We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.

中文翻译：

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规范基和簇对偶性的多面体参数化

我们建立了由 Gross-Hacking-Keel-Kontsevich's 和 Berenstein-Kazhdan's 修饰函数构造的势函数在一个简单的、单连通的基仿射空间 $G/\mathcal{N}$ 上的开双 Bruhat 元胞上的关系，简单的带状代数群 $G$。作为一个副产品，我们推导出了 $G/\mathcal{N}$ 上正则函数环的规范基的多面体参数化的明确识别，这是由势函数和装饰函数与经典字符串和 Lusztig 参数化的热带化引起的。