Positivity of wall exponents in quantum cluster scattering diagrams

Establish necessary and sufficient conditions under which every wall element in a quantum cluster scattering diagram D(B) with structure group Gq can be realized with only positive exponents s in the quantum dilogarithm elements Ψa,b[hn], including nonskew-symmetric cases. Classify precisely when positivity holds and when counterexamples occur for skew‑symmetrizable exchange matrices B.

Background

Quantum cluster scattering diagrams (QCSDs) extend classical cluster scattering diagrams by incorporating quantum dilogarithm elements Ψa,b[hn] in a completed Lie/exponential group Gq. In the classical case, Gross–Hacking–Keel–Kontsevich proved a positivity realization: every wall element can be chosen with a positive exponent, and a diagram with minimal support exists.

For QCSDs, a natural analog of positivity is that all wall elements carry strictly positive exponents s in Ψa,b[hn]. This positivity is known to hold in the skew‑symmetric case and for certain rank‑2 nonskew‑symmetric examples (types B2 and G2). However, counterexamples also exist in the nonskew‑symmetric setting, so the exact scope of positivity is presently unclear.