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.

中文翻译：

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通过镜像对称性对集群品种的Littlewood–Richardson系数

我证明了完整的Fock-Goncharov猜想对于$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）$成立—装饰标记处于一般位置。作为该证明的关键要素，我对原始种子的颤抖表现出最大的绿色序列。我计算了与部分极小值相关的$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）^ \ vee $的Landau-Ginzburg势$ W $模型$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）\ subset \ mathrm {Conf} _3（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）$。关联的“圆锥”的积分点$ {\ 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）} $参数化$ \ 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 $并编码Littlewood-Richardson系数$ c ^ \ gamma _ {\ alpha \ beta} $。在初始种子中，定义$ \ Xi $的不等式正是Zelevinsky的尾部阳性条件。我展示了一个单模$ p ^ * $映射，该映射以$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）$和$ \ Xi $和Knutson-Tao蜂巢状圆锥体。定义$ \ Xi $的不等式正是Zelevinsky的尾部阳性条件。我展示了一个单模$ p ^ * $映射，该映射以$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）$和$ \ Xi $和Knutson-Tao蜂巢状圆锥体。定义$ \ Xi $的不等式正是Zelevinsky的尾部阳性条件。我展示了一个单模$ p ^ * $映射，该映射以$ \ mathrm {Conf} _3 ^ \ times（\ widetilde {\ mathcal {F} \ hspace {-1.6pt} \ ell}）$和$ \ Xi $和Knutson-Tao蜂巢状圆锥体。