A new Garside structure for braid groups of type (e,e,r)
aa r X i v : . [ m a t h . G R ] F e b A NEW GARSIDE STRUCTUREFOR BRAID GROUPS OF TYPE ( e, e, r ) RUTH CORRAN AND MATTHIEU PICANTIN
Abstract.
We describe a new presentation for the complex re-flection groups of type ( e, e, r ) and their braid groups. A diagramfor this presentation is proposed. The presentation is a monoidpresentation which is shown to give rise to a Garside structure. Adetailed study of the combinatorics of this structure leads us todescribe it as post-classical . Introduction
A complex reflection group is a group acting on a finite-dimensionalcomplex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise. Anyreal reflection group becomes a complex reflection group if we extendthe scalars from R to C . In particular all Coxeter groups or Weyl groupsgive examples of complex reflection groups, although not all complexreflection groups arise in this way. One would like to generalise asmuch as possible from the theory of Weyl groups and Coxeter groupsto complex reflection groups.For instance, according to Brou´e–Malle–Rouquier [BMR], one can de-fine the braid group B ( W ) attached to a complex reflection group G ( W )as the fundamental group of the space of regular orbits. When G ( W )is real, the braid group B ( W ) is well understood owing to Brieskorn’spresentation theorem and the subsequent structural study by Deligneand Brieskorn–Saito [Br, Del, BS]: their main combinatorial results ex-press that B ( W ) is the group of fractions of a monoid in which divisi-bility has good properties, and, in addition, there exists a distinguishedelement whose divisors encode the whole structure: in modern termi-nology, such a monoid is called Garside. The group of fractions of aGarside monoid is called a Garside group. Garside groups enjoy manyremarkable group-theoretical, cohomological and homotopy-theoreticalproperties. Finding (possibly various) Garside structures for a givengroup becomes a natural challenge. Date : October 31, 2018.
The groups G ( e, e, r ) and B ( e, e, r ) . The classification of (ir-reducible) finite complex reflection groups was obtained by Shephardand Todd [ST]: • an infinite family G ( de, e, r ) where d, e, r are arbitrary positiveintegral parameters; •
34 exceptions, labelled G , . . . , G .The infinite family includes the four infinite families of finite Cox-eter groups: G (1 , , r ) ∼ G ( A r − ), G (2 , , r ) ∼ G ( B r ), G (2 , , r ) ∼ G ( D r ) and G ( e, e, ∼ G ( I ( e )). For all other values of the parame-ters, G ( de, e, r ) is an irreducible monomial complex reflection group ofrank r , with no real structure.In the infinite family, one may consider, in addition to the real groups,the complex subfamily G ( e, e, r )—note that this subseries contains the D -type and I -type Coxeter series—and our objects of interest are thepossible Garside structures for the braid group B ( e, e, r ).The reflection groups of type ( e, e, r ) are defined in terms of positiveintegral parameters e, r : G ( e, e, r ) = r × r monomial matrices( x ij ) over { } ∪ µ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y x ij =0 x ij = 1 , that is, as the group of r × r matrices consisting of: • monomial matrices (each row and column has a unique non-zeroentry), • with all non-zero entries lying in µ e , the e -th roots of unity, and • for which the product of the non-zero entries is 1.The group G ( e, e, r ) is generated by reflections of C r . There are hy-perplanes in C r corresponding to the reflections of the reflection group.The corresponding braid group B ( e, e, r ) is defined in terms of the fun-damental group of a quotient of the hyperplane complement. We donot make recourse to this definition; our starting point will be knownpresentations for these braid groups.1.2. Brou´e–Malle–Rouquier presentation.
Such a presentation forthe braid group B ( e, e, r ) may be found in [BMR]: • Generators: { t , t } ∪ S r with S r = { s , . . . , s r } , and NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 3 • Relations:( R ) s i s j s i = s j s i s j for | i − j | = 1 , ( R ) s i s j = s j s i for | i − j | > , ( P ) h t t i e = h t t i e ( P ) s t i s = t i s t i for i = 0 , , ( P ) s j t i = t i s j for i = 0 , , and 4 ≤ j ≤ r, and( P ) ( s t t ) = ( t t s ) , where h ab i e denotes the alternating product of a and b with e terms.The collections of relations ( R ) and ( R ) are the usual braid relationson those generators in S r .Furthermore, it is shown in [BMR] that by adding the relation a = 1for all generators a , a presentation for the reflection group G ( e, e, r ) isobtained. The generators in this case are all reflections in G ( e, e, r ). et t s s s r − s r Figure 1.
The diagram of type ( e, e, r ) by [BMR].A diagram shown in Figure 1 is proposed in [BMR] for this presentation.This diagram is interpreted, where possible, as a Coxeter diagram. Thevertices correspond to generators, and the edges to relations: for eachpair of vertices a and b , • no edge connecting the vertices corresponds to a relation ab = ba , • an unlabelled edge connecting the vertices corresponds to aba = bab , • an edge labelled e connecting the vertices corresponds h ab i e = h ba i e .The first two of these give the usual braid relations and the rela-tions ( P ) and ( P ); the third gives the relation ( P ). It remains tointerpret the triangle with short double-line in the interior; in thediagram above, this represents the relation ( P ): s ( t t ) s ( t t ) =( t t ) s ( t t ) s . (This would be a relation corresponding to an edgelabelled 4 between nodes s and t t , if the latter were a node. Con-ventionally, edges labelled by 4 in Coxeter diagrams are designated bydouble-lines.)In the case of finite real reflection groups—that is, finite Coxeter groups—an enormous amount of understanding about the reflection and braidgroups arises from the Coxeter presentations coming from the choice ofgenerators corresponding to a simple set of roots in a root system. Inthis paper we describe presentations of the reflection groups G ( e, e, r )and their braid groups B ( e, e, r ) which have some properties like thoseof Coxeter presentations. RUTH CORRAN AND MATTHIEU PICANTIN
Classical vs dual braid monoids. The success of [BKL], whichdescribes an alternative braid monoid for the ordinary braid group B ( A n − ),provided the impetus to unify the different approaches by introducinga general framework: the Garside theory (see [DP, D2, BDM, B2]).This terminology refers to the fact that, although dealing only withthe ordinary n -strand braid group B ( A n − ), the pioneer paper by Gar-side [G] stands out in which the foundation is laid for a more systematicstudy of the divisibility theory in a well-chosen submonoid of the braidgroup.Garside structures (see Subsection 3.1 for details) are desirable becausethey allow fast calculation in the group (solution to word and conjugacyproblems) by convenient canonical or normal forms. A given Garsidegroup admits possibly several Garside structures, each providing an as-sociated biautomatic structure, etc. Known examples of Garside groupsare braid groups, torus link groups, one-relator groups with center, etc.In the particular case which concerns us here, that is, in the case ofbraid groups, two Garside structures—when defined—seem to be mostnatural: we will use here the term of classical braid monoid (short forArtin–Brieskorn–Deligne–Garside–Saito–Tits monoid) and the term of dual braid monoid proposed by Bessis in [B2] (corresponding to thosemonoids studied in [BKL, B2, P, BC1, BC2, ...]. Given a reflectiongroup G ( de, e, r ), when defined and when no confusion is possible, wewill write B + ( de, e, r ) for the classical braid monoid and B × ( de, e, r )for the dual braid monoid.The presentation in [BMR] does not give rise to a Garside structure . Apresentation giving rise to a Garside monoid for B ( e, e, r ) was obtainedin [BC2]; this monoid fits into the context of dual braid monoids andwill be denoted by B × ( e, e, r ). In that case, the generators are in bijec-tion with the reflections in G ( e, e, r ). In this paper we introduce a newpresentation for B ( e, e, r ) which again gives rise to a Garside monoid,but which has more in common with the classical braid monoids thanthe dual braid monoids.The organization of the rest of the paper is as follows. In Section 2, anew presentation (with an associated diagram) for B ( e, e, r ) is shown(Theorem 2.1). In Section 3 we prove that this new presentation givesrise to a Garside monoid B ⊕ ( e, e, r ) (Theorem 3.2). The underlyingGarside structure is then investigated (Theorem 3.7). Finally, thisallows us to situate B ⊕ ( e, e, r ) as well as possible with respect to thedichotomy between classical and dual braid monoids (Subsection 3.4). In particular, this presentation can be viewed as a monoid presentation; the asso-ciated monoid is not cancellative, so does not embed in a group (see [Co, BC2]).
NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 5 A new presentation
In this section we first introduce the new presentation for the braidgroup B ( e, e, r ), propose a diagram for the presentation, and then dis-cuss its relationship to the reflection group and to other braid groups.Finally, after considering the notion of circle, we prove that the givenpresentation does present the group B ( e, e, r ).2.1. New presentation of type ( e, e, r ) . Let P ⊕ ( e, e, r ) denote thepresentation given by: • Generators: T e ∪ S r with T e = { t i | i ∈ Z /e } and S r = { s , . . . , s r } , and • Relations:( R ) s i s j s i = s j s i s j for | i − j | = 1,( R ) s i s j = s j s i for | i − j | > R ) s t i s = t i s t i for i ∈ Z /e ,( R ) s j t i = t i s j for i ∈ Z /e and 4 ≤ i ≤ r , and( R ) t i t i − = t j t j − for i, j ∈ Z /e .We will show in Subsection 2.4: Theorem 2.1.
The presentation P ⊕ ( e, e, r ) is a group presentation forthe braid group B ( e, e, r ) . Furthermore, adding the relations a = 1 forall generators a gives a presentation of the reflection group G ( e, e, r ) .In particular, the generators of this presentation are all reflections. The new generating set is a superset of the generating set of [BMR].The new generators t i for 2 ≤ i ≤ e − t i = t i − t i − t − i − , and so are just conjugates of the original generators.The presentation P ⊕ ( e, e, r ) can be viewed as a monoid presentation.The corresponding monoid B ⊕ ( e, e, r ) will be the starting point forconstructing the Garside structure for B ( e, e, r ), and we will see: Proposition 2.2.
The submonoid of B ( e, e, r ) generated by T e ∪ S r is isomorphic to the monoid B ⊕ ( e, e, r ) , that is, it can be presentedby P ⊕ ( e, e, r ) considered as a monoid presentation. New diagram of type ( e, e, r ) . We propose the diagram shownin Figure 2 as a type ( e, e, r ) analogy to the Coxeter diagrams for thereal reflection group case.This diagram is again to be read as a Coxeter diagram where possible,that is, when vertices a and b are joined by an (unlabelled) edge, thereis a relation aba = bab . The circle with e vertices at the left of thediagram corresponds to the circle { t i | i ∈ Z /e } (see Subsection 2.4.1). RUTH CORRAN AND MATTHIEU PICANTIN t t e − t t e − t s s s r − s r Figure 2.
The new diagram of type ( e, e, r ): there are e nodes on the circle.Whenever two vertices a and b lie on this circle, there is a relationof the form aa (cid:3) = bb (cid:3) where a (cid:3) and b (cid:3) are the nodes immediatelypreceding a and b respectively on the circle. If two nodes a and b areneither connected by an edge nor both lie on the disc, then there isa relation of the form ab = ba —that is, the corresponding generatorscommute. The diagram automorphism (cid:3) and its inverse (cid:2) . Define the map (cid:3) by s (cid:3) j = s j for all 3 ≤ j ≤ r and t (cid:3) i = t i − for all i ∈ Z /e . Since ρ (cid:3) = ρ (cid:3) itself is a defining relation whenever ρ = ρ is a defining relation,then (cid:3) is a well-defined monoid morphism of B ⊕ ( e, e, r ). Furthermore,since the whole set of relations defining B ⊕ ( e, e, r ) is stable under (cid:3) ,the map (cid:3) is an automorphism of B ⊕ ( e, e, r ). The automorphism (cid:3) rotates the circle in the negative direction by a turn of πe .The samecan be said for its inverse (cid:2) . These diagram automorphisms give rise toautomorphisms of the braid group B ( e, e, r ) as well as of the reflectiongroup G ( e, e, r ). Moreover, these diagram automorphisms send (braid)reflections to (braid) reflections. Proposition 2.3.
The (cid:3) -trivial subgroup of the braid group B ( e, e, r ) is isomorphic to the braid group B ( B r − ) .Proof. The proof follows the one of [DP, Proposition 9.4]. (cid:3)
The diagram anti-isomorphism rev . Let rev( P ⊕ ( e, e, r )) denotethe presentation on the same generators as P ⊕ ( e, e, r ), and relationsobtained by reversing all its relations. This presentation has a dia-gram corresponding to the mirror image of the diagram for P ⊕ ( e, e, r ).Let rev( B ⊕ ( e, e, r )) be the monoid defined by rev( P ⊕ ( e, e, r )). Lemma 2.4.
The monoid B ⊕ ( e, e, r ) is isomorphic to rev( B ⊕ ( e, e, r )) by the isomorphism ϕ which sends t i t − i and s j s j . NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 7
Proof.
The map ϕ permutes the generators T e ∪ S r , and is bijectivebetween the relations of P ⊕ ( e, e, r ) and those of rev( P ⊕ ( e, e, r )): thelatter is clear for all types of relation possibly except ( R ), and in thiscase we find: ϕ ( t i t i − ) = t − i t − i = rev( R ) t − j t − j = ϕ ( t j t j − ) . So ϕ is a well-defined monoid homomorphism, which is both surjective(as it permutes the generators) and injective (as it is bijective on therelations). Hence it is an isomorphism of monoids. (cid:3) Thus ‘mirror flipping’ the diagram corresponds to a group isomorphismbut not an equality. Unlike for braid groups of Coxeter groups, thisdiagram morphism does not give rise to an automorphism of B ( e, e, r ).2.3. Natural maps between different types.
