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Fock-Goncharov conjecture and polyhedral cones for $U \subset SL_n$ and base affine space $SL_n /U$

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.