Acyclic cluster algebras, reflection groups, and curves on a punctured disc

Abstract

We establish a bijective correspondence between certain non-self-intersecting curves in an $n$-punctured disc and positive $\mathbf{c}$-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, we obtain a proof of a conjecture by K.-H. Lee and K. Lee (arXiv:1703.09113) on the combinatorial description of real Schur roots for acyclic quivers with multiple arrows, and give a combinatorial characterization of seeds in terms of curves in an $n$-punctured disc.

To appear in Adv. Math., accepted version; 27 pages, many figures

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