Fock-Goncharov coordinates for rank two Lie groups

TL;DR

This paper develops explicit Fock-Goncharov coordinates for configurations of affine flags in rank 2 complex Lie groups, linking geometric actions to quiver mutations and applying these to higher Teichmüller spaces and 3-manifold representations.

Contribution

It provides explicit coordinate systems for affine flag configurations in rank 2 Lie groups and relates geometric actions to quiver mutations, enabling new computations in higher Teichmüller theory.

Findings

Explicit Fock-Goncharov coordinates for rank 2 Lie groups

Connection between permutations and quiver mutations

Computed boundary-unipotent representations for the figure-eight knot

Abstract

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 Teichmuller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in Sp(4,C) for the figure-eight knot complement.