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

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.

57 pages, 26 figures. v2: added a comment on the Roger--Yang relations in p.42. arXiv admin note: text overlap with arXiv:2204.08947