A determinantal formula for cluster variables in cluster algebras from surfaces

TL;DR

The paper tackles computing cluster variables for cluster algebras arising from surfaces by transforming MSW's perfect-matching expansions into a determinantal formula. It builds a determinantal framework using the weighted biadjacency matrix $B(w_\bullet)$ of the snake graph $\mathcal{G}_\gamma$ associated to a plain arc $\gamma$, establishing $x_\gamma = \det B(w_\bullet)/\operatorname{cross}(\gamma, T_0)$. This relies on a pfaffian orientation of snake graphs and a principal weighting that encodes the specialized height monomials, thereby bypassing explicit enumeration of matchings while preserving exact expansions. The approach strengthens the combinatorial-algebraic toolkit for surface cluster algebras by linking cluster variables to determinant computations and pfaffian structures, with potential computational benefits and conceptual connections to determinant/pfaffian theory.

Abstract

For cluster algebras of surface type, Musiker, Schiffler and Williams gave a formula for cluster variables in terms of perfect matchings of snake graphs. Building on this, we provide a simple determinantal formula for cluster variables via the weighted biadjacency matrix of the associated snake graphs, thus circumventing the enumeration of their perfect matchings.

Figures (2)

• Figure 1: To the left, a snake graph $\mathcal{G}$ with the pfaffian orientation $\mathcal{P}(\mathcal{G})$. To the right, the inductive step in the proof of Proposition \ref{['mio']}
• Figure 2: Example in type $A$. To the right, a principal weighting of $\mathcal{G}_\gamma$ with $M_{-}$ depicted in blue.

Theorems & Definitions (29)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.2
• Lemma 2.1
• Lemma 2.2
• proof
• Corollary 2.1
• Theorem 2.1