Parabolic subgroupsof (braid groups of) Coxeter groups may be realized by considering sub-diagrams of the corresponding diagrams. We describe here parabolics oftype ( e, e, r ) and the corresponding subdiagrams of the diagram shownon Figure 2, as well as maps which arise by taking diagram quotientsinstead.2.3.1.
Maps related to parabolic subdiagrams.
Following [BMR], for agiven diagram, consider the equivalence relation on nodes defined by s ∼ s , and for s = ts ∼ t ⇔ s and t are not in a homogeneous relation with support { s, t } . Thus, for the diagram of Figure 2, the equivalence classes have 1 or e elements, and there is at most one class with e elements.An admissible subdiagram is a full subdiagram of the same type, thatis, with 1 or e elements per class.An admissible subdiagram of a diagram of type ( e, e, r ) must be of theform the union of a diagram of type ( e , e , r ) along with k diagramsof type (1 , , r i ) where e ∈ { , , e } and P ki =0 r i ≤ r .Particular examples are considered below, which show the relationshipwith braid groups of some real reflection groups. • P ⊕ ( e, e, r ′ ) with r ′ ≤ r : the case of ‘chopping off the tail’ of theparachute. This corresponds to reducing the dimension from r to r ′ .A special case of this is P ⊕ ( e, e, B × ( I ( e )) (see Remark 1 on page 10). • P ⊕ (0 , , r ) is a presentation of the classical braid monoid B + ( A r − ). • P ⊕ (1 , , r ) is a presentation of the classical braid monoid B + ( A r − ). • P ⊕ (2 , , r ) is a presentation of the classical braid monoid B + ( D r ). RUTH CORRAN AND MATTHIEU PICANTIN
These sub-presentations will be used in Subsection 3.2.1 in the contextof cube condition calculations.2.3.2.
Maps related to foldings (diagram quotients). (1) Epimorphism B ( e , e , r ) ։ B ( e , e , r ) for e dividing e .The map induced by t j t j mod e and s j s j defines an epi-morphism ν : B ( e , e , r ) ։ B ( e , e , r ). There is an analogousmap between the corresponding monoids and reflection groups.This corresponds to a folding of the ‘parachute’ part of the di-agram.(2) Type B embedding: B (2 , , r − ֒ → B ( e, e, r ).The type (2 , , r −
1) corresponds to the Artin-Tits/Coxetertype B r − . The associated Coxeter diagram is: q q q q r − q r − (the double bar between nodes labelled q and q is equivalentto an edge labelled 4).Whether by an easy adaptation of [Cr, Lemma 1.2 & Theo-rem 1.3] or a direct application of [D1, Proposition 5.4], severalembedding criteria can be applied successfully within the cur-rent framework. We obtain that the map induced by q t i t i − and q j s j +1 for j > B + ( B r − ) ֒ → B ⊕ ( e, e, r ), hence an injection B ( B r − ) ֒ → B ( e, e, r ). This em-bedding will be used in Subsection 3.2.2.2.4. The new presentation is B ( e, e, r ) . Our aim in this subsectionis to prove the theorem announced in the opening subsection:
Theorem 2.1.
The presentation P ⊕ ( e, e, r ) is a group presentationfor the braid group B ( e, e, r ) . Furthermore, adding the relations a = 1 for all generators a gives a presentation of the reflection group G ( e, e, r ) .In particular, the generators of this presentation are all reflections. We will use the presentation of [BMR] as our starting point, givenon page 2. To this presentation we will add generators t i for 2 ≤ i ≤ e corresponding to conjugates of t and t which may be definedinductively by: t i = t i − t i − t − i − for i ≥ . We then verify that the new relations given are both necessary andsufficient. To do this, we introduce the notion of a circle of elementsin a group , as T e = { t i | i ∈ Z /e } turns out to be the circle on ( t , t )in B ( e, e, r ). NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 9
Circles of elements in a group.
Let G be a group and g , g ele-ments of G . Define elements g i for i ∈ Z inductively by: g i = (cid:26) g i − g i − g − i − if i > , and g − i +1 g i +2 g i +1 if i < . Then for all i, j ∈ Z , the relation g i g i − = g j g j − is satisfied. The element thus represented is g g ; denote it by γ , andcall it the disk element . We call the set { g i | i ∈ Z } the circle of elementson ( g , g ), and denote it C ( g , g ). Observe that for any i ∈ Z , g i = γg − i − = g − i +1 γ. Conversely, suppose that a group has a set of elements K = { h i | i ∈ Z } (possibly with doubling up, that is, with h i = h j for distinct i and j )such that h i h i − = h j h j − for all i, j ∈ Z . Then K is C ( h p , h p − ) forany p ∈ Z .From now on, suppose that C ( g , g ) is a circle with disk element γ . Lemma 2.5.
We have γg i = g i +2 γ for all i ∈ Z .Proof. For all i ∈ Z , we have γg i = g i +2 g i +1 g i = g i +2 γ . (cid:3) In general, the circle of elements obtained may be infinite: for example,in the rank two free group generated by { g , g } , the circle C ( g , g ) isinfinite. Obviously, if the group is finite, then any circle of elements isfinite. Lemma 2.6.
If there exist p ∈ Z and e ∈ N satisfying g p = g p + e , thenwe have g i = g i + e for all i ∈ Z , and | C ( g , g ) | divides e .Proof. The proof goes by induction in two directions. Suppose first q ≥ p and g j = g j + e for all j with p ≤ j ≤ q . Then we have g q +1 = γg − q = γg − q + e = g q + e +1 , so the result is true for all j ≥ p . Similarly,for q ≤ p , g j = g j + e for all j ≥ q implies g q − = g − q γ = g − q + e γ = g q + e − . Thus C ( g , g ) is { g i | i ∈ Z /e } and is of cardinality dividing e . (cid:3) Lemma 2.7.
The circle C ( g , g ) is of finite cardinality if and only if h g g i e = h g g i e holds for some e ∈ N . The smallest e for which thisrelation holds is the cardinality of C ( g , g ) .Proof. Suppose first | C ( g , g ) | = e < ∞ . So g i = g i + e holds forall i ∈ Z /e . Using Lemma 2.5 above, we find, for e odd, h g g i e = γ e − g = g e − γ e − = g h g g i e − = h g g i e , and, for e even, h g g i e = g γ e − g = g g e − γ e − = g g − γ e − = γ e = h g g i e . Now assume that g , g are elements satisfying h g g i e = h g g i e . Weshow g q = h g g i q (cid:0) h g g i q − (cid:1) − by induction on q >
1. It is certainly true for q = 1 and q = 2.Suppose q ≥
2. Then we obtain: g q +1 = g q g q − g − q = h g g i q (cid:0) h g g i q − (cid:1) − h g g i q − (cid:0) h g g i q − (cid:1) − h g g i q − (cid:0) h g g i q (cid:1) − = h g g i q (cid:0) h g g i q − (cid:1) − h g g i q − (cid:0) h g g i q (cid:1) − = h g g i h g g i q − (cid:0) h g g i q (cid:1) − = h g g i q +1 (cid:0) h g g i q (cid:1) − , which concludes the induction. In particular, we find g e = h g g i e (cid:0) h g g i e − (cid:1) − = h g g i e (cid:0) h g g i e − (cid:1) − = g , so by Lemma 2.6, g q = g q + e holds for all q ∈ Z and | C ( g , g ) | divides e . (cid:3) Remark 1.
