  I prove several conjectures of \cite{GHKK} on the cluster structure of
$SL_n$, which in particular imply the full Fock-Goncharov conjecture for the
open double Bruhat cell $\mathcal{A} \subset SL_n/U$, for $U \subset SL_n$ a
maximal unipotent subgroup. This endows the mirror cluster variety
$\mathcal{X}$ with a canonical potential function $W$, and determines a
canonical cone $W^T \geq 0 \subset \mathcal{X}\left(\mathbb{R}^T\right)$ of the
mirror tropical space, whose integer points parametrize a basis of
$H^0\left(SL_n/U,\mathcal{O}_{SL_n/U}\right)$, canonically determined by the
open subset $\mathcal{A} \subset SL_n/U$. Each choice of seed identifies
$\mathcal{X}\left(\mathbb{R}^T\right)$ with a real vector space, and $W^T \geq
0$ with a system of linear equations with integer coefficients, cutting out a
polyhedral cone. We obtain in this way (generally) infinitely many
parameterizations of the canonical basis as integer points of a polyhedral
cone. For the usual initial seed of the double Bruhat cell, we recover the
parametrizations of Berenstein-Kazhdan\cite{BKaz,BKaz2} and
Berenstein-Zelevinsky\cite{BZ96} by integer points of the Gelfand-Tsetlin cone.
