Unbounded $\mathfrak{sl}_3$-laminations around punctures

TL;DR

This paper advances the understanding of unbounded $rak{sl}_3$-laminations near punctures by classifying webs, analyzing Weyl group actions, and connecting various approaches, with implications for general Lie algebras.

Contribution

It provides a classification of signed $rak{sl}_3$-webs around punctures and details the tropicalization of Weyl group actions, linking multiple existing frameworks.

Findings

Classification of signed $rak{sl}_3$-webs around punctures

Detailed description of tropicalized Weyl group actions

Discussion on generalization to other semisimple Lie algebras

Abstract

We continue to study the unbounded $\mathfrak{sl}_3$-laminations [IK22], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $\mathfrak{sl}_3$. After giving a classification of signed $\mathfrak{sl}_3$-webs around a puncture, we describe the tropicalization of the Goncharov--Shen's Weyl group action in detail. We also clarify the relationship with several other approaches by Shen--Sun--Weng [SSW23] and Fraser--Pylyavskyy [FP21]. Finally, we discuss a formulation of unbounded $\mathfrak{g}$-laminations for a general semisimple Lie algebra $\mathfrak{g}$ in brief.