The braid group B ( I ( e )), with reflection group the dihe-dral group of order 2 e , may be presented by (cid:10) a, b | h ab i e = h ba i e (cid:11) . Thispresentation gives rise to a Garside structure (see Subsection 3.1 fordetails about Garside structures; this fact was proved in [BS, Del]) cor-responding to the classical braid monoid B + ( I ( e )). Lemma 2.7 impliesthe known fact that B ( I ( e )) also has the presentation (cid:10) a i , i ∈ Z /e | a i a i − = a j a j − for all i, j ∈ Z /e (cid:11) , which gives rise to an alternativeGarside structure, corresponding to the dual braid monoid B × ( I ( e )). Lemma 2.8.
Every element b satisfying bg i = g i b for i ∈ { , } satis-fies bg i = g i b for all i ∈ Z .Proof. Clearly, all the elements of C ( g , g ) lie in the subgroup gener-ated by g and g . Thus if there is an element which commutes with g and g , then it commutes with the entire circle. (cid:3) The last property below describes how certain relations on g and g may be extended to the entire circle C ( g , g ). Lemma 2.9.
Every element a satisfying ag i a = g i ag i for i ∈ { , } satisfies ( a ) ag i a = g i ag i for all i ∈ Z , and ( b ) aγaγ = γaγa .Proof. (a) The proof is again by induction in two directions. We provethe case i ≥
1, the case i <
NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 11 form g i g i − = g j g j − and those of the form g j ag j = ag j a for 0 ≤ j ≤ i ,we have (see Figure 3) g i +1 ag − i +1 = g i g i − g − i ag i g − i − g − i = g i g i − ag i a − g − i − g − i = g i g i − ag i − g i − g − i − a − g − i − g − i = g i ag i − ag i − a − g − i − a − g − i = g i ag i − g − i − ag i − g − i − a − g − i = g i ag − i g i − ag − i − g i a − g − i = a − g i aa − g i − aa − g − i a = a − g i +1 a. • g i g i − g i +1 g i • aa g i a g i • g i +1 g i g i − • a g i aa g i • g i +1 g i g i − • aaa g i • g i − ag i − • g i − g i − • a g i − • g i − g i − • a g i − • g i − • Figure 3.
Proof of g i +1 ag i +1 = ag i +1 a . Theword g i +1 ag i +1 can be read around the top, theword ag i +1 a around the bottom. The interior cells arebounded by words corresponding to relators in the group.The positive relations can be read in opposite directions,starting from the corner of a given cell with the • symbol.(b) By the first part we have g ag = ag a . We find (see Figure 4) aγaγ = ag g ag g = ag g ag g = ag ag ag = g ag g ag = g ag g ag = g g ag g a = g g ag g a = γaγa. (cid:3) Proof of Theorem 2.1.
We now have enough to prove the theo-rem. • a g ag a g • g a g g a • a g g ag • g g a g • γ γ γγ Figure 4.
Proof of γaγa = aγaγ . The word γaγa canbe read starting at the left around the top, the word aγaγ around the bottom. The interior cells are bounded byrelators. Proof of Theorem 2.1.
Let J denote the group presented by P ⊕ ( e, e, r ).Relation ( R ) says that T e is the circle C ( t , t ). By Lemma 2.7, ( P )holds in J . By definition, Relation ( P ) ( resp. ( P )) is a particular caseof ( R ) ( resp. ( R )). Relation ( P ) is precisely a case of Lemma 2.9(b)with a = s . Thus all the relations of B ( e, e, r ) hold in J .On the other hand, ( P ) says that if T = { t i | i ∈ Z } = C ( t , t ) holdsin B ( e, e, r ), then by Lemma 2.7, t i = t i + e holds for all i , which im-plies ( R ). Lemma 2.9(a) implies that ( R ) then holds for all t i ∈ T .Lemma 2.8 ensures that ( R ) holds. Thus J and B ( e, e, r ) are isomor-phic.The new presentation has the same generators as the original, as wellas some conjugates of the originals. Since it is the case for the presen-tation in [BMR], adding the relations a = 1 for all generators a in thenew presentation gives a presentation of the reflection group G ( e, e, r ).Since conjugates of reflections are reflections, the generators of this pre-sentation are all reflections. Denote by the natural map B ( e, e, r ) ։ G ( e, e, r ). The generating reflections in the new presentation of G ( e, e, r )are the matrices: t i = ζ − ie ζ ie I r − and s j = matrix of ( j − j ) , where ζ e is a primitive e -th root of unity. (cid:3) NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 13 A new Garside structure
In this section, we first give a proof that the monoid B ⊕ ( e, e, r )—defined by the presentation studied in the previous section—is a Gar-side monoid. Then we find a precise description of the combinatorics ofthe underlying Garside structure. Finally we produce some argumentsin order to convince the reader that this structure could be namedpost-classical.3.1. Background on Garside theory.
In this preliminary subsec-tion, we list some basic definitions and summarize results by Dehornoy& Paris about Garside theory. For all the results quoted here, we referthe reader to [DP, D1, D2, D4].For x, y in a monoid M , write x y if there exists z ∈ M satisfying xz = y , and say either that x left-divides y or that y is a right multiple of x .There are similar definitions for right division and left multiplication(with the notation y < x if there exists z satisfying y = zx ). Write x ∨ y for the right lcm of x and y , and write x ∧ y for the left gcd. When M is cancellative, elements ( x \ y ) and ( y \ x ) are uniquely defined by: x ∨ y = x ( x \ y ) = y ( y \ x ) . A Garside monoid M is a cancellative monoid with lcm’s and gcd’s andadmitting a Garside element , namely an element whose left and rightdivisors coincide, are finite in number and generate M . There existsa minimal Garside element—usually denoted by ∆ and then called the Garside element—whose divisors are called the simples of M .By ¨Ore’s conditions, a Garside monoid embeds in a group of fractions.A Garside group is a group that is the group of fractions of (at least)one Garside monoid.Recognizing a Garside monoid from a presentation and computing ina Garside group given by a presentation are natural questions whichcan be solved by using word reversing , a syntactic method relevant forsemigroup presentations.Let ε denote the empty word. For h A | R i a semigroup presen-tation and w, w ′ words on A ∪ A − , we say that w reverses to w ′ —written w y R w ′ —if w ′ is obtained from w by (iteratively) • deleting some x − x for x ∈ A , • replacing some x − y with uv − for xu = yv a relation in R .This can be represented diagrammatically as shown in Figure 5. Finiteness is a quite technical condition which can be relaxed in some contexts. x x ε y x y y u v Figure 5.
