Simple dual braids, noncrossing partitions and Mikado braids of type D n
aa r X i v : . [ m a t h . G R ] J u l SIMPLE DUAL BRAIDS, NONCROSSING PARTITIONS ANDMIKADO BRAIDS OF TYPE D n BARBARA BAUMEISTER AND THOMAS GOBET
Abstract.
We show that the simple elements of the dual Garside structure of anArtin group of type D n are Mikado braids, giving a positive answer to a conjectureof Digne and the second author. To this end, we use an embedding of the Artingroup of type D n in a suitable quotient of an Artin group of type B n noticed byAllcock, of which we give a simple algebraic proof here. This allows one to givea characterization of the Mikado braids of type D n in terms of those of type B n and also to describe them topologically. Using this topological representation andAthanasiadis and Reiner’s model for noncrossing partitions of type D n which can beused to represent the simple elements, we deduce the above mentioned conjecture. AMS 2010 Mathematics Classification : : 20F36, 20F55.
Keywords.
Coxeter groups, Artin-Tits groups, dual braid monoids, Garside theory,noncrossing partitions.
Contents
1. Introduction 22. Artin groups of type D n inside quotients of Artin groups of type B n B n and D n B n D n inside e A B n
74. Dual braid monoids 74.1. Noncrossing partitions 74.2. Dual braid monoids 74.3. Standard Coxeter elements in W D n
85. Simple dual braids of type D n are Mikado braids 95.1. Outline of the proof 95.2. Graphical model for noncrossing partitions 95.3. The diagram N x and the braid β x The second author was partially funded by the ANR Geolie ANR-15-CE40-0012.
BARBARA BAUMEISTER AND THOMAS GOBET Introduction
The dual braid monoid B ∗ c of a Coxeter system ( W, S ) of spherical type was intro-duced by Bessis [5] and depends on the choice of a standard Coxeter element c ∈ W (a product of all the elements of S in some order). It is generated by a copy T c of theset T of reflections of W , that is, elements which are conjugates to elements of S . Asa Garside monoid, it embeds into its group of fractions, which was shown by Bessisto be isomorphic to the Artin group A W corresponding to W . Unfortunately, this iso-morphism is poorly understood, and the proof of its existence requires a case-by-caseargument [5, Fact 2.2.4].The aim of this note is to study properties of the simple elements Div( c ) in B ∗ c viewedinside A W in case W is of type D n and to show that they are Mikado braids , that is,that they can be represented as a quotient of two positive canonical lifts of elements of W . These braids appeared in work of Dehornoy [13] in type A n and in work of Dyer [14]for arbitrary Coxeter systems and have many interesting properties. For example, theysatisfy an analogue of Matsumoto’s Lemma in Coxeter groups [14, Section 9]. We referthe reader to [14, Section 9], [12, Section 4] (there the Mikado braids are called rationalpermutation braids , while the terminology Mikado braids rather refers to braids viewedtopologically; it is shown however in [12] that both are equivalent) or [15, Section 3.2]for more on the topic. Another important property is that their images in the Iwahori-Hecke algebra H ( W ) of the Coxeter system ( W, S ) have positivity properties; let usbe more precise. There is a natural group homomorphism a : A W −→ H ( W ) × . If β ∈ A W is a Mikado braid and if we express its image a ( β ) in the canonical basis { C w | w ∈ W } of the Hecke algebra, then the coefficients are Laurent polynomialswith positive coefficients (see [12, Section 8]). This is one of the main motivations forstudying Mikado braids, and showing that simple dual braids are Mikado braids. Thislast property was conjectured for an arbitrary Coxeter system ( W, S ) of spherical typein [12], and shown to hold in all the irreducible types different from D n [12, Theorems5.12, 6.6, 7.1].In the classical types A n and B n , the conjecture is proven using a topological char-acterization of Mikado braids: it can be seen on any reduced braid diagram (resp.symmetric braid diagram in type B n ) whether a braid is a Mikado braid or not. Thepresent paper gives topological models for Mikado braids of type D n , similar to thosegiven in types A n and B n in [12], and solves the above conjecture in the remaining type D n : Theorem 1.1.
Let c be a standard Coxeter element in a Coxeter group ( W, S ) of type D n . Then every element of Div( c ) is a Mikado braid. As a consequence, every simple dual braid in every spherical type Artin group isa Mikado braid, the reduction to the irreducible case being immediate. Licata andQueffelec recently informed us that they also have a proof of the conjecture in types
A, D, E with a different approach using categorification [18].To prove the conjecture, we proceed as follows. Firstly, we explicitly realize the Artingroup A D n of type D n as an index two subgroup of a quotient of the Artin group A B n of type B n . The existence of such a realization, which is of independent interest, isnot new: it was noticed by Allcock [1, Section 4]. We give a simple proof of it here(Proposition 2.7). This allows to realize elements of A D n topologically by Artin braids.We then characterize Mikado braids of type D n as the images of those Mikado braidsof type B n which surject onto elements of W D n ⊆ W B n under the canonical map from A B n onto W B n (Theorem 3.3). This implies that Mikado braids of type D n satisfya nice topological condition, and gives a model for their study in terms of symmetric IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n Artin braids, because elements of A B n can be realized as symmetric Artin braids on n strands (see Section 3.1). Using Athanasiadis and Reiner’s graphical model [3] for c -noncrossing partitions of type D n (which are in canonical bijections with the simpleelements Div( c ) of B ∗ c ; we denote this bijection by x x c , where x is a c -noncrossingpartition), we attach to every such noncrossing partition x an Artin braid β x of type B n ,whose image in the above mentioned quotient is precisely the element x c ∈ A D n (Sec-tion 5). Using the topological characterization of Mikado braids of type B n from [12],we then prove that β x is a Mikado braid of type B n (Proposition 5.3), which concludesby the above mentioned characterization of Mikado braids of type D n (Theorem 5.5). Acknowledgments.
We thank Luis Paris for useful discussions with the second authorand Jon McCammond for pointing out the reference [20].2.
Artin groups of type D n inside quotients of Artin groups of type B n Coxeter groups and Artin groups.
