Interval groups related to finite Coxeter groups I

We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements. In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.


Introduction
The philosophy of interval Garside theory is that starting from suitable intervals in a given group, we construct an interval Garside monoid and group, along with a complex whose fundamental group is the interval Garside group, such that the divisibility relations of the interval provide relevant information about the interval Garside group.Part of the information we obtain are efficient solutions to the word and conjugacy problems, as well as important group-theoretical properties [13].Interval Garside groups also enjoy important homological, and homotopical properties [12].
Garside theory is relevant in the context of Coxeter and Artin groups.Actually, Garside structures first arose out of observations of properties of Artin's braid group that were made in Garside's Oxford thesis [18] and his article [19].It was then realised that Garside's approach extend to all Artin groups of spherical type, independently by Brieskorn-Saito and Deligne in two adjacent articles in the Inventiones [9] and [14].This approach is called the standard approach to Coxeter and Artin groups.
The dual approach consists of analysing the Coxeter group as a group generated by all its reflections.Spherical Artin groups are constructed from intervals called the generalised non-crossing partitions.This approach was established by Bessis in [6].These intervals consist of elements lying below the so-called Coxeter elements that play a prominent role within the dual approach.The Coxeter elements are all conjugate to one another and some of them can be found by taking the product of the elements in the standard generating set in any order.
Coxeter elements are of maximal length over the set of reflections, but they do not exhaust all the elements of maximal length.Quasi-Coxeter elements [3] are of maximal length such that the reflections in a certain reduced decomposition generate the Coxeter group.Among them are Coxeter elements.We call a proper quasi-Coxeter element a quasi-Coxeter element that is not a Coxeter element.Amongst the infinite families of finite Coxeter groups, proper quasi-Coxeter elements exist only in type D n .Carter [11] classified the conjugacy classes in Weyl groups.Among them are the conjugacy classes of quasi-Coxeter elements.He also defined diagrams related to these classes that we call Carter diagrams.Cameron-Seidel-Tsaranov [10] defined presentations of Weyl groups defined on Carter diagrams by adding cycle commutator relators.
We establish presentations of the interval groups related to all quasi-Coxeter elements.Our presentations are compatible with the analysis of Carter [11].Actually, they are always nicely defined on Carter diagrams by adding either cycle commutator relators or twisted cycle commutator relators depending whether the quasi-Coxeter element is a Coxeter element or not.Twisted cycle and cycle commutator relators can be written as relations between positive words.For Coxeter elements, where the interval group is the Artin group, some of our group presentations also arise from cluster algebras (see [1,20] and also [21]).For almost all the other proper quasi-Coxeter elements, we can establish that the interval group related to each of them is not isomorphic to the corresponding Artin group.Although we obtain nice presentations of these groups, the intervals of proper quasi-Coxeter elements are not lattices in almost all the cases, hence not giving rise to Garside structures.This classifies the interval Garside structures one obtains for quasi-Coxeter elements within the dual approach.Along with the description of the presentations of interval groups, we describe important properties for quasi-Coxeter elements, their divisors, and their lifts to the interval groups.
We define the Artin group A(W ) associated with a Coxeter system (W, S) as follows.
Definition 2.2.The Artin group A(W ) associated with a Coxeter system (W, S) is defined by a presentation with generating set S in bijection with S and the braid relations: sts . . .mst = tst . . .mst for s, t ∈ S and s = t, where m st ∈ Z ≥2 is the order of st in W .
These presentations are often represented graphically using a Coxeter diagram Γ.This is a graph with vertex set S, in which the edge {s, t} exists if m st ≥ 3, and is labelled with m st when m st ≥ 4. Let Γ be such a diagram.We denote by W (Γ) and A(Γ) the related Coxeter and Artin groups W and A(W ).The finite Coxeter groups are precisely the real reflection groups, and the spherical Artin groups are the Artin groups related to the finite Coxeter groups.The corresponding Coxeter diagrams are the three infinite families of types A, B, and D, and the exceptional cases of types E 6 , E 7 , E 8 , F 4 , H 3 , H 4 , and I 2 (e).In the remainder of the article, W will always be a finite Coxeter group.
Recall that the Coxeter group of type A n (n ≥ 1) is the symmetric group Sym(n + 1) and the related Artin group is the usual braid group B n+1 = s 1 , . . ., s n | s i s i+1 s i = s i+1 s i s i+1 for 1 ≤ i ≤ n − 1 and s i s j = s j s i for |i − j| > 1 .

Quasi-Coxeter elements
Let (W, S) be a finite Coxeter system, and let T := ∪ w∈W S w be the set of all its reflections.As each w ∈ W is a product of reflections in T , we can define ℓ T (w) := min{k ∈ Z ≥0 | w = t 1 t 2 . . .t k ; t i ∈ T }, the reflection length of w.If w = t 1 t 2 . . .t k with t i ∈ T and k = ℓ T (w), we call (t A Coxeter element is a conjugate of any element that is written as the product of the simple generators of W in any order.Note that every Coxeter element is a quasi-Coxeter element.A quasi-Coxeter element is called proper if it is not a Coxeter element.
It is shown in [5] that the quasi-Coxeter elements in simply laced Coxeter groups are precisely those elements that admit a reduced decomposition into reflections such that the roots related to these reflections form a basis of the related root lattice.In the non-simply laced case, it is also required that the system of coroots generates the coroot lattice.
Recall that a parabolic subgroup of W is a subgroup generated by a conjugate of a subset of S. Note that a more general definition of parabolic subgroups, which is in fact equivalent to our definition for finite Coxeter systems, is used in [2,3].We call an element in W a parabolic quasi-Coxeter element if it is a quasi-Coxeter element in a parabolic subgroup of W .
Since the set T of reflections is closed under conjugation, there is a natural way to obtain new reflection decompositions from a given one.The braid group B n acts on the set T n of n-tuples of reflections via the so-called Hurwitz action of B n on T n .It is readily observed that this action restricts to the set of all reduced reflection decompositions of a given element w ∈ W .If the latter action is transitive, then we say that the dual Matsumoto property holds for w.
Theorem 2.4.An element w ∈ W is a parabolic quasi-Coxeter element if and only if the dual Matsumoto property holds for w.
We recall the following fact, and thereby introduce the notation P w for parabolic quasi-Coxeter elements w ∈ W .The result is an easy consequence of Theorem 6.1 of [3].
Lemma 2.5.Let w ∈ W be a parabolic quasi-Coxeter element and w = t 1 t 2 . . .t k be a reduced decomposition into reflections.Then P w := t 1 , . . ., t k is a parabolic subgroup and the definition of P w is independent of the choice of the reduced reflection decomposition of w.

Decomposition diagrams
We introduce diagrams related to reduced decompositions.Definition 2.6.Let t 1 t 2 . . .t k be a reduced decomposition of w ∈ W .We define a decomposition diagram related to t 1 t 2 . . .t k as follows.The vertices of the diagram correspond to the reflections t 1 , t 2 , . . ., t k .If two reflections commute, we put no edge between the related vertices.Otherwise, we put an edge, and we label it by the order of the product of the two reflections when this order is strictly bigger than 3.In Carter's classification of the conjugacy classes in the Weyl groups [11], it is shown that every element w in W is the product w = w 1 w 2 of two involutions, and that each involution is the product of commuting reflections, which then provides a bipartite decomposition of w.Carter exhibited the list of conjugacy classes of proper quasi-Coxeter elements by describing for each class a diagram related to a bipartite decomposition for a representative of the class (see Table 2 in [11]) which we call a Carter diagram.Note that Definition 2.6 generalises the notion of Carter diagrams.

Interval groups of quasi-Coxeter elements
We start by defining left and right division.Definition 2.7.Let v, w ∈ W .We say that v is a (left) divisor of w, and write v w, if w = vu with u ∈ W and ℓ T (w) = ℓ T (v) + ℓ T (u), where ℓ T (w) is the length over T of w ∈ W .The order relation is called the absolute order relation on W .
The interval [1, w] related to an element w ∈ W is defined to be the set of divisors of w for , that is Similarly, we define the division from the right.We say that v is a right divisor of w, and write v r w, if w = uv with u ∈ W and ℓ T (w) = ℓ T (v) + ℓ T (u).Similarly, we also define the interval [1, w] r of right divisors of an element w ∈ W . Remark 2.8.A quasi-Coxeter element has the inductive property that every left divisor of it is a parabolic quasi-Coxeter element (see Corollary 6.11 in [3]).Therefore, if w is a quasi-Coxeter element, then every element in the interval [1, w] is a parabolic quasi-Coxeter element.Now we introduce the definition of an interval group related to quasi-Coxeter elements in W .Let w be a quasi-Coxeter element in W . Consider the interval [1, w] of divisors of w.Definition 2.9.We define the group G( [1, w]) by a presentation with set of generators [1, w] in bijection with the interval [1, w], and relations corresponding to the relations in [1, w], meaning that uv = r if u, v, r ∈ [1, w], uv = r, and u r, i.e. ℓ T (r) = ℓ T (u) + ℓ T (v).
By transitivity of the Hurwitz action on the set of reduced decompositions of w (see Lemma 2.4), the next result follows immediately.Proposition 2.10.Let w ∈ W be a quasi-Coxeter element, and let T ⊂ [1, w] be the copy of the set of reflections T in W . Then is a presentation of the interval group with respect to w.
Notice that the relations described in Proposition 2.10 are the relations that are visible in the poset ([1, w], ) in heights one and two.They are called the dual braid relations (see [6]).
The following result due to Bessis-Digne-Michel [7] is the main theorem in interval Garside theory.
Since T is stable under conjugation, quasi-Coxeter elements are always balanced.The only obstruction to obtaining interval Garside groups is the lattice property.In the case when the quasi-Coxeter element is a Coxeter element, Bessis [6] showed the following.Theorem 2.12.Let W be a finite Coxeter group.The interval group G([1, w]) for w ∈ W a Coxeter element is an interval Garside group isomorphic to the corresponding Artin group A(W ).
The main purpose of our work is to continue the analysis of the interval groups related to all quasi-Coxeter elements.
We introduce the following notation, which we shall use in the remainder of the article.

