A geometric model for the non-homogeneous tubes of the cluster category of affine type D

TL;DR

The paper addresses the problem of geometrically modeling the non-homogeneous tubes in the cluster category of affine type $D$ by using unoriented arcs on a twice-punctured disk. It extends prior work on the tube of rank $n-2$ to include the two rank-$2$ tubes through the introduction of interior arcs and a two-component quiver structure, realized via a cylinder-based representation of interior arcs. A precise bijection is established between indecomposables in the rank-$2$ tubes and classes of interior arcs, with the Auslander–Reiten translate corresponding to explicit tag changes and endpoint moves. This geometric framework clarifies the regular component's decomposition in affine type $D$, providing a concrete visualization that complements the algebraic Auslander–Reiten theory and sets the stage for further extensions to other tube configurations.

Abstract

In this article, we give a geometric model for non-homogeneous tubes of the cluster category of the affine type $D$. This model is given in terms of homotopy classes of unoriented arcs in the twice punctured disk. In particular, we extend the geometric model for the tube of rank $n-2$ given in arXiv:2407.11232 to the two tubes of rank $2$.

Figures (9)

• Figure 1: Example of a triangulation and its associated tagged triangulation
• Figure 2: Labelling of the marked points of $\mathbb{D}(n)$
• Figure 3: Components of the module category schematically
• Figure 4: Four types of arcs
• Figure 5: Part of the Auslander-Reiten quiver of the tube $\mathcal{T}_1$

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 3.1
• Definition 4.1
• Definition 4.2
• Proposition 4.3
• Remark 4.4