Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras

Abstract

Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Both 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 $\mathcal{B}(Q)$ which is conjectured to be the canonically positive basis of the acyclic cluster algebra $\mathcal{A}(Q)$. In this article, we provide a geometric realization of the elements in $\mathcal{B}(Q)$ 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 transverse quiver Grassmannian.

26 pages. Section 4 was slightly changed according to the comments of the referee. The rest of the paper remains essentially unchanged. To appear in Algebra and Number Theory.

References in corpus (4)

• On triangulated orbit categories
• Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras
• Integral bases of cluster algebras and representations of tame quivers
• Cluster multiplication in regular components via generalized Chebyshev polynomials