[Paper Review] The 2-braid group and Garside normal form

Lars Thorge Jensen|arXiv (Cornell University)|May 20, 2015

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper establishes a categorical realization of the Garside normal form for positive braids via the 2-braid group acting on a categorified left cell module of the Hecke algebra. By analyzing perverse cohomology and anchors in the homotopy category of Soergel bimodules, the authors prove the faithfulness of the 2-braid group in finite type and provide a new proof of Paris' theorem on the injectivity of the canonical map from the positive braid monoid to the braid group in arbitrary type.

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.

Motivation & Objective

• To understand the categorical shadow of the Garside normal form for positive braids within the 2-braid group framework.
• To establish the faithfulness of the 2-braid group action on the categorified left cell module in finite type Coxeter groups.
• To provide a new proof of Paris’ theorem on the injectivity of the canonical map from the positive braid monoid to the braid group in arbitrary type.
• To demonstrate that the 2-braid group encodes strictly more information than its decategorified counterpart, as shown by a faithful action on a module that categorifies a non-faithful representation.

Proposed method

• Using Soergel’s Hom-formula to analyze degree-1 morphisms between indecomposable Soergel bimodules and re-express the Kazhdan-Lusztig basis multiplication in this setting.
• Constructing a categorified left cell module for the Hecke algebra via the category of Soergel bimodules, mimicking the cell module construction.
• Introducing a perverse filtration on the homotopy category of Soergel bimodules to analyze cohomological structure and detect Garside factors.
• Defining the notion of an 'anchor' as a bimodule in the highest non-vanishing perverse cohomology group of a Rouquier complex, which encodes information about Garside factorization.
• Applying induction on the length of Garside factors to reconstruct the Garside normal form from the action of positive braids on the categorified module.
• Using the autoequivalence structure of Rouquier complexes and the braid relations to recover the braid word from the action on the module.

Experimental results

Research questions

• RQ1 How is the Garside normal form of a positive braid reflected in the action of the 2-braid group on a categorified left cell module?
• RQ2 Can the faithfulness of the 2-braid group action on the categorified left cell module be established in finite type Coxeter groups?
• RQ3 Can the canonical map from the positive braid monoid to the braid group be proven injective using categorical methods?
• RQ4 Does the categorified braid group action contain strictly more information than its decategorified version?

Key findings

• The Garside normal form of a positive braid is encoded in the highest non-vanishing perverse cohomology group of its action on the categorified left cell module.
• The number of Garside factors in a positive braid equals the degree of the highest non-vanishing perverse cohomology group of its Rouquier complex.
• The 2-braid group acts faithfully on the categorified left cell module in finite type, confirming a conjecture of Rouquier.
• The faithfulness of the 2-braid group in finite type follows from the ability to reconstruct the Garside normal form from the action on the module.
• A new proof is given of Paris’ theorem: the canonical map from the positive braid monoid to the braid group is injective in arbitrary type.
• The categorified left cell module admits a faithful braid group action, even though it categorifies a twisted reduced Burau representation that is not faithful for $ n \geq 5 $, demonstrating that the categorified action contains strictly more information than the decategorified one.