This section is devoted to recalling basicfacts on Coxeter groups and their Artin groups. We refer the reader to [9, 17] or [8]for more on the topic. A
Coxeter system ( W, S ) is a group W generated by a set S of involutions subject to additional braid relations , that is, relations of the form st · · · = ts · · · for s, t ∈ S , s = t . Here st · · · denotes a strictly alternating product of s and t , and the number m st of factors in the left hand side equals the number m ts of factors in the right hand side. We have m st ∈ { , , . . . } ∪ {∞} , the case m st = ∞ meaning that there is no relation between s and t . Let ℓ : W → Z ≥ be the lengthfunction with respect to the set of generators S .Finite irreducible Coxeter groups are classified in four infinite families of types A n , B n , D n , I ( m ) and six exceptional groups of types E , E , E , F , H , H . If X is agiven type, we denote by ( W X , S X ) a Coxeter system of this type.The Artin group A W attached to the Coxeter system ( W, S ) is generated by a copy S of the elements of S , subject only to the braid relations. This gives rise to a canonicalsurjection π : A W ։ W induced by s s . If W has type X , we simply denote A W by A X .The canonical map π has a set-theoretic section W ֒ → A W built as follows: let w = s s · · · s k be a reduced expression for w , that is, we have s i ∈ S for all i = 1 , . . . , k and k = ℓ ( w ) . Then the lift s s · · · s k in A W is independent of the chosen reducedexpression, and we therefore denote it by w . This is a consequence of the fact thatin every Coxeter group, one can pass from any reduced expression of a fixed element w to any other just by applying a sequence of braid relations. The element w is the canonical positive lift of w .2.2. Embeddings of Coxeter groups.
Let ( W B n , S B n ) be a Coxeter system of type B n . We will identify it with the signed permutations group as follows: let S − n,n be thegroup of permutations of [ − n, n ] = {− n, − n + 1 , . . . , − , , . . . , n } and define W B n := { w ∈ S − n,n | w ( − i ) = − w ( i ) , for all i ∈ [ − n, n ] } . Then setting s := ( − , and s i = ( i, i + 1)( − i, − i − for all i = 1 , . . . , n − we getthat S B n = { s , s , . . . , s n − } is a simple system for W B n (see [8, Section 8.1]).Let ( W D n , S D n ) be a Coxeter group of type D n . Recall that W D n can be realized as anindex two subgroup of W B n as follows: setting t = s s s , t i = s i for all i = 1 , . . . , n − we have that S D n := { t , t , . . . , t n − } is a simple system for the Coxeter group W D n = h t , t , . . . , t n − i of type D n (see [8, Section 8.2]). In the following, a Coxeter group oftype D n will always be viewed inside W B n , with the above identifications. BARBARA BAUMEISTER AND THOMAS GOBET
Embeddings of Artin groups.
We assume the reader to be familiar with Artingroups attached to Coxeter groups and refer to [11, Chapter IX] for basic results. Noticethat there are two surjective maps q B : A B n −→ A A n − , q D : A D n −→ A A n − definedas follows: if we denote by { σ , . . . , σ n − } the set of standard Artin generators of the n -strand Artin braid group A A n − , then q B ( s ) = 1 , q B ( s i ) = σ i for i = 0 , while q D ( t ) = σ , q D ( t i ) = σ i for all i = 0 (see [10, Section 2.1]). Both maps q B and q D aresplit and one can write A X n ∼ = ker( q X ) ⋊ A A n − for X ∈ { B, D } .Crisp and Paris showed that the embedding of W D n in W B n which we recalled inSubsection 2.2 does not come from an embedding ϕ : A D n −→ A B n such that q D = q B ◦ ϕ [10, Proposition 2.6]. In this section we show that there is an embedding of A D n inside a quotient e A B n of A B n ; this embedding can be seen as a natural lift of theembedding of Coxeter groups and has the expected properties (see Lemma 2.8). Thisis mostly a reformulation of results of Allcock [1, Sections 2 and 4], but we will give asimple algebraic proof of this fact here. Definition 2.1.
Define e A B n to be the quotient of A B n by the smallest normal subgroupcontaining s .It follows immediately from this definition that the canonical map π n : A B n ։ W B n factors through e A B n via two surjective maps π n, : A B n ։ e A B n and π n, : e A B n ։ W B n . Remark 2.2.
In [20, Definition 3.3], a similar group, called the middle group , is consid-ered. It is defined as the quotient of A B n by the smallest normal subgroup containing s (as a consequence, every s i for i ≥ is equal to in the quotient since s i lies in thesame conjugacy class as s ).Denote by s ′ i , i = 0 , . . . , n − the image of s i ∈ A B n in e A B n , for all s i ∈ S B n . Set t ′ = s ′ s ′ s ′ and t ′ i = s ′ i for i = 1 , . . . , n − . Lemma 2.3.
The elements t ′ , t ′ , . . . t ′ n − satisfy the braid relations of type D n , that is,we have t ′ t ′ = t ′ t ′ , t ′ t ′ t ′ = t ′ t ′ t ′ , t ′ i t ′ i +1 t ′ i = t ′ i +1 t ′ i t ′ i +1 for all i = 1 , . . . , n − ,t ′ i t ′ j = t ′ j t ′ i if | i − j | > and { i, j } 6 = { , } . Proof.
All the relations except the second one are immediate consequences of the type B n braid relations satisfied by the s ′ , s ′ , . . . , s ′ n − . For the second relation we have t ′ t ′ t ′ = s ′ s ′ s ′ s ′ s ′ s ′ s ′ = s ′ s ′ s ′ s ′ s ′ s ′ = s ′ s ′ s ′ s ′ s ′ = s ′ s ′ s ′ s ′ s ′ = s ′ s ′ s ′ s ′ s ′ = t ′ t ′ t ′ . (cid:3) An immediate corollary is
Corollary 2.4.
There is a group homomorphism ι n : A D n −→ e A B n defined by ι n ( t i ) = t ′ i for all i = 0 , . . . , n − . We have the following situation
Lemma 2.5.
There is a commutative diagram A B n π Bn " " " " ❊❊❊❊❊❊❊❊❊ π n, / / / / e A B n π n, (cid:15) (cid:15) (cid:15) (cid:15) h t ′ , . . . , t ′ n − i _? o o π Dn (cid:15) (cid:15) (cid:15) (cid:15) A D n o o o o W B n W D n _? o o where π Dn : h t ′ , . . . , t ′ n − i −→ W D n is defined by π Dn ( t ′ i ) = t i for all i = 0 , . . . , n − . IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n Remark 2.6.
In Proposition 2.7 below we will show that the map ι n is injective; hence π Dn is in fact simply the canonical surjection A D n ։ W D n . Proof.
We have to show that the composition of π n, and h t ′ , . . . , t ′ n − i ֒ → e A B n factorsthrough W D n . It suffices to show that the image of t ′ i under this composition is precisely t i (viewed inside W B n via the embedding W D n ֒ → W B n ) for all i = 0 , . . . , n − , whichis immediate. (cid:3) Proposition 2.7.