Strategy of the proof
We describe here our general strategy for the proof of Theorem A (see also Theorem 5.1).We are also going to mention some important results that we established within the proof, as they are interesting in themselves.
Let W be the Coxeter group of type D n .We employ the description of W as a group of monomial matrices as will be explained in Section 3.1.Let w be a quasi-Coxeter element of the Coxeter group W of type D n .Actually, there exists a reduced decomposition of w whose reflections s 1 , s 2 , . . ., s n satisfy the relations that can be described by the Carter diagram ∆ (see Figure 1 in Section 3.3).The quasi-Coxeter elements in type D n are characterised in Proposition 3.8.From now on, we let S := {s 1 , s 2 , . . ., s n }.
By [10], the Coxeter group W admits a presentation on the set S of generators whose relations are the quadratic relations (s 2 i = 1 for 1 ≤ i ≤ n) along with the relations of the diagram ∆ and the cycle commutator relator cc(s i , s j , s k , s l ) in correspondence with the unique 4-cycle (s i , s j , s k , s l ) of ∆.Note that in W , the cycle commutator and the twisted cycle commutator relators associated with the 4-cycle are the same.All this is described in Section 3.
Consider the interval group G( [1, w]).By Proposition 2.10, the group G( [1, w]) is generated by a copy T of T along with the dual braid relations tt ′ = t ′ t ′′ (t ∈ T corresponds to t ∈ T ), whenever tt ′ = t ′ t ′′ w, t = t ′ and t, t ′ , t ′′ ∈ T .
We want to prove that G( [1, w]) is isomorphic to the group G that is defined by a presentation on the set of generators S = {s 1 , s 2 , . . ., s n } ⊂ T corresponding to the subset S of T with the corresponding relations described by ∆, along with the twisted cycle commutator relator tc(s i , s j , s k , s l ) in correspondence with the 4-cycle (s i , s j , s k , s l ), where s i , s j , s k , s l correspond to s i , s j , s k , s l , respectively.
Step 1: Definition of f .We define a map f from G to G( [1, w]) by setting f (s i ) = s i for each i.It will follow from Proposition 3.12 and Lemma 5.12 that the braid relators b(s i , s j ) and the twisted cycle commutator relator tc(s i , s j , s k , s l ) specified by the presentation given for G hold in G( [1, w]) as well.Hence f extends to a homomorphism from G to G( [1, w]).
Step 2: Reduced decompositions and their diagrams.Let w 0 be a divisor of length n − 1 of w.We describe a particular reduced decomposition of w 0 = t 1 t 2 . . .t n−1 in Sections 4.2 and 4.3, and characterise whether w 0 is a Coxeter element or a proper quasi-Coxeter element in the subgroup P w0 := t 1 , . . ., t n−1 ⊆ W (see for instance Propositions 4.5,4.15,and 4.16).In order to describe these reduced decompositions, we describe a combinatorial technique (in Section 4.1) by using the description of the elements of W as monomial matrices.
The reduced decomposition t 1 t 2 . . .t n−1 of w 0 corresponds to a decomposition diagram (see Definition 2.6) that we denote by ∆ 0 .We show that ∆ 0 is a disjoint union of Coxeter diagrams of types A or D or of the same type as ∆ (but with fewer generators) with a (single) 4-cycle (see Propositions 4.5,4.15,and 4.16).Thereby we are able to determine the type of w 0 .
If w 0 is a Coxeter element in P w0 , then by [6], the dual braid relation tt ′ = t ′ t ′′ is satisfied in G([1, w 0 ]) where tt ′ = t ′ t ′ w is a consequence of the relators b(t i , t j ) for t i , t j ∈ {t 1 , t 2 , . . ., t n−1 } (i = j).If w 0 is a proper quasi-Coxeter element, then induction on n implies that the dual braid re- In this way, we have shown that all the dual braid relations are consequences of the relations between t 1 , t 2 , . . ., t n−1 in correspondence with the relations between t 1 , t 2 , . . ., t n−1 implied by each decomposition diagram ∆ 0 corresponding to a divisor w 0 of length n − 1 of w.
Step 3: Decomposition of elements in T and T .In order to find a homomorphism g : G([1, w]) → G, we describe a decomposition of each element t in T in terms of elements in S.This can be done because of the dual Matsumoto property for w, i.e. the transitive Hurwitz action on the set of reduced decompositions over T of w.This is based on particular decompositions over S of the reflections in T .This is done in Propositions 5.7 and 5.10.
Let g be the map that sends t i ∈ G([1, w]) to its decomposition over S as given in Proposition 5.10.We show that the map g is a homomorphism.Suppose that tt ′ = t ′ t ′′ is a dual braid relation of G( [1, w]).We need to check that the image of this relation under g holds within G.There exists a divisor w 0 of length n − 1 of w such that tt ′ is a prefix of w 0 .Therefore, tt ′ = t ′ t ′′ holds in G([1, w 0 ]).So rather than checking that the images by g of all the dual braid relations hold in G, our strategy is to shift the analysis to the groups G([1, w 0 ]), from which we can establish the desired homomorphism.This analysis was done in Step 2.
Step 4: Lift of the relations.In order to conclude homomorphism for g, we finally need to show that the image by g of all the defining relations between the elements t 1 , t 2 , . . ., t n−1 of G([1, w 0 ]) in correspondence with the relations between t 1 , t 2 , . . ., t n−1 implied by each decomposition diagram ∆ 0 can be derived from the relations between the elements of S that are implied by the diagram ∆.This is proved in Section 6.3 by induction on n.The base of our induction is the cases n = 4 and n = 5 (see Sections 6.1 and 6.2).Note that we also separate n = 5 as a base of induction so that we do not need anymore to show twisted cycle commutator relators in Section 6.3.This is possible since, apart from one special case (Equation 11 with i = n − 1), in the reduced decomposition t 1 t 2 . . .t n−1 of each w 0 , the only reflection t i such that g(t i ) contains s n in its decomposition over S is precisely the last one, that is t n−1 .Section 6.4 concludes the proof that g is a homomorphism.Isomorphism between G( [1, w]) and G is proved once the composites f • g and g • f have been shown to be identity maps.
Note that it might be possible to apply this strategy more generally, but this article only deals with the case where W is of type D n .
3 Dual approach to the Coxeter group of type D n

