Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Their constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain “imaginary” element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver Q a set (Q), which is conjectured to be the canonically positive basis of the acyclic cluster algebra A(Q).

In this article, we provide a geometric realization of the elements in (Q) in terms of the representation theory of Q. This is done by introducing an analogue of the Caldero–Chapoton cluster character, where the usual quiver Grassmannian is replaced by a constructible subset called the transverse quiver Grassmannian.

cluster algebras, canonical bases, Chebyshev polynomials, cluster characters, quiver Grassmannians
Mathematical Subject Classification 2000
Primary: 16G99
Secondary: 13F99
Received: 26 October 2009
Revised: 17 March 2010
Accepted: 16 April 2010
Published: 10 July 2010
Grégoire Dupont
Université de Sherbrooke
Département de Mathématiques
2500 Boulevard de l’université
Sherbrooke J1K 2R1