The Dual Braid Monoid

We describe a new monoid structure for braid groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid its crucial algebraic properties: the monoid satisfies Öre’s conditions and embeds in its group of fractions, it admits a nice normal form, it can be used to construct braid group actions on categories... It also provides a new presentation for braid groups; the conjugation by a Coxeter element is a “diagram automorphism” of the new presentation. In the type A case, one recovers the Birman-Ko-Lee presentation. Introduction Let (W,S) be a finite Coxeter system. The group W is given by the group presentation < S|∀s ∈ S, s = 1 ; ∀s, t ∈ S, sts . . . } {{ } ms,t = tst . . . } {{ } ms,t >group . Let B(W,S) be the corresponding Artin group. To have simple yet precise notations, it is convenient to introduce a formal copy S ≃ S. For each s ∈ S, we write s the corresponding element of S. With this convention, B(W,S) is defined as the abstract group B(W,S) :=< S|∀s, t ∈ S, sts . . . } {{ } ms,t = tst . . . } {{ } ms,t >group . The map s 7→ s extends to a surjective morphism p : B(W,S) → W . Since the defining relations are between positive words, the presentation of B(W,S) can also be seen as a monoid presentation. We set B+(W,S) :=< S|∀s, t ∈ S, sts . . . } {{ } ms,t = tst . . . } {{ } ms,t >monoid . This monoid is often called the positive braid monoid. We prefer here the term of classical braid monoid (short for Artin-Brieskorn-DeligneGarside-Saito-Tits monoid). The structure of B(W,S) and B+(W,S) has been studied in great detail in [D1] and [BS]. One of the main results is that B+(W,S) satisfies the embedding property, i.e., the morphism B+(W,S) → B(W,S) is injective (since the solution of the presentation universal problem