Littlewood-Richardson coefficients via mirror symmetry for cluster varieties

I prove that the full Fock-Goncharov conjecture holds for ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_3^{\times }(\widetilde{\mathcal{F}\mathrm{\ell }})$– 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 ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_3^{\times }(\widetilde{\mathcal{F}\mathrm{\ell }}{)}^{\vee }$ associated to the partial minimal model $$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}$_3^\times(\widetilde{$\mathcal{F}$\hspace{-1.6pt}\ell}) \subset $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}$3(\widetilde{$\mathcal{F}$\hspace{-1.6pt}\ell})$. The integral points of the associated “cone” ${\Xi:=\left{W^T \geq 0\right} \subset $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}$3^\times(\widetilde{$\mathcal{F}$\hspace{-1.6pt}\ell})^\vee\left({$\mathbb{R}$^T}\right)}$parametrizeabasisfor$$\mathcal{O}$\left($\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}$3(\widetilde{$\mathcal{F}$\hspace{-1.6pt}\ell})\right) = \bigoplus \left(V\alpha \otimes V\beta \otimes V\gamma \right)^G$ and encode the Littlewood-Richardson coefficients $c_{\alpha \beta }^{\gamma }$. In the initial seed, the inequalities defining $\mathrm{\Xi }$ are exactly Zelevinsky’s tail positivity conditions. I exhibit a unimodular $p^{\ast }$ map that identifies $W$ with the potential of Goncharov-Shen on ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}}_3^{\times }(\widetilde{\mathcal{F}\mathrm{\ell }})$ and $\mathrm{\Xi }$ with the Knutson-Tao hive cone.