Word reversing diagrams for x − x y ε and x − y y uv − .First, remark that, for all u, v ∈ A ∗ , u − v y R ε implies u ≡ + R v ,where ≡ + R denotes the monoid congruence generated by R . A semi-group presentation h A | R i is said to be complete (for reversing) whenthe converse holds, that is, when word reversing detects equivalence.Technically, h A | R i is complete if and only if every triple ( u, v, w ) ofwords over A satisfies the cube condition (CC) modulo R : u − ww − v y R uv − for u, v ∈ A ∗ implies ( xu ) − yv y R ε .The CC can be represented diagrammatically as shown in Figure 6. u ww v y v ′ u ′ = ⇒ uv ′ vu ′ y ε Figure 6.
Cube condition: u − ww − v y v ′ u ′− ⇒ ( uv ′ ) − vu ′ y ε .In the general case, the cube condition has to be checked for all triplesof words on A , or for all triples of words in a superset of A closedunder y . However, in the homogeneous case (that is, when everyrelation preserves the length of words), check the cube condition for alltriples of generators suffices to decide completeness.A semigroup presentation h A | R i is complemented if, for all gener-ators x, y in A , there is at most one relation of the type x · · · = y · · · and no relation of the type x · · · = x · · · . We will use the followingcriterium: Theorem 3.1. [D4]
Every monoid defined by a complemented completepresentation and admitting a Garside element is a Garside monoid.
The braid monoid B ⊕ ( e, e, r ) is Garside. The aim of this sub-section is to show:
NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 15
Theorem 3.2.
The braid monoid B ⊕ ( e, e, r ) is Garside. We prove the theorem in two parts: first completeness, then the Garsideelement.3.2.1.
Completeness.
Taking advantage from knowledge of complete-ness for presentations associated to certain parabolic subdiagrams, wecan check completeness of P ⊕ ( e, e, r ) after computing only few cases. Lemma 3.3.
The presentation P ⊕ ( e, e, r ) is complemented and com-plete.Proof. The presentation P ⊕ ( e, e, r ) is complemented and homogeneous.Now, it suffices to check whether every triple ( x, y, z ) of generatorsin T e ∪ S r satisfies the CC.From Subsection 2.3.1 about parabolic subdiagrams, we deduce:( + ) Every triple in S r satisfies the CC because it holds for B + ( A r − ).( + ) Every triple in ( T e × S r ) ∪ ( S r × T e × S r ) ∪ ( S r × T e ) satisfiesthe CC because it holds for B + ( A r − ).( + ) Every triple in T e satisfies the CC, because it holds for B × ( I ( e )).Thus we need only verify the cube condition on triples of type ( + ),that is, containing two generators from T e and one generator from S r .This case can be decomposed into two subcases depending on whetherthis generator from S r is s (say case ( a )) or not (case ( b )). FromSubsection 2.3.1 again, with S − r = S r \ { s } , we find:( + ) Every triple in ( T e × S − r ) ∪ ( T e × S − r × T e ) ∪ ( S − r × T e ) satisfiesthe CC because it holds for B × ( I ( e )) × B + ( A r − ).Therefore we need only verify the CC on triples of type ( + ), thatis, containing two generators from T e and the generator s from S r .Now, various symmetry considerations reduce again the number ofcases that need be considered. On the one hand, triples of the form ( x, y, x ) and ( x, x, y ) always satisfythe CC.Case ( x, y, x ): x x x y y u v = ⇒ xu y v y u vε ε Case ( x, x, y ): x y y x y u v y v u = ⇒ xu x uε On the other hand, a triple ( x, y, z ) satisfies the CC if and only if ( y, x, z )satisfies the CC: this may be seen by reflecting the word reversing di-agram through an axis at π .This results in the following two cases: • case ( x, y, z ) = ( t i , t j , s ) with i and j distinct; • case ( x, y, z ) = ( t i , s , t j ) with i and j distinct.The calculations are shown in Figures 7 and 8 respectively. (cid:3) t j s t j s t i s t i s t (cid:3) j s t (cid:3) i t (cid:3) i s t (cid:3) j ε s t j t (cid:3) j s t (cid:3)(cid:3) j s t i t (cid:3) i s t (cid:3)(cid:3) i = ⇒ t j s t j t (cid:3) j s t (cid:3)(cid:3) j t i s t i t (cid:3) i s t (cid:3)(cid:3) i t (cid:3)(cid:3) j t (cid:3) i s t (cid:3) j t (cid:3)(cid:3) j s t (cid:3)(cid:3) i t (cid:3) j s t (cid:3) i t (cid:3)(cid:3) i s ε εεt (cid:3) i s t (cid:3)(cid:3) i t (cid:3) j s t (cid:3)(cid:3) j ε t (cid:3) j s t (cid:3)(cid:3) j ε ε εt (cid:3) i s t (cid:3)(cid:3) i ε ε ε Figure 7.
Proof of completeness: the case ( x, y, z ) = ( t i , t j , s ) with i = j .3.2.2. Garside element.
Let τ be the element t i t i − of B ⊕ ( e, e, r ). Since t i t i − = t j t j − holds for all i, j , τ is independent of i . It is a common NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 17 s t j s t j t i t j t (cid:3) j t (cid:3) j s t (cid:3)(cid:3) j t j s t (cid:3) j s t (cid:3)(cid:3) j t (cid:3) i s t (cid:3) j t (cid:3)(cid:3) j s = ⇒ s t j s t (cid:3) j s t (cid:3)(cid:3) j t i t (cid:3) i s t (cid:3) j t (cid:3)(cid:3) j s t i s t (cid:3) i s t (cid:3)(cid:3) i ε ε εt (cid:3) j s t (cid:3) i εt (cid:3) i s t (cid:3) j s t i t (cid:3) i s t (cid:3)(cid:3) i t (cid:3) j s t (cid:3)(cid:3) j ε ε εt (cid:3) i s t (cid:3)(cid:3) i t (cid:3)(cid:3) i s t (cid:3) i t (cid:3)(cid:3) i s ε εεt (cid:3)(cid:3) j ε Figure 8.
