Transverse quiver Grassmannians and bases in affine cluster algebras

Dupont, Grégoire

doi:10.2140/ant.2010.4.599

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Transverse quiver Grassmannians and bases in affine cluster algebras

Grégoire Dupont

Vol. 4 (2010), No. 5, 599–624

DOI: 10.2140/ant.2010.4.599

Abstract

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 $ℬ (Q)$ , 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 $ℬ (Q)$ 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.

Keywords

cluster algebras, canonical bases, Chebyshev polynomials, cluster characters, quiver Grassmannians

Mathematical Subject Classification 2000

Primary: 16G99

Secondary: 13F99

Milestones

Received: 26 October 2009

Revised: 17 March 2010

Accepted: 16 April 2010

Published: 10 July 2010

Authors

Grégoire Dupont
Université de Sherbrooke Département de Mathématiques 2500 Boulevard de l’université Sherbrooke J1K 2R1 Canada
http://pages.usherbrooke.ca/gdupont2

Vol. 4, No. 5, 2010