1 Introduction
This paper is concerned with the so-called dual approach to Coxeter groups. A dual Coxeter system is a Coxeter group together with a generating set consisting of all the reflections in the group, that is, the set of conjugates of all the elements of a simple system.
There are several motivations for studying dual Coxeter systems. They were introduced by Bessis [3] and independently by Brady and Watt [8, 10]. The dual Coxeter systems are crucial in the theory of dual braid monoids, which are alternative braid monoids embedding in the Artin–Tits group attached to a finite Coxeter group (that is, a spherical Artin–Tits group) and providing an alternative Garside structure of it.
The latter allows a new presentation of the group and thereby for instance to get better solutions to the word problem in the spherical Artin–Tits groups (see [5, 3]). Each dual braid monoid depends on a choice of a Coxeter element and its poset of simple elements ordered by left-divisibility is isomorphic to the generalized noncrossing partition lattice with respect to that Coxeter element (see [3]).
Bessis [4] later generalized these notions to complex reflections groups and their braid groups where the dual approach is natural. Indeed, there is in general no canonical choice of a simple system for a complex reflection group.
Having replaced the set of simple generators of a Coxeter group by the whole set of reflections
in the Coxeter group (and in the Artin–Tits group), one needs to find new sets of relations between these new generators that define the respective groups. The idea is to take the so-called dual braid relations [3]. Unlike the classical braid relations, a dual braid relation can involve three generators and has the form ab=ca (or ba=ac), where a,b and c are reflections.
In the classical case Matsumoto’s Lemma [30] allows one to pass from any reduced decomposition of an element to any other one by successive applications of braid relations. The same question can be asked for reduced decompositions with respect to the new set of generators, and can be studied using the so-called Hurwitz action on reduced decompositions.
Let us say a bit more on this action. Let G be an arbitrary group, n≥2. There is an action of the braid group ℬn on n strands on Gn where the standard generator σi∈ℬn which exchanges the i-th and (i+1)-th strands acts as
σi⋅(g1,…,gn):=(g1,…,gi-1,gigi+1gi-1,gi,gi+2,…,gn).
Notice that the product of the entries stays unchanged and that all the tuples in a given orbit generate the same subgroup of the Coxeter group. This action is called the Hurwitz action since it was first studied by Hurwitz in 1891 ([23]) in the case where G=𝔖n.
Two elements g,h∈Gn are called Hurwitz equivalent if there is a braid β∈ℬn such that β⋅g=h. It has been shown by Liberman and Teicher (see [29]) that the question of whether two elements in Gn are Hurwitz equivalent or not is undecidable in general. Nevertheless, there are results in many cases (see for instance [19] and [35]).
The Hurwitz action also plays a role in algebraic geometry, more precisely in the braid monodromy of a projective curve (e.g., see [28, 11] or [25]).
In the case of finite Coxeter groups, the Hurwitz action can be restricted to the set of minimal length decompositions of a given fixed element w into products of reflections. Given a reduced decomposition (t1,…,tk) of w where t1,…,tk are reflections (that is, w=t1⋯tk with k minimal), the generator σi∈ℬn then acts as
σi⋅(t1,…,tk):=(t1,…,ti-1,titi+1ti,ti,ti+2,…,tk).
The right-hand side is again a reduced decomposition of w. In fact, we see that the braid group generator σi acts on the i-th and (i+1)-th entries by replacing (ti,ti+1) by (titi+1ti,ti) which corresponds exactly to a dual braid relation. Hence determining whether one can pass from any reduced decomposition of an element to any other just by applying a sequence of dual braid relations is equivalent to determining whether the Hurwitz action on the set of reduced decompositions of the element is transitive.
The transitivity of the Hurwitz action on the set of reduced decompositions has long been known to be true for a family of elements commonly called parabolic Coxeter elements (note that there are several inequivalent definitions of these in the literature). For more on the topic we refer to [2], and the references therein, where a simple proof of the transitivity of the Hurwitz action was shown for (suitably defined) parabolic Coxeter elements in a (not necessarily finite) Coxeter group. See also [24]. The Hurwitz action in Coxeter groups has also been studied outside the context of parabolic Coxeter elements (see [37, 22, 31] – be aware that there are mistakes in [22]).
The aim of this paper is to provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions. We call an element of a Coxeter group a parabolic quasi-Coxeter element if it admits a reduced decomposition which generates a parabolic subgroup. A quasi-Coxeter element is an element admitting a reduced decomposition which generates the whole Coxeter group.
The main result of this paper is the following characterization.
Theorem 1.1Let (W,T) be a finite, dual Coxeter system of rank n and
let w∈W.
The Hurwitz action on RedT(w) is transitive if and only if w is a parabolic quasi-Coxeter element for (W,T).
The following theorem is an immediate consequence.
Theorem 1.2Let (W,T) be a finite, dual Coxeter system of rank n and
let w∈W. If w is a quasi-Coxeter element for (W,T), then for each(t1,…,tn)∈RedT(w) one has
W=〈t1,…,tn〉.
The proof that being a parabolic quasi-Coxeter element is a necessary condition to ensure the transitivity of the Hurwitz action is uniform.
The other direction is case-by-case. In the simply laced types we first prove Theorem 1.2
(see Theorem 6.1) and use it to prove the second direction of Theorem 1.1.
Another consequence of Theorem 1.1 and of work of Dyer [16, Theorem 3.3] is the following.
Corollary 1.3Let (W,T) be a dual Coxeter system and let w∈W. Further, let
(t1,…,tm)∈RedT(w), set W′:=〈t1,…,tm〉
and T′:=W′∩T. If W′ is finite, then the Hurwitz action on RedT′(w) is transitive.
It is an open question whether this statement remains true if there is no finiteness assumption on W′.
The structure of the paper is as follows.
We adopt the approach and terminology introduced in [2] (see Section 2).
In particular, we use an unusual definition of parabolic subgroup, which we show in
Section 4 (after recalling some well-known facts on root systems in Section 3) to be equivalent to the classical
definition for finite Coxeter groups and (as a consequence of results in [18]) for a large family of irreducible infinite Coxeter groups,
the so-called irreducible 2-spherical Coxeter groups. As a byproduct, we obtain some results on parabolic subgroups of finite Coxeter groups.
In particular, we show the following:
Proposition 1.4Let (W,S) be a finite Coxeter system and S′⊆T such that (W,S′) is a simple system. Then the parabolic subgroups with respect to S
coincide with those with respect to S′.
Given a root subsystem Φ′ of a given
root system Φ, we discuss in Section 5 the relationship between the corresponding Coxeter groups and
root lattices, especially in the simply laced types. These results are needed later in Section 6.
It is known for the Coxeter groups of type An that all the elements are parabolic Coxeter elements in the sense of [2].
For types Bn and I2(m), the sets of parabolic Coxeter elements and parabolic quasi-Coxeter elements coincide as it is shown in Section 6.
In particular, Theorem 1.1 is true for An,Bn and I2(m) as a consequence of [2]. For the other types it is in general
false that parabolic quasi-Coxeter elements coincide with parabolic Coxeter elements. In Section 6 we also show:
Theorem 1.5Let w be a quasi-Coxeter element in a finite dual Coxeter system (W,T) of rank n and (t1,…,tn)∈RedT(w) such that
W=〈t1,…,tn〉. Then the reflection subgroup W′:=〈t1,…,tn-1〉 is parabolic.
The proof of this theorem is uniform for the simply laced types, but case-by-case for the other ones. As a corollary we obtain a new characterization of
the maximal parabolic subgroups of a finite dual Coxeter system (see Corollary 6.10). Moreover, it follows from
Theorem 1.5 that an element is a parabolic quasi-Coxeter element if and only if it is a prefix of a quasi-Coxeter element
(see Corollary 6.11).