The homomorphism ι n is injective and h t ′ , . . . , t ′ n − i is a subgroupof e A B n of index two. Hence A D n can be identified with the subgroup of e A B n generatedby the t ′ i , i = 0 , . . . , n − .Proof. We first notice that, as an immediate consequence of Lemma 2.5, the subgroup U := h t ′ , . . . , t ′ n − i ⊆ e A B n is proper since W D n is a proper subgroup of W B n .As s ′ interchanges t ′ and s ′ t ′ s ′ = t ′ , and as s ′ commutes with t ′ i for i = 2 , . . . n − ,the involution s ′ normalizes U and induces on U an automorphism of order (whichis in fact an outer automorphism). Therefore, U = ι n ( A D n ) is of index in e A B n .Next we determine a presentation of U using the Reidemeister-Schreier algorithm(see for instance [19]). We take as a Schreier-transversal T := { , s ′ } for the rightcosets of U in e A B n . This yields the generating set { ts ′ i ts ′ i − | t ∈ T and ≤ i ≤ n − } = { t ′ i | ≤ i ≤ n − } where x is the representative of U x in T for x ∈ e A B n . Application of this algorithm andof Tietze-transformations (see [19]) then precisely yields the braid relations as statedin Lemma 2.3. This shows that ι n is injective. (cid:3) From now on we identify the subgroup h t ′ , t ′ , . . . , t ′ n − i ⊆ e A B n with A D n and we set t i = t ′ i for all i = 0 , . . . , n − . Note that by definition of e A B n , the map q B factorsthrough e A B n , giving rise to a surjection e q B : e A B n −→ A A n − . Then we have Lemma 2.8.
The map ι n satisfies e q B ◦ ι n = q D .Proof. We have q D ( t ) = σ and ( e q B ◦ ι n )( t ) = e q B ( s ′ s ′ s ′ ) = q B ( s ) q B ( s ) q B ( s ) = q B ( s ) = σ . For i ≥ we have q D ( t i ) = σ i = q B ( s i ) = e q B ( s ′ i ) = ( e q B ◦ ι n )( t i ) . (cid:3) Definition 2.9.
Given x ∈ W D n , we denote by x D the canonical positive lift of x in A D n (which we will systematically view inside e A B n ) and by x B the canonical positivelift of x in A B n . Proposition 2.10.
Let x ∈ W D n . We have π n, ( x B ) = x D .Proof. Let t i t i · · · t i k be an S D n -reduced expression of x in W D n . Replacing t by s s s and t i by s i for i = 1 , . . . , n − we get a word in the elements of S B n for x . Notethat this may not be a reduced expression for x in W B n . It suffices to show that onecan transform the above word into a reduced expression for x in W B n just by applyingbraid relations of type B n and the relation s = 1 .We prove the above statement by induction on k . If k = 1 then the claim holds since t i , i ≥ is replaced by s i while t is replaced by s s s which is S B n -reduced. Henceassume that k > . By induction the claim holds for x ′ = t i · · · t i k . By [8, Propositions8.1.2, 8.2.2] one has that s j , j ≥ is a left descend of x ′ in W B n if and only if it is aleft descent of x ′ in W D n . Hence we can assume that t i = t and that it is the onlyleft descent of x in W D n .Firstly, assume that s is a left descent of x ′ in W B n , hence s is not a left descentof s x ′ . We claim that it suffices to show that s is not a left descent of s x ′ : indeed, BARBARA BAUMEISTER AND THOMAS GOBET it implies that ℓ ( s s s x ′ ) = ℓ ( s x ′ ) + 2 (where ℓ is the length function in W B n ) by thelifting property (see [8, Corollary 2.2.8(i)]). Moreover by induction we can get every S B n -reduced decomposition of x ′ using only the claimed relations, hence we can byinduction get a reduced expression for x ′ starting with s with these relations. Theonly additional relation to apply to get a reduced decomposition of x is the deletionof the s = 1 which appears when appending s s s at the left of such a reducedexpression of x ′ . Hence assume that s s x ′ < s x ′ in W B n , i.e., that s is a left descentof s x ′ . By [8, Proposition 8.1.2] it follows that x ′− s (1) > x ′− s (2) which impliesthat x ′− ( − > x ′− (2) , hence − x ′− (2) > x ′− (1) . But by [8, Proposition 8.2.2] itprecisely means that t is a left descent of x ′ , a contradiction.Now assume that s is not a left descent of x ′ in W B n . Then s is not a left descentof x ′ in W B n , otherwise using [8, Proposition 8.1.2] again it would be a left descent of t i x ′ in W B n , hence in W D n by [8, Proposition 8.2.2], a contradiction. It follows thata reduced expression for y = s s x ′ in W B n is obtained by concatenating s s at theleft of a reduced expression for x ′ (which we can obtain by induction). If s y > y thenwe are done, while if s y < y then by Matsumoto’s Lemma we can obtain a reducedexpression of y starting with s just by applying type B n braid relations. Deleting the s at the beginning of the word we then have a reduced expression of x . (cid:3) Remark 2.11.
The fact that reduced expressions of an element x ∈ W D n can betransformed into reduced expressions in W B n as we did in the proof above had beennoticed by Hoefsmit in his thesis [16, Section 2.3] without a proof. The fact that A D n can be realized as a subgroup of e A B n also implies that the corresponding Iwahori-Heckealgebra H ( W D n ) of type D n embeds into the two-parameter Iwahori-Hecke algebra H ( W B n ) of type B n where the parameter corresponding to the conjugacy class of s is specialized at . This is precisely what Hoefsmit uses to study representations ofIwahori-Hecke algebras of type D n using the representation theory of those algebras intype B n . 3. Mikado braids of type B n and D n Mikado braids of type B n . We recall from [12] the following
Definition 3.1.
Let ( W, S ) be a finite Coxeter system with Artin group A W . Anelement β ∈ A W is a Mikado braid if there exist x, y ∈ W such that β = x − y . Wedenote by Mik( W ) (or Mik( X ) if W is of type X ) the set of Mikado braids in A W .We briefly recall results from [12, Section 6.2] on topological realizations of Mikadobraids in type B n which will be needed later on. The Artin group A B n embeds into A A n − , which is isomorphic to the Artin braid group on n strands. Labeling thestrands by − n, . . . , − , , . . . , n , every simple generator in S B n ⊆ S n, − n is then lifted toan Artin braid as follows. The generator s exchanges the strands and − , while thegenerator s i , i = 1 , . . . , n − exchanges the strands i and i + 1 as well as the strands − i and − i − (in both crossings, the strand coming from the right passes over the strandcoming from the left, like in the right picture in Figure 3). Those braids in A A n − which are in A B n are precisely those braids which are fixed by the automorphism whichexchanges each crossing i, i + 1 by a crossing − i, − i − of the same type, for all i . Wecall these braids symmetric .There is the following graphical characterization of Mikado braids in A B n Theorem 3.2 ([12, Theorem 6.3]) . Let β ∈ A B n . The following are equivalent(1) The braid β is a Mikado braid, that is, there are x, y ∈ W B n such that β = x − y . IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n (2) There is an Artin braid in A A n − representing β , such that one can inductivelyremove pairs of symmetric strands, one of the two strands being above all theother strands (so that the symmetric one is under all the other strands). Note that in the second item above, we remove pairs of strands instead of singlestrands so that at each step of the process, the obtained braid is still symmetric (hencein A B n ).3.2. Mikado braids of type D n inside e A B n . The aim of this subsection is to provethe following result, relating Mikado braids of type D n to Mikado braids of type B n : Theorem 3.3.
