2023/10/05 by Christof Geiß, Daniel Labardini-Fragoso, Geiß, Christof +3
Mathematics · #05C70 #13F60 #16G20 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2310.03306
openalex publication_date 2023/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for any possibly-punctured surface with non-empty boundary \mathbfΣ=(Σ, \mathbbM, ℙ), and any tagged triangulation T of \mathbfΣ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential (Q(T), W(T)) associated to T by Cerulli Irelli and the second author. When the set of boundary marked points \mathbbM has at least two elements, Schröer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra A(\mathbfΣ)=U(\mathbfΣ). So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis.