Broken lines and compatible pairs for rank 2 quantum cluster algebras

TL;DR

This paper constructs a quantum-weighted bijection between compatible pairs and broken lines in rank 2 quantum cluster algebras, providing a combinatorial proof of their equivalence and addressing an open problem in the field.

Contribution

It introduces a quantum-weighted bijection for rank 2 quantum cluster algebras, extending combinatorial proofs to quantum settings and finite and affine types.

Findings

Constructed a quantum-weighted bijection for type A2 and Kronecker cluster algebras.

Extended the bijection to skew-symmetric cluster algebras of finite and affine types.

Established a bijection for positive compatible pairs and broken lines with negative angular momentum.

Abstract

There have been several combinatorial constructions of universally positive bases in cluster algebras, and these same combinatorial objects play a crucial role in the known proofs of the famous positivity conjecture for cluster algebras. The greedy basis was constructed in rank $2$ by Lee-Li-Zelevinsky using compatible pairs on Dyck paths. The theta basis, introduced by Gross-Hacking-Keel-Kontsevich, has elements expressed as a sum over broken lines on scattering diagrams. It was shown by Cheung-Gross-Muller-Musiker-Rupel-Stella-Williams that these bases coincide in rank $2$ via algebraic methods, and they posed the open problem of giving a combinatorial proof by constructing a (weighted) bijection between compatible pairs and broken lines. We construct a quantum-weighted bijection between compatible pairs and broken lines for the quantum type $A_2$ and the quantum Kronecker cluster algebras. By specializing the quantum parameter, this handles the problem of Cheung et al. for skew-symmetric cluster algebras of finite and affine type. For cluster monomials in skew-symmetric rank-$2$ cluster algebras, we construct a quantum-weighted bijection between positive compatible pairs (which comprise almost all compatible pairs) and broken lines of negative angular momentum.