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Fock-Goncharov coordinates for rank two Lie groups

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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 Teichmüller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in \({{\,\mathrm{Sp}\,}}(4,{\mathbb {C}})/\langle -I\rangle \) for the figure-eight knot complement.

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Acknowledgements

The author wishes to thank Matthias Goerner, Stavros Garoufalidis, and Dylan Thurston for helpful comments.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    Christian K. Zickert

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  1. Christian K. Zickert
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Correspondence to Christian K. Zickert.

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The author was supported in part by DMS-13-09088.

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Zickert, C.K. Fock-Goncharov coordinates for rank two Lie groups. Math. Z. 294, 251–286 (2020). https://doi.org/10.1007/s00209-019-02307-8

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