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Non-crossing partitions of type (e,e,r)
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Abstract
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 (resp. ) 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.
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OUTLINE. In the first half of the thesis, we introduce the class of'chainable monoids' and describe general techniques for solving the word and division problems and for obtaining normal forms for monoids in this class. Their growth functions are shown to be rational, and calculable. We also show that a monoid with a certain central 'fundamental element' embeds in a group or related monoid. Motivating examples of monoids in this class are positive braid monoids and positive singular braid monoids. Much of the rest of the thesis is devoted to applying the techniques described to singular Artin monoids, and we obtain further results regarding conjugacy and parabolic submonoids. In the final chapter we apply the techniques of chainable monoids to braid groups of complex reflection groups. A chainable monoid is a monoid which may be defined by a finite presentation with four properties; three of which can be determined immediately from the length of the relations and their starting letters, and the fourth, the reduction property, which is more involved to verify. It has been shown that positive Artin monoids, including braid monoids, have these properties. Other examples of monoids in this class include free monoids, free commutative monoids, positive singular braid monoids (see later), a large class of one relation monoids, trace monoids and divisibility monoids. The class of chainable monoids is closed under taking direct and free products. Chainable monoids are always left cancellative. The method of chains enables the construction of a calculable partial map on pairs of words, which returns the left quotient of the first word by the second, whenever it exists, thus giving a solution to the division problem, and hence word problem, in a chainable monoid. Again using chains, we construct another partial map on pairs of words which is shown to return their least common left multiple, whenever it exists. We show that in a chainable monoid, greatest common left divisors always exist, and least common left multiples exist whenever common left multiples exist. [The author is grateful to Dietrich Kuske for recently pointing out that chainable monoids may be algebraically characterised as left cancellative monoids that
A new presentation of then-string braid groupBnis studied. Using it, a new solution to the word problem inBnis obtained which retains most of the desirable features of the Garside–Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.
We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em non-crossing cumulant of type B;} the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of ``non-commutative probability space of type B'', and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of ``vanishing of mixed cumulants of type B'', we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.
We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for Artin braid groups and some other cases. Our proof makes use of results by Birman-Ko-Lee. We give a new intrinsic account of these results. Notations If G is a group acting on a set X, we denote by X G the subset of X of elements fixed by all elements of G. If (X, x) is a pointed topological space, we denote by Ω(X, x) the corresponding loop space, by ∼ the homotopy relation on Ω(X, x) and by π1(X, x) the fundamental group. For all n ∈ N, we denote by µn the set of n-th roots of unity in C.
Nous généralisons au cas des groupes de réflexions complexes la définition de Rouquier des “familles de caractères” des groupes de Weyl (définies antérieurement par Lusztig). Nous mettons en évidence le fait que cette notion dépend d'une manière essentielle du choix de la déformation de l'algèbre de groupe appelée “algèbre cyclotomique”, et nous calculons ces familles pour toutes les algèbres cyclotomiques des familles infinies de groupes de réflexions complexes qui généralisent les familles infinies de groupes de Weyl.
Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space U n of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.
L'article définitions partitious d'un ensemble fini structuré en un cycle qui possédent la propriété qu'une paire de points appartenant à une classe et une paire de points appartenant à une autre classe ne puissent jamais étre en disposition croisée. On établit que ces partitions forment un treillis et l'on précise quelques-unes des propriétés descriptives et énumératives de ce treillis, on en calcule en particulier la fonction de Möbius.
It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lower common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right l.c.m.'s satisfy an additional symmetry condition. A Gaussian group and a Garside group are respectively the group of fractions of a Gaussian monoid and of a Garside monoid. Braid groups and, more generally, finite Coxeter type Artin groups are Garside groups. We determine algorith-mic criterions in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that theses algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter type Artin group that can be explicitely determined.
We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair (W,S) by (W,T), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.RésuméNous étudions une nouvelle structure de monoı̈de pour les groupes d'Artin associés aux systèmes de Coxeter finis. Ce nouveau monoı̈de est, tout comme le classique monoı̈de des tresses positives, un monoı̈de de Garside. Nous en donnons différentes constructions : algébriquement, le nouveau monoı̈de apparaı̂t quand on étudie les systèmes de Coxeter avec un point de vue “dual”, qui consiste à remplacer la paire (W,S) par (W,T), où T est l'ensemble de toutes les réflexions ; géométriquement, il apparaı̂t quand on observe l'arrangement de réflexions depuis un point-base particulier. Pour les systèmes de type A, nous retrouvons le monoı̈de construit par Birman, Ko et Lee.
The first part of this paper investigates a class of homogeneously presented monoids. Constructions which enable division and multiplication to be computed are described. The word problem and the division problem are solved, and a unique normal form is given for monoids in this class. The second part deals with the motivating examples of this class—the positive singular Artin monoids. Finally, the singular Artin monoids are introduced. The word and division problems are solved for all singular Artin monoids of finite type, including, for example, the singular braid monoids introduced by Baez and Birman.
A combinatorial property of positive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the associated monoid and group from a complete presentation: cancellativity or existence of common multiples in the case of the monoid, or isoperimetric inequality in the case of the group. In particular, we obtain a new criterion for recognizing that a monoid embeds in a group of fractions. Typical presentations eligible for the current approach are the standard presentations of the Artin–Tits groups and the Heisenberg group.
We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of types B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain decompositions (self-dual for type B), whose rank-generating functions, zeta polynomials, rank-selected chain numbers have simple closed forms.
Presentations "à la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.
A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(\pi,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(\pi,1)s enjoy Bestvina's weak non-positive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually abelian.
Non-crossing partitions for the group Dn
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45 rue d'Ulm, 75230 Paris cedex 05, France E-mail address: david dot bessis at ens dot fr
- École Dma
- Normale Supérieure
DMA,École normale supérieure, 45 rue d'Ulm, 75230 Paris cedex 05, France
E-mail address: david dot bessis at ens dot fr
11 rue Pierre et Marie Curie, 75231 Paris cedex 05, France E-mail address: corran at ihp dot jussieu dot fr
- Henri Institut
- Poincaré
Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75231 Paris cedex 05, France
E-mail address: corran at ihp dot jussieu dot fr