Tensor diagrams and cluster combinatorics at punctures

TL;DR

This paper explores the algebraic and combinatorial structures of cluster algebras associated with punctured surfaces, extending tagged triangulations to higher dimensions using tensor diagrams and skein algebra tools.

Contribution

It introduces a higher-dimensional analogue of tagged triangulations via tensor diagrams and develops skein algebra methods for computations in these advanced cluster algebras.

Findings

Extension of tagged triangulations to higher k using tensor diagrams

Development of skein algebra tools for cluster algebra calculations

Detailed analysis of finite mutation type examples

Abstract

Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SL(k)-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 is 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.