A Module-Theoretic Interpretation of Schiffler's Expansion Formula

Abstract

We give a module-theoretic interpretation of Schiffler's expansion formula which is defined combinatorially in terms of complete (Γ, γ)-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.

Key Words:

• Cluster algebras
• Cluster categories
• Marked surfaces
• Representations

2000 Mathematics Subject Classification:

• 13F60
• 13E10
• 16G20
• 18G55

ACKNOWLEDGEMENT

Thomas Brüstle is supported by NSERC. Jie Zhang is supported by China Scholarship Council.

Notes

Communicated by D. Zacharia.