Theorem 1.5 allows us to argue by induction to prove the main theorem. For type Dn we need to show that every two maximal
parabolic subgroups intersect non-trivially provided that n≥6 (see Section 7) which then allows us to conclude Theorem 1.1
by induction. For the types E6,E7 and E8 we first check by computer that every reflection occurs in the Hurwitz orbit of every reduced decomposition
of a quasi-Coxeter element, and then argue by induction. Theorem 1.1 is verified for types F4, H3 and H4 by computer. All this is
done in Section 8.
2 Dual Coxeter systems and Hurwitz action
2.1 Dual Coxeter systems
Let (W,T) be a dual Coxeter system of finite rank n in the sense
of [3]. This is to say that there is a subset S⊆T with |S|=n such that (W,S) is a (not necessarily finite) Coxeter system, and T={wsw-1∣w∈W,s∈S} is the set of reflections for the Coxeter system (W,S) (unlike Bessis, we specify no Coxeter element). We call (W,S) a simple system for (W,T) and S a set of simple reflections. If S′⊆T is such that (W,S′) is a Coxeter system, then {wsw-1∣w∈W,s∈S′}=T (see [9, Lemma 3.7]). Hence a set S′⊆T is a simple system for (W,T) if and only if (W,S′) is a Coxeter system. The rank of (W,T) is defined as |S| for a simple system S. This is well-defined by [9, Theorem 3.8]. It is equal to the rank of the corresponding root system.
Simple systems for (W,T) have been studied by several authors (see [18]). Clearly, if S is a simple system for (W,T), then so is wSw-1 for
any w∈W. Moreover, it is shown in [18] that for an important class of infinite Coxeter groups including the irreducible affine Coxeter groups, all simple
systems for (W,T) are conjugate to one another in this sense.
The following result is well known and follows from [16].
Proposition 2.1Let (W,S) be a (not necessarily finite) Coxeter system of rank n. Then W cannot be generated by less than n reflections.
Proof.
Assume that W=〈t1,…,tk〉, with k≤n. Following [16], write
χ(W)={t∈T∣{u∈T∣ℓ(ut)<ℓ(t)}={t}}
for the set of canonical Coxeter generators of W where ℓ is the length function in (W,S). It follows from [16, Corollary 3.1 (i)] that |χ(W)|≤k. But since W is of rank n the set χ(W) contains S hence we have |χ(W)|≥n, which concludes the proof.
∎
A reflection subgroupW′ is a subgroup of W generated by reflections.
It is well known that (W′,W′∩T) is again a dual Coxeter system (see [16]).
For w∈W, a reduced T-decomposition of w is a shortest length decomposition of w into reflections, and we denote by RedT(w) the set
of all such reduced T-decompositions. When the context is clear, we drop the T and elements of RedT(w) are referred to as reduced decompositions of w. The length of any element in RedT(w) is called the reflection length (or absolute length) of w and we denote it by ℓT(w). The absolute length function ℓT:W→ℤ≥0 can be used to define the absolute order≤T on W as follows. For u,v∈W:
u≤Tv if and only if ℓT(u)+ℓT(u-1v)=ℓT(v).
The reflection subgroup generated by {s1,…,sm} is called a parabolic subgroup for (W,T) if there is a simple system S={s1,…,sn} for (W,T)
with m≤n. Notice that this differs from the usual notion of a parabolic subgroup generated by a conjugate of a subset of a fixed simple system S (see [21, Section 1.10]). However we prove in Section 4 the equivalence of the definitions for finite Coxeter groups.
2.2 Coxeter and quasi-Coxeter elements
We now define (parabolic) Coxeter elements and (parabolic) quasi-Coxeter elements. The second item of the definition below is borrowed from [2]. The third one is a generalization of Voigt’s original definition in [37], see also Remark 2.8.