The Mikado braids of type D n viewed inside e A B n are precisely theimages of those Mikado braids of type B n which surject onto elements of W D n , that is,we have Mik( D n ) = { π n, ( β ) | β ∈ Mik( B n ) and π Bn ( β ) ∈ W D n } . Proof.
Let γ ∈ Mik( D n ) ⊆ e A B n . Then there exist x, y ∈ W D n such that γ = ( x D ) − y D .Note that by Lemma 2.5 we have π n, ( γ ) = x − y ∈ W D n . But by Proposition 2.10 wehave γ = π n, ( β ) where β = ( x B ) − y B ∈ Mik( B n ) , which shows the first inclusion.Conversely, let β ∈ Mik( B n ) such that π Bn ( β ) ∈ W D n . We have to show that π n, ( β ) ∈ Mik( D n ) . By definition there are x, y ∈ W B n such that β = ( x B ) − y B .Since π Bn ( β ) = x − y ∈ W D n , if either x or y is in W D n then both of them are in W D n in which case we are done by Proposition 2.10. Hence assume that x, y / ∈ W D n . Since W D n is a subgroup of W B n of index two and s / ∈ W D n there are x ′ , y ′ ∈ W D n such that x = s x ′ , y = s y ′ . If follows that x B = s ± x ′ B (the exponent depending on whether s x > x or not) and y B = s ± y ′ B . Hence since the image of s in e A B n has order two,using Proposition 2.10 again we have π n, ( β ) = ( x ′ D ) − y ′ D which concludes. (cid:3) Dual braid monoids
Noncrossing partitions.
Let ( W, S ) be a Coxeter system of spherical type. Let T = S w ∈ W wSw − denote the set of reflections in W and ℓ T : W −→ Z ≥ the corre-sponding length function. A standard Coxeter element in ( W, S ) is a product of all theelements of S . Given u, v ∈ W , we can define a partial order ≤ T on W by u ≤ T v ⇔ ℓ T ( u ) + ℓ T ( u − v ) = ℓ T ( v ) . In this case we say that u is a prefix of v .Let c be a standard Coxeter element. The set NC ( W, c ) of c-noncrossing partitions consists of all the x ∈ W such that x ≤ T c . The poset ( NC ( W, c ) , ≤ T ) is a lattice,isomorphic to the lattice of noncrossing partitions when W = W A n ∼ = S n +1 . See [2] formore on the topic. Remark 4.1.
There are several (unequivalent) definitions of Coxeter elements (see forinstance [4, Section 2.2]). The above definitions still make sense for more general Cox-eter elements, but for the realization of the dual braid monoids (which are introducedin the next section) inside Artin groups the Coxeter element is required to be standard(see [12, Remark 5.11]).4.2.
Dual braid monoids.
We recall the definition and properties of dual braidmonoids. For a detailed introduction to the topic the reader is referred to [5, 12]or [11]. Dual braid monoids were introduced by Bessis [5], generalizing definitions ofBirman, Ko and Lee [7] and Bessis, Digne and Michel [6] to all the spherical types.Let ( W, S ) be a finite Coxeter system. Denote by T the set of reflections in W and by BARBARA BAUMEISTER AND THOMAS GOBET A W the corresponding Artin-Tits group. Let c be a standard Coxeter element in W .Bessis defined the dual braid monoid attached to the triple ( W, T, c ) as follows. Takeas generating set a copy T c := { t c | t ∈ T } of T and set B ∗ c := h t c ∈ T c | t c ∈ T c , t c t ′ c = ( tt ′ t ) c t c if tt ′ ≤ T c i The defining relations of B ∗ c are called the dual braid relations with respect to c . Wemention some properties of B ∗ c , which can be found in [5]. The monoid B ∗ c is infiniteand embeds into A W . In fact, B ∗ c is a Garside monoid, hence it embeds into its groupof fractions Frac( B ∗ c ) and the word problem in Frac( B ∗ c ) is solvable. Bessis showed that Frac( B ∗ c ) is isomorphic to A W , but his proof requires a case-by-case analysis (see [5,Fact 2.2.4]) and the isomorphism is difficult to understand explicitly.More precisely, the embedding B ∗ c ⊆ A W sends s c to s for every s ∈ S . In [12,Proposition 3.13], a formula for the elements of T c (which are the atoms of the monoid B ∗ c ) as products of the Artin generators is given, but it does not give in general a braidword of shortest possible length. Example 4.2.
Let ( W, S ) be of type A and c ∈ W be the Coxeter element s s where s i = ( i, i + 1) . Then we have the dual braid relation ( s ) c ( s ) c = ( s s s ) c ( s ) c .Hence inside A W , the atom ( s s s ) c corresponding to the non-simple reflection s s s is equal to s s s − .As every Garside monoid, B ∗ c has a finite set of simple elements , which form alattice under left divisibility. They are defined as follows. For x ∈ NC ( W, c ) , let x = t t · · · t k be a T -reduced expression of x , that is, a reduced expression as productof reflections. Then Bessis showed that the element x c := ( t ) c ( t ) c · · · ( t k ) c ∈ B ∗ c isindependent of the choice of the reduced expression of x and therefore well-defined asa consequence of a dual Matsumoto property [5, Section 1.6]. The Garside elementis the lift c c of c and the set Div( c ) of simple elements (that is, of (left) divisors of c c ) is given by Div( c ) := { x c | x ∈ NC ( W, c ) } . There is an isomorphism of posets ( NC ( W, c ) , ≤ T ) ∼ = (Div( c ) , ≤ ) , x x c , where ≤ is the left-divisibility order in B ∗ c . Ingeneral, we are only able to determine the elements of Div( c ) as words in the classicalArtin generators S of A W by an inductive application of the dual braid relations. Itis therefore difficult to study properties of elements of Div( c ) viewed inside A W . Notethat the composition B ∗ c ֒ → A W ։ W sends every product ( t ) c ( t ) c · · · ( t k ) c , t i ∈ T to t t · · · t k .4.3. Standard Coxeter elements in W D n . In this subsection, we characterize stan-dard Coxeter elements in W D n in terms of signed permutations. This will be neededto introduce graphical representations of c -noncrossing partitions of type D n in Sec-tion 5.2.Recall that W D n ⊆ W B n and that w ( − i ) = − w ( i ) , for all i ∈ [ − n, n ] and all w ∈ W B n . In W B n , cycles of the shape ( i , . . . , i r , − i , . . . , − i r ) are abbreviated by [ i , . . . i r ] and called balanced cycles , and those of type ( i , . . . , i r )( − i , . . . , − i r ) by (( i , . . . , i r )) and called paired cycles . The set of reflections in W D n is T := T D n := { ( i, j )( − i, − j ) | i, j ∈ {− n, . . . , n } , i = ± j } , and every w ∈ W D n can be written as a product of disjoint cycles in which there is aneven number of balanced cycles (see [3, Section 2]). Lemma 4.3.
