Séminaire Lotharingien de Combinatoire, 93B.33 (2025), 12 pp.
Amanda Burcroff, Kyungyong Lee and Lang Mou
A Combinatorial Formula for Rank 2 Scattering Diagrams
Abstract.
Cluster algebras are celebrated for their intriguing positivity properties. Combining two distinct approaches to positivity, we give a directly computable, manifestly positive, and elementary (yet highly nontrivial) formula describing generalized cluster scattering diagrams in rank 2. This formula enumerates new combinatorial objects called tight gradings on maximal Dyck paths, inspired by the greedy basis construction for cluster algebras. Using the positivity of rank 2 generalized cluster scattering diagrams, we prove the Laurent positivity of generalized cluster algebras of all ranks, resolving a conjecture of Chekhov-Shapiro from 2014.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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