Definition 2.2Let (W,T) be a dual Coxeter system and S={s1,…,sn} be a simple system for (W,T).
We say that c∈W is a classical Coxeter element if c is conjugate to some sπ(1)⋯sπ(n) for π a permutation of the symmetric group 𝔖n. An element w∈W is a classical parabolic Coxeter element if w≤Tc for some classical Coxeter element c.
An element c∈W is called a Coxeter element if there exists a simple system S′={s1′,…,sn′} for (W,T) such that c=s1′⋯sn′. An element w∈W is a parabolic Coxeter element if there exists a simple system S′={s1′,…,sn′} for (W,T) such that w=s1′⋯sm′ for some m≤n.
An element w∈W is called a quasi-Coxeter element for (W,T) if there exists (t1,…,tn)∈RedT(w) such that W=〈t1,…,tn〉. An element w∈W is a parabolic quasi-Coxeter element for (W,T)
if there is a simple system S′={s1′,…,sn′} and (t1,…,tm)∈RedT(w) such that
〈t1,…,tm〉=〈s1′,…,sm′〉
for some m≤n.
Remark 2.3Let us point out a few facts about these definitions.
An element of the form sπ(1)⋯sπ(n) obtained as product of elements of a fixed simple system in some order as in Definition 2.2 (a) is usually called a standard Coxeter element. In the case where the Dynkin diagram of the Coxeter system is a tree, every two standard Coxeter elements are conjugate to each other by a sequence of cyclic conjugations (see [7, V, 6.1, Lemme 1]). Hence the set of classical Coxeter elements forms in that case a single conjugacy class. In particular, this holds for all the finite Coxeter groups.
If the Coxeter group is finite, then an element w∈W is a parabolic Coxeter element if and only if w≤Tc for some Coxeter element c (see [14, Corollary 3.6]).
It is clear that a classical Coxeter element is a Coxeter element and hence it follows from Remark 2.3 (b) that a classical parabolic Coxeter element is a parabolic Coxeter element. The difference between classical Coxeter elements and Coxeter elements is somewhat subtle. For finite Weyl groups (see
Section 3 for the definition), the two definitions are equivalent, as a consequence of [34, Theorem 1.8 (ii) and Remark 1.10]. It seems however that no case-free proof of this fact is known. An example where these two definitions differ is the dihedral group I2(5) (see [2, Remark 1.1]).
We will show in Corollary 6.11 the same statement as the one given in Remark 2.3 (b) but for parabolic quasi-Coxeter elements, namely that w∈W is a parabolic quasi-Coxeter element if and only if there exists a quasi-Coxeter element w′∈W such that w≤Tw′.
Example 2.4In type D4 with simple system {s0,s1,s2,s3} where s2 does not commute with any other simple reflection, the element
c:=s1(s2s1s2)(s2s0s2)s3
is a quasi-Coxeter element. It has a reduced decomposition generating the whole group since if we write
(t1,…,t4)=(s1,s2s1s2,s2s0s2,s3)
we have that t1t2t1=s2 and s2t3s2=s0. Using the permutation model for a group of type D4 (see Section 7), it can be shown that there is no reduced decomposition of this element yielding a simple system for the group. By computer we checked that the poset {w∈W∣w≤Tc} has 54 elements and is not a lattice. There is a single conjugacy class of quasi-Coxeter elements which are not Coxeter elements in that case.
Therefore we can not define a new Garside structure on the Artin–Tits group of type D4 by replacing the Coxeter element by a quasi-Coxeter element.
2.3 Rigidity
Finite Coxeter groups are reflection rigid, that is, if (W,S) and (W,S′) are simple systems for (W,T), then both systems determine the same diagram (see [9, Theorem 3.10]). We define (W,T) to be irreducible if (W,S) is irreducible and its type to be the type of (W,S) for some (equivalently each) simple system S⊆T. In most cases, the type is determined by the group itself. There are only two exceptions, namely
WB2k+1≅WA1×WD2k+1(k≥1),
WI2(4k+2)≅WA1×WI2(2k+1)(k≥1),
which follows from the classification of the finite irreducible Coxeter groups and [32, Theorem 2.17, Lemma 2.18 and Theorem 3.3].