Proof of completeness: the case ( x, y, z ) = ( t i , s , t j ) with i = j .multiple of T e ; and since no word of length one could be a multiple ofall the t i , τ is the lcm of T e .The classical braid monoid B + (2 , , n ) for B (2 , , n ) (indeed, B ( d, , n )for any d ≥
2) is defined by the following Coxeter diagram: q q q q n − q n The Garside element ∆ B n of B + (2 , , n ) ∼ B + ( B n ) is the lcm of Q n = { q , q , . . ., q n } , which can be written in the various forms:∆ B n = q ( q q q ) · · · ( q n · · · q q q · · · q n )= ( q n · · · q q q · · · q n ) · · · ( q q q ) q = ( q q · · · q n ) n = ( q n · · · q q ) n . This is a central element of B + ( B n ).Define ψ : Q ∗ r − → ( T e ∪ S r ) ∗ by ψ ( q i ) = (cid:26) τ for i = 1, and s i +1 for i > ψ induces an injection B + ( B r − ) ֒ → B ⊕ ( e, e, r ); in particular, the poset structures with respect to co-incide on B + ( B r − ) and B ⊕ ( e, e, r ). We deduce that the element Λ = ψ (∆ B r − ) in B ⊕ ( e, e, r ) has the following decompositions:Λ = τ ( s τ s ) · · · ( s r · · · s s τ s s · · · s r ) = ( τ s · · · s r ) r − = ( s r · · · s s τ s s · · · s r ) · · · ( s τ s ) τ = ( s r · · · s τ ) r − and is precisely the least common multiple of ψ ( Q r − ) = { τ } ∪ S r .Since τ is the lcm of T e , we deduce: Lemma 3.4.
The element Λ is the lcm of T e ∪ S r . Also, by centrality of ∆ B r − in B + ( B r − ), we haveΛ τ = τ Λ and Λ s p = s p Λ for 3 ≤ p ≤ r. Let Λ = τ , and Λ p = s p · · · s τ s · · · s p for 3 ≤ p ≤ r . We find:Λ = Λ Λ · · · Λ r . A balanced element in a monoid is an element β such that x β holdsprecisely when β < x holds.The following result could be deduced from older results (see for in-stance [DP] or [D3]), but the proof is straightforward and we includeit to make the current work self-contained. Proposition 3.5.
Suppose that M is a cancellative monoid and β isan element in M such that for all x ∈ M there exists an element φ ( x ) satisfying βx = φ ( x ) β . If φ is surjective then β is balanced.Proof. Suppose there exist x, y ∈ M satisfying φ ( x ) = φ ( y ). Then βx = βy holds, hence x = y by left cancellation, so φ is injective.Thus φ is an automorphism of M .For x β , denote by β x the unique element of M satisfying x β x = β (uniqueness comes from left cancellation). Similarly, for β < x thenwrite x β x = β .Suppose x β . Then we have β = φ ( β ) = φ ( x ) φ ( β x ), hence φ ( x ) β .By the same argument but using φ − instead, we deduce that φ ( x ) β implies x β . Thus x β holds precisely when φ ( x ) β holds. Asymmetric argument shows that β < x holds precisely when β < φ ( x )holds.So finally, suppose x β , which implies φ ( x ) β . Then we have φ ( x ) β φ ( x ) x = βx = φ ( x ) β. Left cancellation then gives β φ ( x ) x = β , hence β < x . A similar ar-gument shows that β < x implies x β . Hence β < x holds preciselywhen x β holds. (cid:3) Proposition 3.6.
The element Λ is balanced.Proof. Let i ∈ Z /e . From ( s τ s ) t i − = s t i t i − s t i − = s t i s t i − s = t i s t i t i − s = t i ( s τ s ), we deduce Λ t i − = t i Λ and Λ p t (cid:3) i = t i Λ p for 3 ≤ p ≤ r , hence Λ t i = t i + r Λ.Defining φ ( s p ) = s p and φ ( t i ) = t i + r for i ∈ Z /e gives rise to anautomorphism φ of the cancellative monoid B ⊕ ( e, e, r ) satisfying Λ x = φ ( x )Λ. The result then follows by Proposition 3.5. (cid:3) NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 19
Proof of Theorem 3.2.
We now have enough to complete theproof of Theorem 3.2, that B ⊕ ( e, e, r ) is a Garside monoid: Proof of Theorem 3.2.
On the one hand, by Lemma 3.3, B ⊕ ( e, e, r ) ad-mits a complemented and complete presentation. On the other hand,the element Λ Λ · · · Λ r —which we will henceforth denote by ∆—is theGarside element of B ⊕ ( e, e, r ). Indeed, Proposition 3.6 states that leftand right divisors of ∆ coincide and Lemma 3.4 insures that ∆ is thelcm of the generators and, in particular, the set of divisors of ∆ gener-ates B ⊕ ( e, e, r ). Now, invoking Theorem 3.1, we obtain that B ⊕ ( e, e, r )is a Garside monoid with Garside element ∆ = Λ Λ · · · Λ r . (cid:3) Hence B ⊕ ( e, e, r ) embeds in the group B ( e, e, r ) defined by the samepresentation. Furthermore, we have for free: Proposition 2.2.
The submonoid of B ( e, e, r ) generated by T e ∪ S r is isomorphic to the monoid B ⊕ ( e, e, r ) , that is, it can be presentedby P ⊕ ( e, e, r ) considered as a monoid presentation. Structure of the lattice of simples in B ⊕ ( e, e, r ) . Here wecompletely describe the structure of the lattice of simples in the Garsidemonoid B ⊕ ( e, e, r ). Though sometimes somewhat technical, our carefulstudy leads to a clear statement (Theorem 3.7) which fully explainsthe combinatorics of the Garside structure and which will allow thecomputation of several related numerical objects and then, in the nextsubsection, to appreciate how classical B ⊕ ( e, e, r ) actually is.Since the only relations from P ⊕ ( e, e, r ) which can be applied to Λ k correspond to applications of ( R ) to τ , there are only four types ofnon-trivial left divisors of Λ k :(1) s k · · · s τ s · · · s j − s j ,(2) s k · · · s τ ,(3) s k · · · s t i , and(4) s k · · · s j +1 s j ,with 3 ≤ j ≤ k and i in Z /e .The following theorem together with the fact that we know preciselythe form of the divisors of Λ k for each k allows us to have precise controlover the simples. Theorem 3.7.
The simples in B ⊕ ( e, e, r ) are precisely the elements ofthe form p · · · p r where p k is a divisor of Λ k for ≤ k ≤ r . Define the polynomial P ⊕ ( e,e,r ) ( q ) = P a n q n where a n is the number oflength n simples in B ⊕ ( e, e, r ). This polynomial is discussed in more detail in Subsection 3.4.2. Theorem 3.7 directly gives a factorizationof it. Corollary 3.8.
We have: P ⊕ ( e,e,r ) ( q ) = r Y k =2 (1 + q + · · · + q k − + eq k − + q k + · · · + q k − ) . Corollary 3.9.
The number of simples in B ⊕ ( e, e, r ) is P ⊕ ( e,e,r ) (1) = r Y k =2 (2( k −
1) + e ) = (2( r −
1) + e )!! e !! , where the notation n !! represents the product n ( n − · · · · for n evenand the product n ( n − · · · · for n odd (see [Slo, sequences A000165and A001147] ). Remark.
For q = 1, we find P ⊕ ( e,e,r ) ( q ) = r Y k =2 q k − + ( e − q k − ( e − q k − − q − . For instance, Figure 9 displays the lattice of simples in B ⊕ (3 , , P ⊕ (3 , , ( q ) = 1 + 4 q + 7 q + 11 q + 7 q + 4 q + q = (1 + 3 q + q )(1 + q + 3 q + q + q ) . t t t Λ Λ s ∆ Figure 9.