The Coxeter group of type D n
We employ the description of the Coxeter group W of type D n (n ≥ 4) as the group of n × n monomial matrices such that the nonzero coefficients are equal to 1 or −1 and their product is equal to 1.This description will help us to describe our combinatorial technique and to easily explain our arguments.
Note that this description of W corresponds to the case d = 1, e = 2 of the infinite series G(de, e, n) of complex reflection groups (see [26]).Notation 3.1.A monomial matrix w ∈ W is associated with a permutation in Sym(n) that has been marked by overlining some elements within its cycles.We call the result, σ w , a marked permutation, where an entry i (1 ≤ i ≤ n) of a cycle indicates that the coefficient in row i of w is equal to 1, while an entry i indicates that this coefficient is equal to −1.
The monomial matrix w is denoted by σ w , so we have w = σ w .When there is no confusion, we remove the cycles (i), for 1 ≤ i ≤ n of length 1 from σ w .
We note that, for w ∈ W , the marked permutation σ w must always have an even number of overlined entries.We also note that the notation σ w is not the cycle decomposition of the permutation π w of the unit vectors ±e i , 1 ≤ i ≤ n of R n that is also naturally associated with w.In fact, each cycle of length k in σ w corresponds to either two cycles of length k or a single cycle of length 2k in π w .
be an element of W of type D 6 .Using Nota- where, for brevity, we label the unit vectors ±i with 1 ≤ i ≤ 6. .
We set the following convention for the remainder of the article.
By convention, we also set i = i for any positive integer i.We also suppose that a decreasing-index cycle of the form (x i , x i−1 , . . ., x i ′ ) is the identity element when i ≤ i ′ and an increasing-index expression of the form (x i , x i+1 , . . ., x i ′ ) is the identity element when i ≥ i ′ .Finally, we also assume that a cycle of length ≥ 3 that contains n should start by n.
Lemma 3.4.The set T of reflections in W is represented by the set of elements Note that the reflection (i, j) represents a transposition matrix whose entries are all 1, while (i, j) represents a matrix derived from the previous one by changing the signs in rows i and j.For 2 ≤ i ≤ n, we denote by s i the reflection (i − 1, i).
The following lemma is straightforward to prove.Lemma 3.5.Let t and t ′ be two reflections in W , with t = (i, j) or (i, j) and t ′ = (k, l) or (k, l).If {i, j} does not intersect {k, l}, then the reflections t and t ′ commute.If the cardinality of the intersection is 1, then we get (i, j)(k, l)(i, j) = (k, l)(i, j)(k, l).

Length function over the set of reflections
Shi computed in [25] the length function over the set of reflections in the infinite series of complex reflection groups.The Coxeter group W of type D n corresponds to the group G(2, 2, n).The length function over the set of reflections in G(2, 2, n) appears in Corollary 3.2 in [25].Let us recall this result.Proposition 3.6.Let w ∈ G(2, 2, n), and suppose that w is represented by a marked permutation σ w as described in Notation 3.1 Suppose that σ w is written as a product of r cycles, and define e to be the number of these cycles that have an even number of overlined entries.Then, the length ℓ T (w) over T of w is equal to n − e.Note also that the n × n identity matrix corresponds to the marked permutation with n cycles each containing a single entry i.Each cycle then contains 0 overlined entries.So e = n and we see that the ℓ T (Id) = 0.

Quasi-Coxeter elements in type D n
Let W be a Coxeter group of type D n .By Carter [11], W contains ⌊ n 2 ⌋ conjugacy classes of quasi-Coxeter elements.We fix an integer m with 1 ≤ m ≤ ⌊n/2⌋; this fixes a conjugacy class of quasi-Coxeter elements in W .The m-th conjugacy class is associated by Carter [11] with the diagram ∆ m,n displayed in Figure 1.When there is no confusion, we denote ∆ m,n by ∆.
In ∆ an edge between two nodes s i and s j describes the relation s i s j s i = s j s i s j , and when there is no edge between s i and s i , this means that the two reflections commute.In the next proposition, we choose a particular representative of each conjugacy class of quasi-Coxeter elements that will be helpful in the description of our main result.Proposition 3.8.Choose the reflections s 1 := (m, m + 1) and The element w can be written as the product s 2 s 3 . . .s m s 1 s m+1 s m+2 s m+3 s m+4 . . .s n .
Proof.See Proposition 25 in [11] for representatives of the conjugacy classes, where Carter defines the notion of signed cycle-type.The second sentence of the proposition is readily checked.The Carter diagram ∆ is the decomposition diagram (see Definition 2.6) related to the reduced decomposition s 2 s 3 . . .s m s 1 s m+1 s m+2 s m+3 s m+4 . . .s n of the quasi-Coxeter element w.We are using this particular decomposition of the quasi-Coxeter element since it will be helpful to describe the divisors of length n − 1 of w in Section 4 and to describe necessary combinatorial techniques for our analysis in Sections 4 and 5.
Example 3.9.The element w = (3, 2, 1)(6, 5, 4) is a representative of a conjugacy class of quasi-Coxeter elements in type D 6 .In this case m = 3.We end this section by the following statement that will be used to construct the homomorphism f introduced in Step 1 of the strategy of our proof in Section 2.5.Proposition 3.12.(1) We have that s i s j w and s i s j is of order 2, for |i − j| > 1.
(2) We have that s i s i+1 w, and s i s i+1 is of order 3, for 2 ≤ i ≤ n − 1.
(3) Let 2 ≤ i ≤ n.We have that s 1 s m w, s m+2 s 1 w, and s 1 s i w.Further the elements s 1 s m and s m+2 s 1 are of order 3, and s 1 s i is of order 2.
Proof.The result is an immediate consequence of the Hurwitz action and of the choice of the elements s i .

Maximal divisors of quasi-Coxeter elements
As we pointed out in the strategy of our proof (Step 2), our method depends on an analysis of maximal divisors of a quasi-Coxeter element w, and in particular of the decomposition of each such as a product of n − 1 reflections.In Section 4.1 we identify 11 different cases for such maximal divisors w 0 , which fall into three types, I, II and III, and then in the following sections, we find reduced decompositions for elements w 0 of type I (in Section 4.2), and of types II and III (in Section 4.3), as well as their decomposition diagrams (see Definition 2.6).

