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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and the quantum Kronecker cluster algebras. By specializing the quantum parameter, this handles the problem of Cheung et al.&#xa0; 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.</p>

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Broken lines and compatible pairs for rank 2 quantum cluster algebras

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Amanda Burcroff,
  • Kyungyong Lee

摘要

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 \(A_2\) 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.