The lattice of simples in B ⊕ (3 , , height of an element of B ⊕ ( e, e, r ) : NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 21
Define a map Ht : ( T e ∪ S r ) ∗ → { , , . . . , r } on generators by Ht( t i ) = 2,Ht( s q ) = q , and, for w = a · · · a k , define Ht( w ) = max ki =1 Ht( a i ).Define the height of the empty word to be Ht( ε ) = 1. If ρ = ρ is adefining relation of B ⊕ ( e, e, r ), it is clear from inspection that Ht( ρ ) =Ht( ρ ) holds. Thus Ht( w ) only depends on the element in B ⊕ ( e, e, r )represented by w . So the height map Ht : B ⊕ ( e, e, r ) → { , , . . . , r } iswell-defined. Lemma 3.10.
Every x ∈ B ⊕ ( e, e, r ) with Ht( x ) < k satisfies x Λ k = Λ k x (cid:3) . Proof.
For j < k , we find: s j Λ k = s j s k · · · s j +1 s j s j − · · · s τ s · · · s k = s k · · · s j +2 s j s j +1 s j s j − · · · s τ s · · · s k = s k · · · s j +2 s j +1 s j s j +1 s j − · · · s τ s · · · s k = s k · · · s s j +1 τ s · · · s k = s k · · · s τ s j +1 s · · · s j − s j s j +1 · · · s k = s k · · · s τ s · · · s j − s j +1 s j s j +1 · · · s k = s k · · · s τ s · · · s j − s j s j +1 s j s j +2 · · · s k = s k · · · s τ s · · · s j − s j s j +1 s j s j +2 · · · s k = s k · · · s τ s · · · s j − s j s j +1 s j +2 · · · s k s j = Λ k s j . For every k and 0 ≤ i < e , we have: t i Λ k = t i s k · · · s τ s · · · s k = s k · · · s t i s t i t i − s · · · s k = s k · · · s s t i s t i − s · · · s k = s k · · · s t i t i − s t i − s · · · s k = s k · · · s t i t i − s s · · · s k t i − = Λ k t i − . The result follows. (cid:3)
Recall that a ∨ b denotes the lcm of a and b and that, by cancellativity,elements ( a \ b ) and ( b \ a ) are uniquely defined by: a ∨ b = a ( a \ b ) = b ( b \ a ) . Lemma 3.11.
Let a be an element in T e ∪ S r and q k be a right divisorof some Λ k . Then a ∧ q k = 1 and Ht( q k \ a ) < k together imply q k \ a ∈ T e ∪ S r and a \ q k = q k .Proof. If q k is trivial, then the result follows directly. Consider theremaining cases: (1) Let q k = s j · · · s k for some 3 ≤ j ≤ k .First, a q k implies a = s j . Next, s j − ∨ q k = q k s j − q k ( resp. t i ∨ q k = q k t i q k for j = 3) and Ht( q k ) = k imply a = s j − ( resp. a = t i for j = 3). The only possible cases are then: a ∨ q k = s l q k = q k s l for a = s l with 3 ≤ l < j − ,s l q k = q k s l − for a = s l with j < l ≤ k, and t i q k = q k t i for a = t i and j = 3 . (2) Let q k = t i s · · · s k for some i ∈ Z /e .First, a q k implies a = t i . Next, t i ∨ q k = q k t i − q k and Ht( q k ) = k imply a = t j for j = i . The possible cases are then: a ∨ q k = (cid:26) s q k = q k t i for a = s , and s l q k = q k s l − for a = s l with 3 < l ≤ k. (3) Let q k = s j − · · · s τ s · · · s k for some 3 ≤ j ≤ k .First, a q k implies a = s j − ( resp. a = t i for j = 3). Next, s j ∨ q k = q k s j q k and Ht( q k ) = k imply a = s j . The possiblecases are then: a ∨ q k = s l q k = q k s l for a = s l with 3 ≤ l < j − ,s l q k = q k s l − for a = s l with j < l ≤ k, and t i q k = q k t i for a = t i and j = 3 . In each case, we find a ∨ q k = aq k = q k a ′ with a ′ = q k \ a ∈ T e ∪ S r and a \ q k = q k . (cid:3) Proof of Theorem 3.7.
We have to prove a double inclusion. First, weshow that if, for each k ∈ { , . . . , r } , p k is a divisor of Λ k , then p p . . . p r divides ∆. For each k , let q k be the unique element of the monoidsatisfying p k q k = Λ k . Lemma 3.10 implies Λ j q (cid:3) k = q k Λ j for each q k andeach j > k . Let q ′ k be the element obtained by applying r − k timesthe map (cid:3) to q k . Then we obtain∆ = ( p q )Λ · · · Λ r = p Λ · · · Λ r q ′ = · · · = p p · · · p r q r q ′ r − · · · q ′ q ′ . Thus p p · · · p r is a divisor of ∆. This completes the first inclusion.Now let k ∈ { , . . . , r } . We prove, by induction on k ≥
2, that if p left-divides Λ · · · Λ k then p = p · · · p k holds for some divisors p j of Λ j with 2 ≤ j ≤ k . The result holds vacuously for k = 2.Assume k >
2. By the induction hypothesis, p ∧ Λ · · · Λ k − can bewritten as p · · · p k − , where Λ j is p j q j for some q j with Ht( q j ) ≤ j .We obtain p = p · · · p k − p for some p q ′ k − · · · q ′ Λ k = Λ k q ′ (cid:3) k − · · · q ′ (cid:3) where q ′ j is, as above, the element obtained by applying the map (cid:3) ,( k − j ) times to q j . Let p k = p ∧ Λ k , p k q k = Λ k and p = p k p ′ .We have to show that p ′ is trivial. Suppose instead that p ′ is not trivial.Then we may write p ′ = ap ′ a for some a ∈ T e ∪ S r . We have p ′ q k q NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 23 p p k − p k q k p k q k p ′ a a bb (cid:2) Lemma3.11 p p k q k q Λ · · · Λ k Figure 10.
