[Paper Review] The dual braid monoid

David Bessis|ArXiv.org|Jan 18, 2001

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper introduces the dual braid monoid, a new Garside monoid structure for Artin groups associated with finite Coxeter systems, replacing the classical generating set $S$ with the set of all reflections $T$. The dual braid monoid shares key algebraic properties with the classical positive braid monoid, including the embedding property and a normal form, and recovers the Birman-Ko-Lee monoid in type $A$, providing a new algebraic and geometric framework for studying braid groups via reflection group symmetries.

ABSTRACT

We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical construction indicates there is a ``dual'' way of studying Coxeter systems, where the pair (W,S) is replaced by (W,T), with T the set of all reflections. In the type A case, we recover the monoid constructed by Birman-Ko-Lee

Motivation & Objective

• To develop an alternative algebraic framework for finite real reflection groups and their braid groups by replacing the classical Coxeter system $(W,S)$ with $(W,T)$, where $T$ is the set of all reflections.
• To construct a new positive braid monoid—called the dual braid monoid—defined by dual braid relations, analogous to the classical positive braid monoid but based on reflections rather than simple reflections.
• To prove that the dual braid monoid is a Garside monoid, ensuring it inherits crucial structural properties such as the embedding property, normal forms, and solvability of the word problem.
• To establish a geometric interpretation of the dual braid monoid via local monoids associated with regular vectors in the complexified hyperplane arrangement.
• To demonstrate that in type $A$, the dual braid monoid coincides with the monoid constructed by Birman, Ko, and Lee, thereby unifying two distinct approaches to braid group structures.

Proposed method

• The dual braid monoid is defined via a positive presentation using dual braid relations derived from the reflection set $T$, forming a pre-monoid structure on the set of elements less than or equal to a Garside element $\Delta$.
• The construction relies on the existence of a Garside element $\Delta$ in the monoid such that the set $P = \{m \in M \mid m \prec \Delta\}$ forms a Garside pre-monoid, enabling the recovery of the full monoid via the free monoid construction $\mathbf{M}(P)$.
• The paper proves that the dual braid monoid is a Garside monoid by verifying the axioms of a Garside pre-monoid, particularly the existence of left and right least common multiples and greatest common divisors for pairs of generators.
• A key technical tool is the use of the Coxeter element $c$ as a candidate for the Garside element, with the poset $(P_c, \prec)$ shown to be a lattice under the divisibility order.
• The geometric interpretation associates a local monoid $M_v$ to each regular vector $v$ in the complexified reflection space, where $M_v$ is isomorphic to the classical braid monoid when $v$ lies in a real chamber and to the dual braid monoid when $v$ lies in a special stratum.
• The paper uses case-by-case verification for exceptional types (e.g., $H_3$, $I_2$) and computer-assisted proofs where necessary, while providing general arguments for classical types $A$, $B$, and $D$.

Experimental results

Research questions

• RQ1 Can a new positive braid monoid be constructed for Artin groups using the full set of reflections $T$ instead of the simple reflections $S$?
• RQ2 Does this new monoid, called the dual braid monoid, satisfy the Garside property, including the embedding property and the existence of a normal form?
• RQ3 How does the dual braid monoid relate to the classical positive braid monoid and to the Birman-Ko-Lee monoid in type $A$?
• RQ4 What is the geometric interpretation of the dual braid monoid in terms of the complexified hyperplane arrangement and local monodromy structures?
• RQ5 Can the dual braid monoid be characterized algebraically and geometrically in all finite Coxeter types, including exceptional ones?

Key findings

• The dual braid monoid is a Garside monoid, which implies it satisfies the embedding property, ensuring injectivity into the corresponding Artin group.
• The dual braid monoid admits a normal form based on the left gcd decomposition with respect to a Garside element $\Delta$, enabling algorithmic solutions to the word problem.
• In type $A_n$, the dual braid monoid is isomorphic to the monoid constructed by Birman, Ko, and Lee, thus unifying two distinct constructions of the same algebraic object.
• The dual braid monoid arises geometrically as the local monoid $M_v$ when the regular vector $v$ lies in a specific stratum of the complexified hyperplane arrangement, corresponding to the eigenspace of a Coxeter element.
• For types $B$ and $D$, the dual braid monoid is shown to be a Garside monoid via the existence of a lattice structure on the set of divisors of a Coxeter element $c$, confirming the general theory in these cases.
• The diagram automorphism induced by conjugation by $\Delta$ has finite order, and $\Delta^d$ is central in the Artin group, a key structural feature of Garside groups.