Dual Braid Research Articles

Coxeter Pop-Tsack Torsing

Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T :W→W by Pop T (w)=w·π T (w) -1 , where π T (w) is the join in the noncrossing partition lattice NC(w,c) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. We also prove that W is coincidental if and only if it has a unique forward orbit of size h. For arbitrary W, we show that the forward orbit of c -1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

Boundary braids

The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points remain pointwise fixed forms a subgroup isomorphic to a braid group with fewer strands. In this article, we generalize this phenomenon by introducing the notion of boundary braids. A boundary braid is a braid that has a representative where some specified subset of the points remains in the boundary cycle of the disk. Although boundary braids merely form a subgroupoid rather than a subgroup, they play an interesting geometric role in the piecewise Euclidean dual braid complex defined by Tom Brady and the second author. We prove several theorems in this setting, including the fact that the subcomplex of the dual braid complex determined by a specified set of boundary braids metrically splits as the direct metric product of a Euclidean polyhedron and a dual braid complex of smaller rank.

Dual Garside structures and Coxeter sortable elements

In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading's bijection from $c$-sortable elements to noncrossing partitions of a Coxeter element $c$, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of $c$-sortable elements.

Asymptotic combinatorics of Artin–Tits monoids and of some other monoids

We introduce methods to study the combinatorics of the normal form of large random elements in Artin–Tits monoids. These methods also apply in an axiomatic framework that encompasses other monoids such as dual braid monoids.

Uniform measures on braid monoids and dual braid monoids

We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by μk the uniform distribution on positive (dual) braids of length k, we prove that the sequence (μk)k converges to a unique probability measure μ∞ on infinite positive (dual) braids. The key point is that the limiting measure μ∞ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a by-product, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids.

Dual braid monoids, Mikado braids and positivity in Hecke algebras

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written \(x^{-1} y\) where x and y are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type \(A_n\) and \(B_n\) and then show that in spherical types different from \(D_n\) the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type \(D_n\)). This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type \(A_n\), it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.

Noncrossing partitions and Bruhat order

We prove that the restriction of Bruhat order to type A noncrossing partitions for the Coxeter element c=s1s2⋯sn defines a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the base change from the graphical basis of the Temperley–Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we introduce a new set of vectors counted by the Catalan numbers and give new bijections–fixing each reflection–between noncrossing partitions associated to distinct Coxeter elements.

Every braid admits a short sigma-definite expression

A result by Dehornoy (1992) says that every nontrivial braid admits a σ -definite expression, defined as a braid word in which the generator σ_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a σ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.

FINITE THURSTON-TYPE ORDERINGS ON DUAL BRAID MONOIDS

For a finite Thurston-type ordering < of the braid group Bn, we introduce a new normal form of a dual positive braid which we call the [Formula: see text]-normal form, which is useful to compute the ordering. This normal form extends Fromentin's rotating normal form and the author's [Formula: see text]-normal form of positive braids. Using the [Formula: see text]-normal form, we give a combinatorial description of the restriction of the ordering < to the dual braid monoids [Formula: see text]. We prove that the restriction to the dual positive monoid [Formula: see text] is a well-ordered set of the order type ωωn-2.

THE WELL-ORDERING OF DUAL BRAID MONOID

We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:math.GR/0811.3902, and then proving that this new ordering coincides with the standard ordering of braids.

Growth function for a class of monoids

In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras. Nous étudions une classe de monoïdes incluant les monoïdes de Garside, et donnons une preuve combinatoire simple d'une formule pour la somme formelle de leurs éléments. Cela mène à une formule pour la fonction de croissance du monoïde dans le cas homogène, et peut être aussi relevé en une résolution de l'algèbre de monoïdes. Ces résultats sont ensuite appliqués aux monoïdes liés aux systèmes de Coxeter : nous donnons la fonction de croissance des monoïdes d'Artin-Tits ainsi que des monoïdes duaux ; pour ces derniers nous montrons que leur algèbre de monoïde en types A et B est une algèbre de Koszul.

A well-ordering of dual braid monoids

Let B n + ∗ denote the dual braid monoid on n strands, i.e., the submonoid of the braid group B n consisting of the braids that can be expressed as positive words in the Birman–Ko–Lee generators. We introduce a new normal form on B n + ∗ , which is based on expressing every braid of B n + ∗ in terms of a certain finite sequence of braids of B n − 1 + ∗ . We deduce an inductive characterization of the Dehornoy ordering of dual braid monoids, and explicitly compute the associated order types. To cite this article: J. Fromentin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

A dual braid monoid for the free group

We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman–Ko–Lee and by the author of new Garside monoids for Artin groups of spherical type.

Non-crossing partitions of type [formula omitted

We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group G ( e , e , r ) . For the particular case e = 2 (resp. r = 2 ), our lattice coincides with the lattice of simple elements for the type D n (resp. I 2 ( e ) ) dual braid monoid. Using this lattice, we construct a Garside structure for the braid group B ( e , e , r ) . As a corollary, one may solve the word and conjugacy problems in this group.

Explicit presentations for the dual braid monoids

Birman, Ko and Lee have introduced a new monoid B ∗ n —with an explicit presentation—whose group of fractions is the n-strand braid group B n . Building on a new approach by Digne, Michel and himself, Bessis has defined a dual braid monoid for every finite Coxeter type Artin–Tits group extending the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the underlying Garside structures. To cite this article: M. Picantin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 843–848.