Proof of Theorem 3.7with q = q ′ (cid:3) k − · · · q ′ (cid:3) and Ht( q ) < k . Thus a ∨ q k must divide q k q ,but a q k holds by gcd-ness of p k . Therefore, by Lemma 3.11, thereexists b ∈ T e ∪ S r satisfying aq k = q k b , thus p k aq k = Λ k b , hence p k a = b (cid:2) p k . We find p = p · · · p k − b (cid:2) p k p ′ a with p · · · p k − b (cid:2) Λ · · · Λ k − ,which contradicts p · · · p k − = p ∧ Λ · · · Λ k − . Therefore, p ′ is trivial,which concludes the induction. (cid:3) A note on the reflection group G ( e, e, r ) and the Garside structure. Thenumber of simples in B ⊕ ( e, e, r ) is (2( r −
1) + e )!! e !! , while the numberof elements in G ( e, e, r ) is e r − r !. For e = 2, we have equality be-tween these two expressions, corresponding to the classical type D r case. For e >
2, we have x + ee < x + 1, hence(2( r −
1) + e )!! e !! = e r − (cid:18) r −
1) + ee (cid:19) · · · (cid:18) ee (cid:19) (cid:18) ee (cid:19) < e r − r ( r − · · · (3)(2) = e r − r !(For example, there are 35 simples in B ⊕ (3 , , G (3 , , G ( e, e, r )may be represented by simples from B ⊕ ( e, e, r ). For example, the ele-ment t t = ζ e ζ − e I r − , may not be represented by a simple from B ⊕ ( e, e, r ).The known classical braid monoids for braid groups of real reflectiongroups all have equality between number of simples and size of reflectiongroup. In this way the monoid B ⊕ ( e, e, r ) appears not to be strictly classical. However in a number of ways it is seen to be dual, or simplydifferent from, the so-called dual braid monoids, and so deserves a namelike post-classical. This is the content of the next subsection.3.4. How classical is B ⊕ ( e, e, r ) ? The braid group B ( e, e, r ) seemsto admit no classical braid monoid, in the sense that its submonoidgenerated by the generators of [BMR]—providing a minimal generatingset—is indeed not finitely presented (see [Co, BC1, BC2]). Recall thatthe braid monoid B ⊕ ( e, e, r ) ( resp. the dual braid monoid B × ( e, e, r ))coincides with the classical braid monoid B + ( D r ) ( resp. the dual braidmonoid B × ( D r )) for e = 2 and with the dual braid monoid B × ( I ( e ))for r = 2.In this subsection we look at various properties of the monoid B ⊕ ( e, e, r ),which mainly deal with enumerative aspects, and consider them in rela-tion to known classical and dual braid monoids for other braid groups.While it cannot be considered as strictly classical, B ⊕ ( e, e, r ) has muchin common with the classical braid monoids than with the dual braidmonoids, and it could be considered as a post-classical braid monoid.The following three observations allow to legitimate this terminology.3.4.1. A kind of duality.
According to [B2], the duality terminology inthe context of Garside monoids for braid groups of finite real reflectiongroups W can be justified by the numerical facts summarized in thefollowing table: B + ( W ) B × ( W )Product of the atoms c w ∆ w c Number of atoms n N
Length of ∆
N n
Order of a a ∆ h Regular degree h Each of the braid group presentations constructed in [B1] corresponds to a regulardegree d . The product of the generators raised to the power d (which is the order ofthe image of this product in the reflection group), is always central. See also [B3]. NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 25
A different kind of duality can be observed between the monoids B ⊕ ( e, e, r )and B × ( e, e, r ): B ⊕ ( e, e, r ) B × ( e, e, r )Number of atoms e + r − e + r − ( r − Length of ∆ r ( r − r Order of a a ∆ ee ∧ r e ( r − e ∧ r Thus the monoid B ⊕ ( e, e, r ) may be considered to be a kind of dualof the dual braid monoid B × ( e, e, r ). The latter fits into the generalframework of dual braid monoids defined in [B2], but it satisfies onlysome of the numerical properties summarized in the first table above.In a parallel way, B ⊕ ( e, e, r ) could be named simply classical. Here, wecould mention that neither B ⊕ ( e, e, r ) nor B × ( e, e, r ) can be producedby [B1, Theorem 0.1], so in particular, the notion of regular degree isnot relevant.3.4.2. Poincar´e polynomial.
For a given Garside monoid M , the poly-nomial P M is defined by P M ( q ) = P a n q n where a n denotes the numberof length n simples in M (see earlier comments preceding Corollary 3.8).In the case of the classical braid monoids associated to finite Coxetergroup W (for example B + ( A n ), B + ( B n ), B + ( D n ), etc), this polynomialcoincides with the Poincar´e polynomial of W , where a n is the numberof length n elements of with respect to a set of simple reflections. Inthese cases, we have: P + W ( q ) = r Y k =1 (1 + q + · · · + q d k − )where the numbers d k denote the reflection degrees. The polyno-mial P ⊕ ( e,e,r ) ( q ) does not satisfy this general formula, except for thecases e = 2 or r = 2. However the similarity of factorization of thePoincar´e polynomial (see below) suggests describing B ⊕ ( e, e, r ) againas a post-classical braid monoid. P ⊕ ( e,e,r ) ( q ) = r Y k =1 (1 + q + · · · + q k − + eq k − + q k + · · · + q k − ) ,P + A n ( q ) = n Y k =1 (1 + q + · · · + q k ) ,P + B n ( q ) = n Y k =1 (1 + q + · · · + q k − ) ,P + D n ( q ) = (1 + q + · · · + q n − ) n − Y k =1 (1 + q + · · · + q k − ) . Zeta polynomial.
For a given Garside monoid M , the zeta poly-nomial Z M can be defined by requiring that Z M ( q ) be the number oflength q − a · · · a q − in the lattice of simples of M .Whenever G ( de, e, r ) is well-generated (which is the case for G ( e, e, r )),the zeta polynomial of the dual braid monoid B × ( de, e, r ) admits a nicefactorization: Z × ( de,e,r ) ( q ) = r Y k =1 d k + d r ( q − d k , where d ≤ · · · ≤ d r are the reflection degrees (see [Ch, R, AR]).On the contrary, the zeta polynomial Z +( de,e,r ) of the classical braidmonoid B + ( de, e, r ) (when defined) is not known to admit any nicefactorization. In this way, B ⊕ ( e, e, r ) has more in common with classi-cal braid monoids than dual braid monoids. For instance, we find: Z ⊕ (3 , , ( q ) = 11 q + 171 q + 985 q + 2585 q + 2964 q + 1444 q + 240240= ( q + 1)( q + 6)(11 q + 94 q + 261 q + 194 q + 40)240 . Conclusion.
While we feel that the new Garside monoid B ( e, e, r )deserves the description post-classical, we do not exclude the possibilitythat this is the best presentation available, and would like to concludewith a motivating question: Question 3.12.
Does B ( e, e, r ) admit other Garside structures? Acknowledgment
The completion of this work was made possible from a collaborationbegun at the GDR Tresses conference in Autrans, 2004 (GDR 2105CNRS, “Tresses et Topologie de basse dimension” [Aut]).
NEW GARSIDE STRUCTURE FOR BRAID GROUPS OF TYPE ( e, e, r ) 27
The first author would also like to thank the European Union for aMarie Curie Postdoctoral Research Award at the time this work wasundertaken.The authors thank Ivan Marin for pointing out to them that, duringhis thesis work supervised by Daan Krammer, Mark Cummings haddiscovered the same Garside structure for B ( e, e, r ), even though proofsand motivations are essentially different. References [AR] Ch. Athanasiadis, V. Reiner,
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