An element c ∈ W D n ( n ≥ ) is a standard Coxeter element if and onlyif c = ( i , − i )( i , . . . , i n , − i , . . . , − i n ) where { i , . . . , i n } = { , , , . . . , n } , i ∈ { , } and the sequence i · · · i n is first increasing, then decreasing. IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n Proof.
The proof is by induction on n . The case n = 3 is easy to check by hand. Let c be a standard Coxeter element in W D n , n ≥ . Then either s n c or cs n is a standardCoxeter element in W D n − , in which case induction and a straightforward computationshows that c is of the required form. Conversely if c is of the above form, then since ( i , − i ) commutes with s n either s n c or cs n is of the above form in W D n − , hence is astandard Coxeter element in W D n − , implying that c is a standard Coxeter element in W D n . (cid:3) Elements in NC ( W D n , c ) will be described below via a graphical representation.5. Simple dual braids of type D n are Mikado braids The aim of this section is to show Theorem 1.1, that is, that simple dual braids oftype D n are Mikado braids.5.1. Outline of the proof.
The proof proceeds as follows. • Step 1.
We describe in Section 5.2 a pictural model for the elements x ∈ NC ( W D n , c ) which is due to Athanasiadis and Reiner [3]. In this model theelement x is represented by a diagram consisting of non-intersecting polygonsjoining labeled points on a circle. The labeling depends on the choice of thestandard Coxeter element c , more precisely, we first require to write the Coxeterelement as a signed permutation (as in Lemma 4.3). • Step 2.
We slightly modify the diagram from Step associated to x ∈ NC ( W D n , c ) to obtain a new diagram N x consisting of non-intersecting polygonsjoining labeled points on a circle. The only difference with the Athanasiadis-Reiner model is that there is a point with two labels in the latter, which we splitin two different points. As we will see, the diagram N x is not unique in general,but we will show that all the information which we will use from the diagram N x is independent of the chosen diagram representing x . From this new dia-gram N x , we build a topological braid β x lying in an Artin group A B n of type B n (viewed inside A A n − , hence β x is a symmetric braid on n strands). Wefirst explain how to define the diagram N x for elements of T D n ⊆ NC ( W D n , c ) and we then do it for all x ∈ NC ( W D n , c ) . • Step 3.
We show that the braids π n, ( β t ) ∈ e A B n , for t ∈ T D n , lie in A D n andsatisfy the dual braid relations with respect to c . This will follow from the moregeneral statement that if x ≤ T xt ≤ T c with t ∈ T D n , then π n, ( β x ) π n, ( β t ) = π n, ( β xt ) . This property and the fact that π n, ( β s ) = s for all s ∈ S D n will beenough to conclude that π n, ( β x ) is equal to the simple dual braid x c for all x ∈ NC ( W D n , c ) (this is explained in the proof of Corollary 5.2). In particularwe also show that π n, ( β x ) does not depend on the choice of the diagram N x . • Step 4.
We show that the braid β x , x ∈ NC ( W D n , c ) is a Mikado braid in A B n by using the topological characterization of [12]. Recall that β x is defined graph-ically, as an Artin braid on n strands. Together with Step and Theorem 3.3,it follows that x c = π n, ( β x ) is a Mikado braid, which proves Theorem 1.1.5.2. Graphical model for noncrossing partitions.
Athanasiadis and Reiner founda graphical model for noncrossing partitions of type D n . We present it here (withslightly different conventions). First we explain how to label a circle depending on thechoice of the standard Coxeter element c .Given a standard Coxeter element c = ( i , − i )( i , . . . , i n , − i , . . . , − i n ) in W D n ,where the notation is as in Lemma 4.3 and where i = − n , we place n − points(labeled by i , . . . , i n , − i , . . . , − i n ) on a circle as follows: point − n is at the top of the circle while point n is at the bottom. The remaining points all have distinct heightdepending on their label: if i < j then point i is higher than point j . Moreover, whengoing along the circle in clockwise order starting at i = − n , the points must be metin the order i i · · · i n ( − i )( − i ) · · · ( − i n ) . Finally, we add a point at the center of thecircle, labeled by ± i .Athanasiadis and Reiner showed that c -noncrossing partitions are those for whichthere exists a graphical representation as follows (in their description, we have i = n ;this corresponds to a choice of Coxeter element which is not standard, however byconjugation we can assume it to be standard an to have i ∈ { , } . The c -noncrossingpartition lattices are isomorphic for all Coxeter elements c ). Given x ∈ NC ( W D n , c ) ,consider its cycle decomposition inside S − n,n and associate to each cycle the polygongiven by the convex hull of the points labeled by elements in the support of the cycle.It results in a noncrossing diagram, i.e., the various obtained polygons do not intersect,with two possible exceptions: if there is a polygon Q of x with i ∈ Q , − i / ∈ Q , then − Q is also a polygon of x . Thus the two polygons Q and − Q will have the middlepoint in common (Note that since x is a signed permutation, for every polygon P of x we have that − P is also a polygon of x , possibly with P = − P ). The second caseappears when the decomposition of x has a product of factors of the form [ j ][ i ] forsome j = ± i . In this case to avoid confusion with the noncrossing representation of thereflection (( j, i )) (or (( j, − i )) ) we have to choose an alternative way of representingthis product. Note that the cycle [ j ] should be considered as a polygon P such that P = − P . By analogy with the situation where there is such a polygon and where thepoint ± i lies inside P , we represent [ j ] by two curves both joining j to − j and notintersecting except at the points ± j , in such a way that the point ± i lies betweenthese two curves.Conversely, to every noncrossing diagram with the above properties, one can associatean element x of NC ( W D n , c ) as follows: we send each polygon P with labels j , j , . . . , j k (read in clockwise order) to the cycle ( j , j , . . . , j k ) except in case P = − P . Each singlepoint with label i is sent to the one-cycle ( i ) except i in case there is a polygon P with P = − P (in which case ± i lie inside P ). In this last case, if P is labeled by j , j , . . . , j k then we send it to the product of cycles ( i , − i )( j , j , . . . , j k ) (like inthe middle example of Figure 1). The element x is then the product of all the cyclesassociated to all the polygons of the noncrossing diagram (note that they are disjoint).Note that when the middle point lies in two different polygons, one has to specify inwhich polygon the label i lies. Examples are given in Figure 1 and we refer to [3] formore details. b bb bb bb bb b bb bb b − − − − − − − − b bb bb bb bb b bb bb b − − − − − − − − b bb bb bb bb b bb bb b − − − − − − − − Figure 1.