Divisors of length n − 1
Let w = (m, m − 1, . . ., 2, 1)(n, n − 1, . . ., m + 1) be a quasi-Coxeter element.The elements of length n − 1 that divide w consist of all the products w(i, j) and w(i, j) for which 1 ≤ i < j ≤ n.We denote by w 0 a divisor of length n − 1 of w.We compute these divisors in Equations 1 to 11 below.We distinguish 3 types that we denote by I, II, and III and that are displayed in the following Tables 1, 2, and 3.The first column of each table represents the cases for i and j.The second column is the divisor w 0 .Notice that we get from type II to type III by applying symmetry.
Remark also that Equation 2 is similar to Equation 1; the difference is that two entries are further overlined in Equation 2. We see the same similarities between Equations 6 and 7, and Equations 9 and 10.
Notice that each element w 0 of the 11 Equations admits exactly one cycle with an even number of overlined elements (we assume that 0 is even).Hence each element is of length n − 1 by Proposition 3.6.In Sections 4.2 and 4.3, we describe a reduced decomposition over the set T of reflections for each divisor w 0 of w of type I and of types II and III, respectively.
We provide an example where we explicitly write the monomial matrices.
Let us multiply w from the right by the transposition (1, 4) as described in Equation ( 1).So we get Using the marked permutation notation introduced in Notation 3.1, we have that w(1, 4) = (5, 1, 2, 4, 3) (the coefficient is equal to −1 on row numbers 1 and 3 of the matrix w 1 ) which is compatible with the result of Equation 1.

Reduced decompositions and diagrams for type I
Suppose that w 0 has type I (see Table 1).As a marked permutation, it is a cycle of the form (x 1 , x 2 , x 3 , . . ., x n ), where each x k is equal to p or p (1 ≤ p ≤ n), with {x 1 , x 2 , . . ., x n } = {1, 2, . . ., n}, and with an even number of overlined entries (see Equations 1 to 5).We will describe how to produce a reduced reflection decomposition of length n − 1 for this element.
We continue the study of Example 4.1 that will help the understanding of a procedure that describes the reduced decompositions.The general idea is to multiply the marked permutation w 0 = (x 1 , x 2 , x 3 , . . ., x n ) from the right by a sequence of reflections in order to obtain the identity matrix.A decomposition of w 0 is given by the product in reverse order of all the reflections used in the procedure.It turns out that this decomposition is reduced.
The general procedure is as follows.
• If x k = p, then whether x k+1 = q or q, we multiply (x k , x k+1 , . . ., x r ) from the right by the reflection (p, q) and get (x k )(x k+1 , x k+2 , . . ., x r ), whose length is one less than the length of (x k , x k+1 , . . ., x r ).
Remark that the entry x r can be equal to u for 1 ≤ u ≤ n in which case it is equal to u, by the convention that i = i for any positive integer i (see Convention 3.3).Proposition 4.4.Let w 0 be a divisor of w of type I.A reduced decomposition of w 0 is obtained as the product in reverse order of the reflections that are applied in Procedure 4.3.
Proof.Let w 0 be a divisor of w of type I.It is of the form (x 1 , x 2 , x 3 , . . ., x n ).Applying Procedure 4.3 for all k from 1 to n − 1, the element (x 1 , x 2 , x 3 , . . ., x n ) is transformed to the identity matrix (x 1 )(x 2 ) . . .(x n ) in n − 1 steps.Since all reflections are of order 2, a decomposition of the element (x 1 , x 2 , x 3 , . . ., x n ) is given by the product in reverse order of all the reflections used in this procedure.
Since ℓ(w 0 ) = n − 1 by Proposition 3.6, the decomposition we obtain is reduced (as it consists of n − 1 reflections).We apply Proposition 4.4 to establish decompositions of type I divisors w 0 of w, as described in the following lemmas.Lemma 4.6.Let w 0 := w(i, j) be a divisor of w of type I, where 1 ≤ i ≤ m, m + 1 ≤ j ≤ n, so that we are in the situation of Equation 1. Then w 0 has one of the following decompositions of length n − 1 over T .
Lemma 4.7.Let w 0 = w(i, j) be a divisor of w of type I, where 1 ≤ i < m, m + 1 ≤ j < n, so that we are in the situation of Equation 2. Then w 0 has one of the following decompositions of length n − 1 over T .
Lemma 4.8.Let w 0 = w(m, j) be a divisor of w of type I, where m + 1 ≤ j < n, so that we are in the situation of Equation 3. Then w 0 has one of the following decompositions of length n − 1 over T .
Similarly, we get the next result.
Lemma 4.9.Let w 0 be as in the situation of Equations 4 and 5. Then w 0 has the following decompositions of length n − 1 over T .
(1) Suppose i = m and j = n, so that we are in the situation of Equation 4: Suppose i = m and j = n, so that we are in the situation of Equation 5:

