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

TL;DR

This paper establishes a correspondence between certain non-self-intersecting curves in a punctured disc and positive c-vectors of acyclic cluster algebras with multiple arrows, providing new combinatorial insights and proofs.

Contribution

It introduces a bijective correspondence linking curves on a punctured disc to c-vectors in acyclic cluster algebras with multiple arrows, proving a conjecture on real Schur roots.

Findings

Proved a bijection between curves and c-vectors.

Provided a combinatorial characterization of seeds.

Confirmed a conjecture on real Schur roots.

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.