Agreement of triangle ice quivers across Dynkin types beyond A

Establish that the ice quivers associated to the basic triangle Δ for higher Teichmüller cluster A-varieties constructed by Goncharov–Shen (GS19) and by Keller–Liu (KL25) agree for all simply-laced Dynkin types I beyond type A with the linear orientation. By amalgamation along triangulations, this agreement would imply that the ice quivers associated with general triangulations of the marked surface coincide.

Background

The paper constructs relative 3-Calabi–Yau categories and relates their cosingularity categories to C_I-valued topological Fukaya categories, aiming at additive categorifications of cluster algebras from higher Teichmüller theory. For the triangle, Keller–Liu (KL25) describe categories tied to an ice quiver with potential and sketch a Morita equivalence with a relative Ginzburg dg category; moreover, the underlying ice quiver is expected to coincide with the initial cluster seed for the triangle. In type A this coincidence is clear, but in other Dynkin types it is not established in the text.

Verifying the agreement of triangle ice quivers produced by the constructions of GS19 and KL25 beyond type A is a necessary step to ensure that, under the Fock–Goncharov amalgamation along triangulations, the global ice quivers for general surfaces align, completing the additive categorification program laid out in the introduction.