We say that a reflection group is strongly reflection rigid if whenever
(W,S) and (W,S′) are simple systems for (W,T), then S and S′ are conjugate in W.
Notice that (W,T) is strongly reflection rigid if and only if every Coxeter element is a classical Coxeter element. Hence in particular let us point out that Remark 2.3 (c) implies:
Remark 2.5The finite Weyl groups are strongly reflection rigid.
2.4 Hurwitz action on reduced decompositions
The braid group on n strands denoted ℬn is the group with generators
σ1,…,σn-1 subject to the relations
σiσj=σjσifor |i-j|>1,
σiσi+1σi=σi+1σiσi+1for i=1,…,n-2.
It acts on the set Tn of n-tuples of reflections as
σi⋅(t1,…,tn)=(t1,…,ti-1,titi+1ti,ti,ti+2,…,tn),
σi-1⋅(t1,…,tn)=(t1,…,ti-1,ti+1,ti+1titi+1,ti+2,…,tn).
We call this action of ℬn on Tn the Hurwitz action and an orbit of this action an Hurwitz orbit.
Lemma 2.6Lemma 2.6 ([2, Lemma 1.2])
Let W′ be a reflection subgroup of W and let T′=T∩W′ be the
set of reflections in W′.
For an element w∈W′ such that ℓT′(w)=n, the braid group on n strands acts on RedT′(w).
We now give the main result of [2].
Theorem 2.7Let (W,T) be a dual Coxeter system of finite rank n and
let c=s1⋯sm be a parabolic Coxeter element in W.
The Hurwitz action on RedT(c) is transitive.
In symbols, for each (t1,…,tm)∈Tm such that
c=t1⋯tm, there is a braid β∈Bm such that
β⋅(t1,…,tm)=(s1,…,sm).
Hence in the case where W is finite, Theorem 1.1 generalizes Theorem 2.7.
Remark 2.8For the case where (W,T) is simply laced and w is a quasi-Coxeter element in W, Voigt observed in his thesis [37] that the Hurwitz action on RedT(w) is transitive. His definition of a quasi-Coxeter element is slightly different. Let Φ be the root system associated to W (see Sections 3 and 5 for definitions and notations on root systems and lattices), then Voigt defined w=sα1⋯sαn∈W to be quasi-Coxeter if spanℤ(α1,…,αn) is equal to the root lattice of Φ. The connection with our definition will be explained in Section 5.
3 Root systems and geometric representation
In this section we fix some notation, and for the convenience of the reader we recall some facts on root systems and the geometric representation of a Coxeter group as can be found for example in [21] or [7].
Let V be a finite-dimensional Euclidean vector space with
positive definite symmetric bilinear form (-∣-). For 0≠α∈V, let
sα:V→V be the reflection in the hyperplane orthogonal to α, that is, the map defined by
v↦v-2(v∣α)(α∣α)α.
Then sα is an involution and sα∈O(V), the orthogonal group of V with respect to (-∣-).
Definition 3.1A finite subset Φ⊆V of nonzero vectors is called root system in V if
The root system is called crystallographic if in addition
〈β,α〉:=2(β∣α)(α∣α)∈ℤ for all α,β∈Φ.
The rankrk(Φ) of Φ is the dimension of V. The group WΦ=〈sα∣α∈Φ〉 associated to the root system Φ is a Coxeter group. In the case where the root system is crystallographic, WΦ is a (finite) Weyl group. A subset Φ′⊆Φ is called a root subsystem if Φ′ is a root system in spanℝ(Φ′).
