Littlewood-Richardson coefficients via mirror symmetry for cluster varieties

Abstract

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