Examples of noncrossing diagrams for x =((1 , − , , − , x = ((8 , , , , , x = ((6 , , − ∈ NC ( W D n , c ) . IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n The diagram N x and the braid β x . To define the diagram N x , we slightlymodify the labeling of the circle given in the previous section by splitting the point ± i into two points placed on the vertical axis of the circle consistently with their labels (allthe points should be placed such that the point i is higher than the point j if i < j ).An example is given in Figure 2 and we call this labeling the c -labeling of the circle.The idea is then to start from Athanasiadis and Reiner’s graphical representation of x ∈ NC ( W D n , c ) and just split the middle point into two points. For convenience wemay represent the polygons by curvilinear polygons since in some cases, because ofthe splitting it might not be possible to have the polygons not intersecting each other.Depending on the situation we will add an edge joining the two points i and − i : weexplain more in details below how to draw the diagrams N x , first when x is a reflection,then in general. b bb bb bb bb bb bb bb b − − − − − − − − Figure 2.
Example of a c -labeling in type D . Here c = t t t t t t t t =(2 , − − , − , − , − , − , , and i = 2 .5.3.1. Pictures for reflections.
Reflections are all of the form t = c c , where c and c are two -cycles with opposite support. If c = ( i, j ) , we will draw a curvilinear“polygon” with two edges both joining i to j . We then orient the polygon in coun-terclockwise order. We do the same for c = ( − i, − j ) in such a way that the secondcurvilinear polygon does not intersect the first one. In some cases, there is not a uniqueway of drawing two such curvilinear polygons with the condition that the resultingdiagram should be noncrossing. We explain how to do it in the next paragraph byseparating the set of reflections into three classes.Firstly, assume that supp ( c ) = { i, j } ⊆ { , . . . , n } , then N t is drawn as in the leftpicture of Figure 3. Now assume that supp ( c ) = { i, − j } with i ∈ { , . . . , n }\{ i } , j ∈ {− , . . . , − n }\{− i } . In that case, we draw the two curvilinear polygons in such away that the two middle points labeled by ± i lie between them, as done in Figure 4.The last case is the case where c = ( i , j ) with j ∈ {− , . . . , − n }\{− i } . In that case,there are two ways of drawing the curvilinear polygon (see the left pictures of Figure 5).We can choose any of the two pictures for N t .Starting from such a noncrossing diagram, we then associate an Artin braid β t on n strands to it, by first projecting the noncrossing diagram to the right (as done inthe left pictures of Figures 3 and 4), i.e., putting all the points on the same verticalline, obtaining a new graph for the noncrossing partition. This new graph can then beviewed as a braid diagram, viewed from the bottom: a curve joining point k to point ℓ corresponds to a k -th strand ending at ℓ , while single points without a curve startingor ending at them correspond to unbraided strands. If a point has nothing at its right(resp. at its left), it means that the corresponding unbraided strand is above all theothers (resp. below all the others). The points lying right to (resp. left to) a curvecorrespond to an unbraided strand lying above (resp. below) the strand corresponding to that curve. See the above mentioned Figures. Note that in the case of Figure 5, thetwo braids β t obtained from the two different diagrams N t are distinct in A B n , but theirimages π n, ( β t ) in the quotient e A B n are the same because we can invert the crossingscorresponding to the generator s (because of the relation s = 1 which holds in thequotient). b bb bb bb bb bb bb bb b bbbbbbbbbbbbbbbb − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b Figure 3.
The diagram N t for t = (3 , − , − and the braid β t . b bb bb bb bb bb bb bb b bbbbbbbbbbbbbbbb − − − − − − − − − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b Figure 4.
The diagram N t for t = (3 , − − , and the braid β t .We now generalize the above picturial process, by associating a (possibly non unique)noncrossing diagram N x and an Artin braid β x to every x ∈ NC ( W D n , c ) .5.3.2. Pictures for noncrossing partitions.
To obtain a noncrossing diagram N x withoriented curvilinear polygons from x as we did for reflections in the previous section,we proceed as follows: we orient every polygon of the noncrossing partition in coun-terclockwise order (note that this is the opposite orientation to the one given by thecorresponding cycle of x , that is, an arrow j → j means that the cycle of x sends j to j ; hence this orientation corresponds to x − ). Polygons reduced to a single edgeare replaced by curvilinear polygons with two edges as we did for reflections in Sec-tion 5.3.1. Again we split the points with labels ± i into two points with labels − i and i respectively as in Figure 2.In the case where the middle point in the Athanasiadis-Reiner model has no edgestarting at it and does not lie inside a symmetric polygon, then the two points i and − i have no edge starting at them in the new diagram. In the case where there are twodistinct polygons P and − P sharing the middle point, they are separated so that each IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n b bb bb bb bb bb bb bb b − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb b − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b Figure 5.
Two diagrams N t for t = (2 , − − , and the correspond-ing Artin braids β t . Note that the two Artin braids on the right areequal in the quotient e A B n .point lies in the correct curvilinear polygon (see Figure 6): there might be several non-isotopic diagrams which work when separating P from − P (in case P is a -cycle weprecisely get what we already noticed and explained in Figure 5). A similar argumentto the one given in Figure 5 shows that the images in e A B n of the various Artin braids β x obtained from the distinct diagrams N x at the end of the process explained belowwill be equal. In case there is a symmetric polygon P = − P or a factor [ j ][ i ] in x , weadd a curvilinear polygon with two edges joining − i to i , oriented in counterclockwiseorder (Recall that in the noncrossing representation of [ j ][ i ] , the factor [ j ] is alreadyrepresented by a curvilinear “polygon” with two edges and the point i inside it. Herewe orient this polygon in counterclockwise order as in all other cases). b bb bb bb bb b bb bb b − − − − − − − − b bb bb bb bb bb bb bb b − − − − − − − − Figure 6.