Conversely, to any finite Coxeter group one can associate a root system. For an infinite Coxeter system (W,S), one can still associate a set of vectors (again called a root system) with slightly relaxed conditions (see for instance [21, Section 5.3–5.7]). In the following, when dealing with root systems it will always be in the sense of the above definition unless otherwise specified since this paper primarily concerns finite Coxeter groups. When results generalize to arbitrary Coxeter groups, we will mention that we work with the generalized root systems.
Let Φ≠∅ be a root system. Then Φ is reducible if Φ=∪˙Φ1Φ2 where Φ1, Φ2 are nonempty root systems such that (α∣β)=0 whenever α∈Φ1, β∈Φ2. Otherwise Φ is irreducible. For an irreducible crystallographic root system Φ, the set
{(α∣α)∣α∈Φ} has at most two elements (see [21, Section 2.9]). If this set has only one element, up to rescaling we can assume it to be 2 and call Φ simply laced. It follows from the classification of irreducible root systems that simply laced root systems are crystallographic. Simply laced root systems have types An (n≥1), Dn (n≥4) or En (n∈{6,7,8}) in the classification. We will sometimes use the notation WXn for the Coxeter group with corresponding root system of type Xn for convenience. For more on the topic we refer the reader to [21].
4 Equivalent definitions of parabolic subgroups
In this section, we show one direction of Theorem 1.1 with a case-free argument. To this end, we show Proposition 1.4, that is, we show that for finite Coxeter groups, the definition of parabolic subgroups given in Section 2 coincides with the usual one. We also mention that they coincide for a large class of infinite Coxeter groups.
Let (W,S) be a finite Coxeter system with root system Φ and V:=spanℝ(Φ).
We say that a subgroup generated by a conjugate of a subset of S is parabolic in the classical sense. It is well known that these are exactly the subgroups of the form
CW(E):={w∈W∣w(v)=vfor all v∈E}
where E⊆V is any set of vectors (see for instance [26, Section 5-2]).
Definition 4.1Given a subset 𝒜⊆W, the parabolic closureP𝒜 of 𝒜 is the intersection of all the parabolic subgroups in the classical sense containing 𝒜. It is again a parabolic subgroup in the classical sense (see [36, 12.2–12.5] or [33]).
We denote by Fix(𝒜) the subspace of vectors in V which are fixed by every element of 𝒜. If 𝒜={w}, then we simply write Fix(w) for Fix(𝒜)=ker(w-1) and Pw for P𝒜. For convenience we also set Mov(w):=im(w-1). Note that V=Fix(w)⊕Mov(w) (see [1, Definition 2.4.6]). It follows from the above description that P𝒜=CW(Fix(𝒜)).
In this section we give a case-free proof that for finite Coxeter groups, the parabolic subgroups as defined in Section 2.1 coincide with the parabolic subgroups in the classical sense. As a consequence we are able to show one direction of Theorem 1.1. We first recall the following result
Lemma 4.2Lemma 4.2 ([3, Lemma 1.2.1 (i)], [1, Theorem 2.4.7] after [12])
Let w∈W, t∈T. Then
Fix(w)⊆Fix(t)if and only if t≤Tw.
Notice that Lemma 4.2 implies [2, Theorem 1.4]
if W is finite.
Proposition 4.3Let (W,S) be a finite Coxeter system, T=⋃w∈WwSw-1 and w∈W. If the Hurwitz action on RedT(w) is transitive, then the subgroup generated by the reflections in any reduced decomposition of w is equal to Pw.
Proof.
We prove the contrapositive of the statement. Let (t1,…,tm)∈RedT(w) and assume that W′:=〈t1,…,tm〉 is not equal to Pw. Since ti≤Tw for each i, we have ti∈Pw for all i=1,…,m by Lemma 4.2. It follows that W′⊊Pw. Since both W′ and Pw are reflection subgroups of W, there exists a reflection t∈Pw with t∉W′. It follows that Fix(w)⊆Fix(t), hence also that t≤Tw by Lemma 4.2. In particular, there exists (q1,…,qm)∈RedT(w) with q1=t. Since the Hurwitz orbit of (t1,…,tm) remains in W′ and t∉W′, the Hurwitz action on RedT(w) can therefore not be transitive.
