Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras

TL;DR

This paper provides a geometric realization of canonical bases in affine cluster algebras using transverse quiver Grassmannians, connecting combinatorial constructions with representation theory.

Contribution

It introduces a geometric interpretation of basis elements via transverse quiver Grassmannians, extending previous combinatorial approaches to affine quivers.

Findings

Established a link between basis elements and representation theory.

Defined transverse quiver Grassmannians as a new geometric tool.

Extended Caldero-Chapoton cluster character to affine quivers.

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.