Splitting of two polygons with common middle point.If one has the diagram N x with oriented curvilinear polygons as in Figure 6 on theright, we proceed exactly as we did for reflections in Section 5.3.1 to obtain β x : firstly,we put all the black points on a vertical line and project the noncrossing diagramto obtain a picture as in the left pictures in Figures 3 and 4; this diagram gives theArtin braid β x viewed from the bottom. We illustrate this process for the noncrossingdiagram of the element x = ((8 , , , , of Figure 1 in Figure 7. Note thatas a consequence of this procedure, the orientation we put on polygons, which as we already noticed at the beginning of the subsection is not the one corresponding to x but to x − , defines the permutation induced by the strands of β x . The fact that thepermutation induced by the strands of β x is x − rather than x comes from the factthat our convention is to concatenate Artin braids from top to bottom.Note that we can always recover the braid from the middle diagram without ambi-guity, because all the strands either strictly go up or down, except possibly in one case:in case i = 2 and x = (1 , − , − ∈ NC ( W D n , c ) , then the strands joining to − and − to do not strictly go up or down. In that case we represent the braid as donein Figure 8 in the next subsection. bbbbbbbbbbbbbbbb − − − − − − − − − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b Figure 7.
The Artin braid β x where x = ((8 , , , , is as inFigure 1. The strands corresponding to the cycle [6 , , are drawn inblue. Note that it is a Mikado braid in A B n . bbbbbbbbbbbbbbbb − − − − − − − − − − − − − − − − b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b Figure 8.
The Artin braid β t for t = (1 , − , − in case i = 2 .In this way, we associate to every noncrossing partition x ∈ NC ( W D n , c ) an Artinbraid β x ∈ A B n . For some x there are several possible β x ∈ A B n as illustrated in IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n Figure 5, but they have the same image under π n, , hence π n, ( β x ) is well-defined. Wehave Proposition 5.1.
Let x ∈ W D n , t ∈ T D n such that x ≤ T xt ≤ T c . Then π n, ( β x β t ) = π n, ( β xt ) . Proof.
The situation x ∈ NC ( W D n , c ) , t ∈ T and x ≤ T xt ≤ T c precisely corresponds toa cover relation in the noncrossing partition lattice of type D n . These covering relationswere described by Athanasiadis and Reiner [3, Section 3]: there are three families ofcovering relations. Setting y = xt , we have that x is obtained from y by replacing oneor two balanced cycles or one paired cycle as follows: [ j , j , . . . , j k ] [ j , . . . , j ℓ ](( j ℓ +1 , . . . , j k )) , ≤ ℓ < k ≤ n − , (( j , j , . . . , j k )) (( j , . . . , j ℓ ))(( j ℓ +1 , . . . , j k )) , ≤ ℓ < k ≤ n − , [ j , . . . , j ℓ ][ j ℓ +1 , . . . , j k ] (( j , . . . , j k )) , ≤ ℓ < k ≤ n − . Note that in the last case, we have either ℓ = 1 and j = ± i or k = ℓ + 1 and j k = ± i since x is a noncrossing partition. Indeed, the noncrossing partition has atmost one polygon P with P = − P , in which case the middle point lies inside P .We have to show that the braid that we obtain by the concatenation β x ⋆ β t has thesame image in e A B n as β xt . It is easy to deduce from the noncrossing representations N x what the result of the concatenation of two such braids is. By the process explainedabove, the noncrossing diagram itself can be considered as an Artin braid, viewed insidea circle or rather a cylinder. An edge of a curvilinear polygon represents a strand, andthe orientation indicates the startpoint and the endpoint of that strand.Consider the case where the cover relation xt x is the first one above, that is, itconsists of breaking a symmetric polygon into a symmetric polygon and two oppositecycles. This means that xt has the two symmetric factors [ i ] and [ j , j , . . . , j k ] while x has the same factors as xt except that the two symmetric factors are replaced by [ i ][ j , . . . , j ℓ ](( j ℓ +1 , . . . , j k )) for some ℓ ∈ { , . . . , k − } and t = (( j ℓ , j k )) . We have k ≥ . All the other polygonsof x and xt have support disjoint from {± j , . . . , ± j k } , hence when concatenating β x ⋆β t it is graphically clear that they will stay unchanged: indeed, these polygons are disjointfrom the two curvilinear polygons associated to the reflection t . Hence we can assumethat xt = [ i ][ j , j , . . . , j k ] and x = [ i ][ j , . . . , j ℓ ](( j ℓ +1 , . . . , j k )) . The situation isdepicted in Figure 9 below.In the concatenated diagram, the strand starting at j first goes to − j ℓ inside β x ,then the strand starting at − j ℓ goes to − j k inside β t . Hence the result is that thestrand starting at j goes to − j k , and can be drawn as in the diagram on the rightsince there is no obstruction for such an isotopy. Similarly, the strand starting at − j ℓ +1 first goes to − j k , then to j ℓ , hence is isotopic to the strand which goes directly from − j ℓ +1 to − j ℓ as drawn in the picture on the right. The same happens on the other side,while all other strands stay unchanged. It follows that the result of the concatenationcorresponds to the diagram on the right, which is precisely the diagram N xt associatedto xt .Hence we have the claim in the case where the cover relation is the one described,with k ≥ . We have to show the same for the other two cover relations. We also treatthe case of the last cover relation and leave the second one to the reader. Note that inthe case where the cover relation is given by [ j , . . . , j ℓ ][ j ℓ +1 , . . . , j k ] (( j , . . . , j k )) , bb bb bb bb bb bb bb − i i ... ......... − j k − − j ℓ +1 − j k − j ℓ j j − j − j j ℓ j k j ℓ +1 j k − − i i ... ......... ...... bb bb bb bb bb bb bb − j k − − j ℓ +1 − j k − j ℓ j j − j − j j ℓ j k j ℓ +1 j k − Figure 9.
