Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SLk-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k=2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.
Bibliographical noteFunding Information:
P. P. was partially supported by NSF grants DMS-1148634 , DMS-1351590 , and Sloan Fellowship . C. F. was supported by NSF grant DMS-1745638 and a Simons Travel Fellowship .
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- Cluster algebras
- Skein algebra
- Tensor diagram