Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary

Abstract

We prove that for any possibly-punctured surface with non-empty boundary $\hat{\mathbf{I}}\pounds =(\hat{I}\pounds ,\mathbb{M},\mathbb{P})$, and any tagged triangulation $T$ of $\hat{\mathbf{I}}\pounds$ 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 $\mathbb{M}$ 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 $\mathcal{A}(\hat{\mathbf{I}}\pounds )=\mathcal{U}(\hat{\mathbf{I}}\pounds )$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis.

47 pages, 12 figures. V3: Comments on "peculiar" ability to evaluate Euler characteristics added (see section 7), references updated