The 2-braid group and Garside normal form

2015/05/20 by Lars Thorge Jensen, Jensen, Lars Thorge
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.CT #math.RT

paper · pdf · doi:10.48550/arxiv.1505.05353

40 pages

openalex publication_date 2015/05/20 · arxiv created 2015/10/02 · arxiv updated 2015/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the relation between the Garside normal form for positive braids and the 2-braid group defined by Rouquier. Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the 2-braid group on a certain categorified left cell module. This allows us to deduce the faithfulness of the 2-braid group in finite type. We also give a new proof of Paris' theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type.

Cited by

• Faithful Burau-like representations of some rank two Garside groups and torus knot groups

Related

• A dual braid monoid for the free group
• Alternating normal forms for braids and locally Garside monoids
• Braid groups are linear
• Explicit Presentations for the Dual Braid Monoids
• Left-Garside categories, self-distributivity, and braids