Polyhedral parametrizations of canonical bases & cluster duality

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 $G\mathrm{/}\mathcal{N}$ 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 $G\mathrm{/}\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.