Reduced decompositions and diagrams for types II and III
In this section, we find reduced decompositions for the maximal divisors w 0 of w that are of types II and III.They are listed in Equations 6 to 11.
We define a combinatorial technique that enables us to obtain a reduced decomposition, whose decomposition diagram ∆ 0 is the union of Coxeter diagrams of type A or D, or a proper Carter diagram of type D.
First, observe that each element w 0 defined in one of the Equations 6-11 is the product of three cycles.Since ℓ(w 0 ) = n− 1, by Proposition 3.6 each w 0 admits exactly one cycle with an even number of overlined elements.The other two cycles contain an odd number of overlined elements.Observing these equations, we recognise that these cycles contain exactly one overlined element at the end of the cycle.Assume that the cycles are x := (x 1 , x 2 , . . ., x p ), y := (y 1 , y 2 , . . ., y q ), z := (z 1 , z 2 , . . ., z r ) such that an even number of entries of (z 1 , z 2 , . . ., z r ) are overlined, x p = u, and y q = v.Also, observe that we always have p + q + r = n.
The combinatorial technique is based on Procedure 4.3.We formulate it in the following procedure.
Furthermore, we impose an additional condition: If one of the two cycles (x 1 , x 2 , . . ., x p ), (y 1 , y 2 , . . ., y q ) contains n, then Step 2 deals with the cycle that contains n.Proposition 4.11.Let w 0 be a divisor of w of type II or III.We continue with the notations introduced at the beginning of the section.Without loss of generality, assume that u < v.A reduced decomposition of w 0 is obtained as the product (u, v)(u, v) followed by the reflections used in Procedure 4.10 in reverse order.
Proof.After application of Procedure 4.10, the monomial matrix w 0 is transformed to the diagonal matrix with diagonal coefficients equal to 1 everywhere apart from diagonal positions [u, u] and [v, v], where the two coefficients are equal to −1.Multiplying this diagonal matrix by (u, v)(u, v), it is transformed to the identity matrix.A decomposition of w 0 is therefore the product of (u, v) by (u, v) followed by the reflections used in Procedure 4.10 in reverse order.
The decomposition is reduced if its length is equal to n − 1.In the first step of Procedure 4.10, the number of reflections that have been used is equal to r − 1, while p − 1 and q − 1 reflections are used in each of Steps 2 and 3.In addition, we multiplied at the end by two reflections: (u, v) and (u, v).Therefore, the number of reflections used in this decomposition is We explain Procedure 4.10 and Proposition 4.11 in the following two examples.The first example corresponds to type II and the second to type III.Example 4.12.We continue with our running Example 4.1, so n = 5, m = 2 and w = (2, 1) (5,4,3).Let w 0 = w(1, 2) = (1)(2) (5,4,3), whose cycle decomposition is given in Equation 6.
We apply Procedure 4.10 to w 0 .
Step 1.The even cycle is (2) and corrsponds to r = 1 in Procedure 4.10.
Step 1 does not apply and we move to Step 2.
The decomposition diagram associated to this reduced decomposition is a Coxeter diagram of type D 4 .This is readily checked by Lemma 3.5.The diagram is then the following.
We apply now Procedure 4.10.The cycle (3) contains only one element.So, we move to Step 2.
By Lemma 3.5, the decomposition diagram associated with this reduced decomposition is a proper Carter diagram of type D 5 .The diagram is the following.
• If p and q are equal to 1, then the diagram of the reduced decomposition is the disjoint union of a diagram of type A r−1 and two nodes.
• If p = 1 and q = 2 (or q = 1 and p = 2), then the diagram of the reduced decomposition is a disjoint union of a type A 3 and type A r−1 diagrams.
• If p = 1 and q > 2 (or q = 1 and p > 2), then the diagram of the reduced decomposition is a disjoint union of a Coxeter diagram of type D q+1 and a Coxeter diagram of type A r−1 (or a disjoint union of diagrams of types D p+1 and A r−1 , respectively).
In all these cases, the element w 0 is a Coxeter element in P w0 and therefore a parabolic Coxeter element in W .
Proof.By Lemma 4.14, xy as well as z are parabolic quasi-Coxeter elements in W .Here we prove that xy and z are Coxeter elements in P xy and P z , respectively.By Proposition 4.11 and Lemma 3.10 the cycle z is a Coxeter element of type A r−1 in P z .As xy and z are disjoint cycles, the two elements commute.
If p = q = 1, then w 0 equals (u, v)(u, v)z.As (u, v) and (u, v) commute, the first bullet of the proposition follows.
If p = 1 and q ≥ 2, then it is straightforward to check that the decomposition diagram of the reduced decomposition of xy given in Proposition 4.11 is a Coxeter diagram of type D q+1 .Thus, by Lemma 3.10 xy is a Coxeter element of type D q+1 in P xy .This yields the other two bullets, as • the decomposition of z ′ is related to its Carter diagram as described in Proposition 3.8, • the element z is a Coxeter element of type A r−1 in P z .
In particular, w 0 is a proper parabolic quasi-Coxeter element in W .
Proof.Since xy and z commute, we can apply Proposition 4.5 to z, and obtain the assertion for z, as well as Procedure 4.10 along with Proposition 4.11 to xy.The latter yields the decomposition whose decomposition diagram is the Carter diagram ∆ q−1,n with p + q vertices.In particular the decomposition diagram is connected, which yields that P xy is either of type A p+q or of type D p+q .By [11, Theorem A], ∆ q−1,n is not a Carter diagram in a group of type A p+q (as in the latter type Carter diagrams contain no cycles).Therefore P xy is of type D p+q .All the parabolic subgroups of type D n , n ≥ 4, are conjugate in W , and all the Coxeter elements in a finite Coxeter group are conjugate.As xy has a different cycle-type than the elements appearing in Proposition 4.15, it is not a Coxeter element in P xy .Thus, in this case xy is a proper quasi-Coxeter element in P xy .Therefore, w 0 is a proper parabolic quasi-Coxeter element in W .
By Proposition 4.10, a reduced decomposition of w(i, n) is Its decomposition diagram is described in Figure 2. 5 Decomposition of the reflections and their lifts

