Algebra & Number Theory
- Algebra Number Theory
- Volume 4, Number 5 (2010), 599-624.
Transverse quiver Grassmannians and bases in affine cluster algebras
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 a set , which is conjectured to be the canonically positive basis of the acyclic cluster algebra .
In this article, we provide a geometric realization of the elements in in terms of the representation theory of . 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.
Algebra Number Theory, Volume 4, Number 5 (2010), 599-624.
Received: 26 October 2009
Revised: 17 March 2010
Accepted: 16 April 2010
First available in Project Euclid: 20 December 2017
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Dupont, Grégoire. Transverse quiver Grassmannians and bases in affine cluster algebras. Algebra Number Theory 4 (2010), no. 5, 599--624. doi:10.2140/ant.2010.4.599. https://projecteuclid.org/euclid.ant/1513729556