Cluster expansions: T-walks, labeled posets and matrix calculations

Kantarcı Oğuz E., Yıldırım E.

Journal of Algebra, vol.669, pp.183-219, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 669
  • Publication Date: 2025
  • Doi Number: 10.1016/j.jalgebra.2025.01.024
  • Journal Name: Journal of Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.183-219
  • Keywords: Cluster algebras, Distributive lattices, Order ideals
  • Galatasaray University Affiliated: Yes

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 T-path models from [43,44] for unpunctured surfaces to general surfaces. To do so, we introduce a new combinatorial way to generate these paths. The second is by using order ideals of labeled posets associated to arcs. In this context, we use the methods introduced in [25,26] 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.