A module-theoretic interpretation of Schiffler's expansion formula

TL;DR

This paper provides a module-theoretic interpretation of Schiffler's combinatorial expansion formula for cluster variables in marked surface cluster algebras, linking it to the cluster category's indecomposable objects.

Contribution

It introduces a module-theoretic perspective that aligns Schiffler's expansion formula with the cluster character, enhancing the understanding of cluster algebra structures from a categorical viewpoint.

Findings

Schiffler's expansion formula coincides with the cluster character of the associated cluster category.

The module-theoretic interpretation clarifies the combinatorial formula through geometric and categorical perspectives.

The approach bridges combinatorial, geometric, and algebraic methods in cluster algebra theory.

Abstract

We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (T,r)-paths in order to get the expansion of the cluster variables in the cluster algebra of a marked surface (S,M). Based on the geometric description of the indecomposable objects of the cluster category of the marked surface (S,M), we show the coincidence of Schiffler-Thomas' expansion formula and the cluster character defined by Palu.