Littlewood–Richardson coefficients via mirror symmetry for cluster varieties. Issue 3 (29th April 2020) - Magee, Timothy
Abstract: I prove that the full Fock–Goncharov conjecture holds for Conf 3 × ( F ℓ ∼ ) — 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 Conf 3 × ( F ℓ ∼ ) ∨ associated to the partial minimal model Conf 3 × ( F ℓ ∼ ) ⊂ Conf 3 ( F ℓ ∼ ) . The integral points of the associated 'cone' Ξ : = { W T ⩾ 0 } ⊂ Conf 3 × ( F ℓ ∼ ) ∨ ( R T ) parametrize a basis for O ( Conf 3 ( F ℓ ∼ ) ) = ⨁ ( V α ⊗ V β ⊗ V γ ) G and encode the Littlewood–Richardson coefficients c α β γ . 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 Conf 3 × ( F ℓ ∼ ) [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.