LAUSR

Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G . We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for the flip (changing the diagonal in a quadrilateral). This gives explicit coordinates on higher Teichmüller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in $${{\,\mathrm{Sp}\,}}(4,{\mathbb {C}})/\langle -I\rangle $$ Sp ( 4 , C ) / ⟨ - I ⟩ for the figure-eight knot complement.