Littlewood–Richardson coefficients via mirror symmetry for cluster varieties

I prove that the full Fock–Goncharov conjecture holds for Conf3×(Fℓ∼) — the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau–Ginzburg potential W on Conf3×(Fℓ∼)∨ associated to the partial minimal model Conf3×(Fℓ∼)⊂Conf3(Fℓ∼). The integral points of the associated ‘cone’ Ξ:={WT⩾0}⊂Conf3×(Fℓ∼)∨(RT) parametrize a basis for O(Conf3(Fℓ∼))=⨁(Vα⊗Vβ⊗Vγ)G and encode the Littlewood–Richardson coefficients cαβγ. 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 on Conf3×(Fℓ∼) [8] and Ξ with the Knutson–Tao hive cone [14]. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.