Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations

TL;DR

The paper tackles the computation of cluster expansions for arcs on surfaces (including punctured and self-folded cases) within cluster algebras. It introduces two complementary combinatorial schemes—$T$-walks (a generalization of $T$-paths) and labeled posets (via order ideals)—and augments them with a matrix framework based on oriented posets to efficiently evaluate expansions. The main results express the cluster variable $x_\gamma$ as $x_\gamma = x(T_0)\mathcal{W}(P_\gamma;xy)$ (or equivalently $x_\gamma = \frac{x(M_-)}{cross(T,\gamma)}\mathcal{W}(P_\gamma;xy)$), and establish a bijection between $T$-walks and poset ideals. Together, these methods unify and extend existing snake/band-graph approaches, accommodate notched and self-folded configurations, and provide scalable computation with potential applicability beyond classical surface settings.

Abstract

We give two new combinatorial methods for computing cluster expansion formulas for arcs coming from possibly punctured surfaces. The first is by using $T$-walks, an extension of the $T$-path model for unpunctured surfaces to general surfaces. We also introduce a new way of generating $T$-paths. The second method is by using order ideals of labeled posets associated to arcs. We also use the theory of oriented posets to give a quick way to calculate the expressions using $2$ by $2$ matrices. The techniques introduced are applicable to different settings in cluster algebras and beyond.

Figures (28)

• Figure 1: An illustration of a self-folded triangle on the left and a tagged arc on the right.
• Figure 2:
• Figure 3: The minimal direction $\vec{v}$ for chosen $\gamma$
• Figure 4:
• Figure 5: An arc $\gamma$ crossing three arcs.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: FT18
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7
• Definition 3.1
• Theorem 3.2
• Lemma 3.3