Interval groups and the claimed presentation
Let w be a quasi-Coxeter element in W of type D n .Consider the interval [1, w] of divisors of w for the absolute order given in Definition 2.7 and the interval group G( [1, w]) with its presentation given in Definition 2.9.We denote by bold symbols the elements in G( [1, w]).The copy [1, w] of the interval [1, w] contains copies of the reflections (i, j) and (i, j) for 1 ≤ i < j ≤ n, which we denote by (i, j) and (i, j).
By Proposition 2.10, the group G( [1, w]) is described by a presentation on the set with relations the dual braid relations.These relations are described as uv = vu if uv w and uv = vu, and as uv = vt = tu (t ∈ T ) if uv w and uvu = vuv for u, v ∈ T and u = v.
Theorem 5.1.The interval group G( [1, w]) is isomorphic to the group G defined by the presentation with generating set S and relations described by the diagram ∆ in Figure 3 together with the twisted cycle commutator relator associated with the cycle (s 1 , s m , s m+1 , s m+2 ), that is, ] .
We will always describe this relator by a curved arrow inside the corresponding cycle; see Figure 3.It is an easy exercise to check that if one of the twisted cycle commutator relators holds, then the three other relators also hold.
Remark 5.5.Suppose that m = 1, i.e. w is a Coxeter element.In this case, the group G is the Artin group of type D n .Our proof of Theorem 5.1 establishes a new proof of a result of Bessis showing that the interval group related to a Coxeter element in type D n is isomorphic to the related Artin group (see [6]).
We end the section by showing that the poset ([1, w], ) of a proper quasi-Coxeter element w in type D n is not a lattice.Hence the monoid defined by the same presentation as G( [1, w]) viewed as a monoid presentation fails to be a Garside monoid.Note that this fact does not mean that the group G([1, w]) does not admit Garside structures.Proof.We check using GAP that for w a proper quasi-Coxeter element in type D 4 , there exists a bowtie in ([1, w], ), hence it is not a lattice, see Proposition 1.10 in [23].(The bowtie consists of two reflections t 1 , t 2 that commute in W such that Let w be a proper quasi-Coxeter element in type D n for n > 4. Then it contains a subword w ′ of w whose Carter diagram is a 4-cycle.According to Theorem 2.1 in [16], all elements below w ′ in ([1, w], ) are in ([1, w ′ ], ) read as a poset in the rank 4 parabolic subgroup P w ′ .Therefore there is still the bowtie coming from the case n = 4 inside ([1, w], ) for n > 4.

Decomposition of the reflections on Carter generators
The purpose of this section is to find decompositions of the elements of T in terms of Carter generators.This corresponds to Step 3 in the strategy of our proof as explained in Section 2.5.
We recall that we fix an integer m with 1 ≤ m ≤ ⌊n/2⌋.We recall from Proposition 3.11 that a presentation of the Coxeter group W is defined on Carter generators s 1 , s 2 , . . ., s n together with the relations described in the diagram presentation illustrated in Figure 1 together with the quadratic relations.
The second column of Table 5 contains the 12 divisors (the w 0 's) of length 3 of the quasi-Coxeter element w of types I, II, and III, where we separate each type by two lines.We follow Section 4.1 in order to produce them.We also follow Sections 4.2 and 4.3 to produce reduced decompositions of these elements and their decomposition diagrams (the ∆ 0 's).The last column produces a Coxeter-like diagram related to ∆ 0 that we call the lift of ∆ 0 .It encodes the relations between the lift to the interval group of two reflections that appear in the reduced decomposition of each w 0 .Proposition 6.1.All the relations that describe the type A 3 diagrams on the last column of Table 5 are consequences of the relations described by the diagram presentation over s 1 , s 2 , s 3 , s 4 illustrated in Figure 3.
We prove the proposition by showing using kbmag [22] within GAP [17] that the relations appearing in the last column of Table 5 are consequences of the relations between s 1 , s 2 , s 3 , s 4 .For example, let us consider a diagram where some twisted cycle commutator relators appear.Consider the element Number 6 of the table.We have to show that (1, 2) commutes with s 4 , where ] that is a consequence of the relations of the claimed presentation.Now we can show that G( [1, w]) is isomorphic to G, that is Theorem 5.1 in the case where n = 4. Proposition 6.2.In the case n = 4, the groups G and G([1, w]) are isomorphic.
Proof.By transitivity of the Hurwitz action on the reduced decompositions over T of w, the group G( [1, w]) is generated by a copy of the set of reflections in T , and subject to the dual braid relations tt ′ = t ′ t ′′ that correspond to relations tt ′ = t ′ t ′′ in W where tt ′ = t ′ t ′′ w.
Consider the map f : G −→ G([1, w]) : s i −→ s i .By Lemma 5.12, the relations of the presentation of G hold in G( [1, w]).

Number Maximal divisor w
Table 5: Reduced decompositions and diagram lifts.Now consider the map g : G([1, w]) −→ G that maps each generator t of G([1, w]) to the expression for it over the generators s 1 , s 2 , s 3 , and s 4 that is given by Proposition 5.10.Let tt ′ , t ′ t ′′ be the two sides of a dual braid relation.Then there exists w 0 w of length 3 such that tt ′ = t ′ t ′′ w 0 .By [6], we know that the group G([1, w 0 ]) is isomorphic to the group defined by a presentation that we have described by Coxeter diagrams of type A 3 in the last column of Table 5. Hence we obtain that the dual braid relation tt ′ = t ′ t ′′ is a consequence of the relations of the corresponding diagram in the table.
In addition, we have already shown in Proposition 6.1 that the relations of these diagrams are consequences of the relations we have associated with the diagram ∆.Hence the map g is a homomorphism.
Clearly, the composition f • g is equal to id G([1,w]) and g • f equal to id G .Therefore, the groups G and G([1, w]) are isomorphic.
We prove that the interval group G( [1, w]) is isomorphic to the group G with five generators s 1 , s 2 , s 3 , s 4 , and s 5 corresponding to s 1 , s 2 , s 3 , s 4 , s 5 , with relations described by the diagram of Figure 3, where the curved arrow describes the twisted cycle commutator relator: tc(s 1 , s 2 , s 3 , s 4 ) = ].
Similarly to Table 4, we provide decompositions of t by applying Proposition 5.10 and we divide them according to the three types I, II, III of elements in T .