Concatenating diagrams corresponding to the cover relation [ i ][ j , . . . , j k ] [ i ][ j , . . . , j ℓ ](( j ℓ +1 , . . . , j k ) . we have either ℓ = 1 and j = ± i or ℓ + 1 = k and j k = ± i . Assume that ℓ + 1 = k and j k = − i , the case where j k = i as well as the cases where ℓ = 1 , j = ± i are similar. We have x = (( j , . . . , j ℓ , − i )) , t = (( j ℓ , i )) . In this case, there are twopossible diagrams N x for x and the same holds for N t (see Figure 5 for an illustrationin the case where the noncrossing partition is a reflection). Since the correspondingbraids β x obtained from the two different diagrams N x have the same image under π n, we can choose any diagrams among the two, but the diagram N t has to be chosen to becompatible with the diagram N x if we want to do the same proof as for the first coverrelation. One of the two situations is represented in Figure 10. Arguing as in the firstcase we then get the diagram on the right of the figure for the concatenation β x ⋆ β t .This diagram is the diagram N xt up to the orientation of the two curves joining i to − i : but changing their orientation corresponds to inverting a middle crossing in β xt which gives rise to a braid which has the same image in e A B n thanks to the relation s = 1 . This proves the claim. bb bb bb bb − i i ...... − j ℓ − j j − j j j ℓ bb b bb b bb j − j − i i ...... − j ℓ − j j j ℓ Figure 10.
Concatenating diagrams corresponding to the cover rela-tion [ j , . . . , j ℓ ][ − i ] (( j , . . . , j ℓ , − i )) . (cid:3) Corollary 5.2.
Let x ∈ NC ( W D n , c ) . Then π n, ( β x ) = x c . IMPLE DUAL BRAIDS, NONCROSSING PARTITIONS AND MIKADO BRAIDS OF TYPE D n Proof.
Recall that S D n = { (1 , − − , } ∪ { ( i, i + 1)( − i, − i − | i = 1 , . . . , n − } .By construction of the braid β t from the diagram N t we have that π n, ( β s ) = s forall s ∈ S D n , and it is a general fact that s c = s for every simple reflection s . Hencewe have the claim in case x is in S D n and in particular π n, ( x ) lies in A D n . Sinceby Proposition 5.1 the elements π n, ( β t ) with t ∈ T D n satisfy the dual braid relationswith respect to c , we claim that π n, ( β t ) = t c for all t ∈ T D n . Indeed, for all t ∈ T D n ,we can always find s ∈ S D n such that either st ≤ T c or ts ≤ T c , say, st ≤ T c , and ℓ S ( sts ) < ℓ S ( t ) (this can be seen for instance using the noncrossing representation of t ). It follows that we have the dual braid relation π n, ( β s ) π n, ( β t ) = π n, ( β sts ) π n, ( β s ) . Arguing by induction on ℓ S ( t ) , we have that π n, ( β q ) = q c for every reflection q oc-curring in the above equality except possibly t . Thanks to the dual braid relation s c t c = ( sts ) c s c we get that π n, ( β t ) = t c and in particular that π n, ( β t ) ∈ A D n .Now for x ∈ N C ( W D n , c ) arbitrary we can use Proposition 5.1 as well as the factthat x ≤ T xt ≤ T c , t ∈ T D n , implies that ( xt ) c = x c t c (see the end of Subsection 4.2)to get by induction on ℓ T ( x ) that π n, ( β x ) = x c . (cid:3) Simple dual braids are Mikado braids.
In all the examples drawn in thefigures given in the previous sections, we see that the Artin braids β x resulting fromsimple dual braids are Mikado braids: they indeed satisfy the topological conditiongiven by the point (2) of Theorem 3.2. This is the main statement which we want toprove here. Proposition 5.3.
Let x ∈ NC ( W D n , c ) . Then β x ∈ A B n is a Mikado braid.Proof. As β x ∈ A B n , it suffices to verify the point (2) of Theorem 3.2. Note that exceptin case x = (1 , − , − and i = 2 (in which case the braid β x which is drawn inFigure 8 is obviously Mikado), the diagram which we obtained from N x by putting allthe dots on the same vertical line (as done in Figures 3 and 4; we call this diagrama vertical diagram ) has the following property: each oriented curve joining two pointseither strictly increases or strictly decreases, and every two such distinct curves nevercross. The first property follows from the fact that the diagram is obtained from N x by projecting to the right a curve which is already either strictly increasing or strictlydecreasing, while the second follows from the fact that the polygons in N x do not cross.In such a diagram, consider a curve joining two points and going up with respect tothe orientation, with no other curve lying at its right. It follows from the discussion inthe paragraph above that it always exists. Every single point lying at the right of sucha curve corresponds to a vertical unbraided strand in β x which lies above all the otherstrands. Therefore, every such point can be removed in the vertical diagram, and thesymmetric point lying at the left of the curve which is symmetric to the original curvecan be removed simultaneously: it corresponds to removing a vertical unbraided strandlying above all the other strands in β x , and simultaneously removing the symmetricunbraided strand lying below all the other strands, giving a new braid β ′ x lying in A B n − since we removed a symmetric pair of strands. After removing all such points in thevertical diagram, the original curve has nothing at its right, hence corresponds to astrand which lies above all the other strands, and we can therefore remove it, as well asits symmetric strand. Again we obtain an element which lies in an Artin group of type B m for a smaller m . Going on inductively, we can remove every strand correspondingto a curve, with a braid which stays symmetric at each step. If after removing thelast curve we still have points, these correspond to vertical unbraided strands whichcan be removed. This concludes by Theorem 3.2. We illustrate the above procedure inExample 5.4 below. (cid:3) Note that we could define more generally vertical diagrams (not necessarily corre-sponding to simple dual braids) and associate to them an Artin braid, which wouldtherefore always be Mikado.
Example 5.4.
We illustrate the procedure given in the proof of Proposition 5.3 in case x is the element x = ((8 , , , , from Figure 1. The vertical diagram and thebraid β x are given in Figure 7. The blue curve joining to − in the vertical diagramhas no other curve lying at its right. There is only the single point , which correspondsin β x to a strand which lies above all the others, with the symmetric strand − lyingbelow all the others. Removing the pair of strands and − , we get a symmetricbraid on strands, hence in A B . We can then remove the strand corresponding tothe original curve joining to − as well as its symmetric strand, since there is noremaining strand lying above it. Going on inductively we eventually remove all pairsof strands.As a corollary we get the main result Theorem 5.5.
Let x ∈ NC ( W D n , c ) . Then x c is a Mikado braid.Proof. By Corollary 5.2 we have that π n, ( β x ) = x c for every x ∈ NC ( W D n , c ) . Butby Proposition 5.3, β x is a Mikado braid in A B n . Applying Theorem 3.3 we get that x c = π n, ( β x ) is a Mikado braid in A D n . (cid:3) References [1] D. Allcock, Braid pictures for Artin groups,
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E-mail address : [email protected] Thomas Gobet, Institut Élie Cartan de Lorraine, Université de Lorraine, site deNancy, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
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