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Abstract

The {\em rank $n$ swapping algebra} is a Poisson algebra defined on the set
of ordered pairs of points of the circle using linking numbers, whose geometric
model is given by a certain subspace of $(\mathbb{K}^n \times
\mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$. For any ideal
triangulation of $D_k$---a disk with $k$ points on its boundary, using
determinants, we find an injective Poisson algebra homomorphism from the
fraction algebra generated by the Fock--Goncharov coordinates for
$\mathcal{X}_{\operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction
algebra for $r=k\cdot(n-1)$ with respect to the (Atiyah--Bott--)Goldman Poisson
bracket and the swapping bracket. This is the building block of the general
surface case. Two such injective Poisson algebra homomorphisms related to two
ideal triangulations $\mathcal{T}$ and $\mathcal{T}'$ are compatible with each
other under the flips.