Number
Decomposition of t (type I) 1.
(1, 2) = s  Table 7 contains the same information as Table 5 in the case n = 4.We have 20 divisors of w of length 4 (the w 0 's) obtained by multiplying w from the right by (i, j) and (i, j) for 1 ≤ i < j ≤ 5.These divisors belong to types I, II, and III.We separate each type by 2 lines in the table.The third column produces the reduced decomposition from Sections 4.2 and 4.3.The last column describes the lift of the diagram ∆ 0 that encodes the relations between the lift to the interval group of two reflections that appear in the reduced decomposition of each w 0 .
We showed, using kbmag within GAP that all the relations described in the diagrams of the last column are consequences of the relations of the claimed presentation (see Theorem 5.1).The only cases that correspond to proper quasi-Coxeter elements are numbers 15, 17, and 19 of Table 7.Let w 0 be one of these elements.We know from Proposition 6.2 that G([1, w 0 ]) is isomorphic to the group defined by a presentation associated to the square diagram with the twisted cycle commutator relator.We conclude with the statement of the result for n = 5, whose proof we omit since it is similar to the proof of Proposition 6.2.Proposition 6.3.In the case n = 5, the groups G and G( [1, w]) are isomorphic.We want to show that g is a homomorphism.Consider a dual braid relation of G([1, w]), meaning a relation of the form tt ′ = t ′ t ′′ for t, t ′ , t ′′ ∈ T .It corresponds to the fact that tt ′ = t ′ t ′′ w.Then we have to prove that the relation g(t)g(t ′ ) = g(t ′ )g(t ′′ ) is a consequence of the relations of the presentation of G.As ([1, w], ) is a graded poset in which all the maximal flags have the same length, there are divisors w 0 of length n − 1 of w such that tt ′ w 0 .By Proposition 6.4, the braid relations and the twisted cycle commutator relator that correspond to the reduced decomposition and the diagram ∆ 0 for w 0 (produced in Sections 4.2 and 4.3) are a consequence of the G-relations (see Step 4 of our strategy).By induction, the relation g(t)g(t ′ ) = g(t ′ )g(t ′′ ) is a consequence of the braid relations and the twisted cycle commutator relator related to the reduced decomposition in the g-image of the lift of P w0 to G. Therefore, g is a homomorphism.
Clearly, the composition f • g is equal to id G([1,w]) and g • f is equal to id G .Therefore, the groups G and G( [1, w]) are isomorphic.
We call the set {s 1 , s 2 , . . ., s n } a Carter generating set.As w is a quasi-Coxeter element, every Carter generating set generates the Coxeter group.Note that the Carter diagram ∆ contains m−2 and n−m−2 vertices on the left-and right-hand sides of the single 4-cycle within ∆, respectively.For m = 1, the element w is a Coxeter element and ∆ 1,n is the Coxeter diagram of type D n , and w becomes (1)(n, n − 1, . . ., 2).We call a proper Carter diagram of type D n a diagram ∆ m,n that is not the Coxeter diagram of type D n , that is with m ≥ 2.
The next result is a consequence of Theorem 3.10 of Cameron-Seidel-Tsaranov[10].Proposition 3.11.The Coxeter group has a presentation with set of generators the Carter generators.The relations are s 2 i = 1 for 1 ≤ i ≤ n and the relations described by ∆ m,n together with the cycle commutator relation [s m , s m+1 s m+2 s 1 s m+2 s m+1 ] = (s m s m+1 s m+2 s 1 s m+2 s m+1 ) 2 = 1.

Proposition 4 . 5 .
The decomposition diagram of each reduced decomposition corresponding to Type I is represented by a Coxeter diagram of type A n−1 , and w 0 is a Coxeter element in P w0 .In particular, w 0 is a parabolic Coxeter element in W . Proof.It is an immediate consequence of Proposition 4.4 that the decomposition diagram of the reduced decomposition of w 0 produced in Procedure 4.3 is a string.Therefore, Lemmas 3.10 and 2.5 yield the assertion.

Lemma 4 . 14 .
3 and 4.10 are tailored in order to obtain a Coxeter diagram of type A or D, or a proper Carter diagram of type D. Observe first the following.We continue to use the notation of x, y and z introduced at the beginning of this section.The elements xy and z are parabolic quasi-Coxeter elements in W . Proof.By Proposition 4.11, we have ℓ T (z) + ℓ T (w 0 z −1 ) = ℓ T (w 0 ) which yields z, xy w 0 w.Therefore z as well as xy are parabolic quasi-Coxeter elements in W , see Corollary 6.11 in [3].

Proposition 4 . 16 .
Suppose p, q ≥ 2. Then the decomposition diagram of w 0 is a disjoint union of a proper Carter diagram of type D p+q and a Coxeter diagram of type A r−1 .Further • the element z ′ := xy is a proper quasi-Coxeter element of type D p+q in P z ′ ,

Figure 2 :
Figure 2: Decomposition diagram in the situation of Equation 11.

Proposition 5 . 6 .
Let w be a proper quasi-Coxeter element in type D n for n ≥ 4. The poset ([1, w], ) is not a lattice.

1 .
The commuting relation between (1, 2) and s 4 is precisely the twisted cycle commutator relator [s 4 , s

Table 6 :
Decompositions of t in the case n = 5.

Table 7 :
Reduced decompositions and diagram lifts.