Littlewood–Richardson coefficients via mirror symmetry for cluster varieties

I prove that the full Fock–Goncharov conjecture holds for Conf×3(F) — the configuration spaceof triples of decorated flags in generic position. As a key ingredient of this proof, I exhibita maximal green sequence for the quiver of the initial seed. I compute the Landau–Ginzburgpotential W on Conf×3(F)∨associated to the partial minimal model Conf×3(F) ⊂ Conf3(F).The integral points of the associated ‘cone’ Ξ := {WT 0}⊂Conf×3(F)∨(RT) parametrize abasis for O(Conf3(F)) =(Vα⊗ Vβ⊗ Vγ)Gand encode the Littlewood–Richardson coefficientscγαβ. In the initial seed, the inequalities defining Ξ are exactly the tail positivity conditions of[18]. I exhibit a unimodular p∗map that identifies W with the potential of Goncharov–Shen onConf×3(F) [8] and Ξ with the Knutson–Tao hive cone [14].This paper relies extensively on colour figures. Some references to colour may not bemeaningful in the printed version, and we refer the reader to the online version which includesthe colour figures.