Rank $n$ swapping algebra for ${\mathrm{PGL}}_n$ Fock-Goncharov $\mathcal{X}$ moduli space

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 ${\mathcal{X}}_{{\mathrm{PGL}}_n,D_k}$ 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 ${\mathcal{T}}^{\mathrm{\prime }}$ are compatible with each other under the flips.