∎
Corollary 4.4Let (W,S) be a finite Coxeter system and T=⋃w∈WwSw-1. A subgroup P⊆W is parabolic if and only if it is parabolic in the classical sense. In particular, if S′⊆T is such that (W,S′) is a simple system, then the parabolic subgroups in the classical sense defined by S coincide with those defined by S′.
Proof.
If P is parabolic, then P=〈s1,…,sm〉 where {s1,…,sn}=S′⊆T is a simple system for W and m≤n. By [2, Theorem 1.3], the Hurwitz action on RedT(w) where w=s1s2⋯sm is transitive. By Proposition 4.3, it follows that P is parabolic in the classical sense.
Conversely, if P is parabolic in the classical sense, then P is generated by a conjugate of a subset of S, and a conjugate of S is again a simple system for W. Hence P is parabolic.
∎
As a corollary we get a proof of one direction of Theorem 1.1:
Corollary 4.5Let (W,T) be a finite, dual Coxeter system of rank n and
let w∈W.
If the Hurwitz action on RedT(w) is transitive, then w is a parabolic quasi-Coxeter element for (W,T).
Proof.
Let w=t1⋯tm∈RedT(w). By Proposition 4.3, W′:=〈t1,…,tm〉 is parabolic in the classical sense. By Corollary 4.4, it follows that W′ is parabolic and hence w is a parabolic quasi-Coxeter element.
∎
If (W,S) is of type A~1, i.e. if W is a dihedral group of infinite order, then the non-trivial parabolic subgroups are precisely those subgroups that are generated by a reflection. Therefore parabolic subgroups in the classical sense coincide with parabolic subgroups in this case.
As an immediate consequence of a theorem of Franzsen, Howlett and Mühlherr [18] we also get the equivalence of the definitions for a large family of infinite Coxeter groups (including the irreducible affine Coxeter groups):
Proposition 4.6Let (W,S) be an infinite irreducible 2-spherical Coxeter system, that is, S is finite, and ss′ has finite order for every s,s′∈S. Then a subgroup of W is parabolic if and only if it is parabolic in the classical sense.
Proof.
Under these assumptions, if (W,S′) is a Coxeter system (without assuming S′⊂T), it follows from [18, Theorem 1 b)] that there exists w∈W such that S′=wSw-1, hence any parabolic subgroup is a parabolic subgroup in the classical sense.
∎
Question 4.7Do parabolic subgroups always coincide with parabolic subgroups in the classical sense?
5 Root lattices
In this section, we study root lattices and their sublattices. The results will be needed for the better understanding of quasi-Coxeter elements in the simply laced types. Parts of this section have been inspired by [37].
Definition 5.1Let V be a Euclidean vector space with symmetric bilinear form (-∣-). A latticeL in V is the integral span of a basis of V. The lattice L is called integral if (α∣β)∈ℤ for all α,β∈L and
even if (α∣α)=2 for all basis elements α.
For a set of vectors Φ⊆V we set L(Φ):=spanℤ(Φ).
Remark 5.2If Φ is a root system, then L(Φ) is a lattice, called the
root lattice. If Φ is a crystallographic root system, then L(Φ) is an integral lattice.
Proposition 5.3For an even lattice L the set
Φ(L):={α∈L∣(α∣α)=2}
is a simply laced, crystallographic root system in spanR(L).
Proof.
The set Φ(L) is contained in the ball around 0 with radius 2, therefore bounded, thus finite. The rest of the proof is straightforward.
∎
Definition 5.4Let Φ be a crystallographic root system in V. The weight latticeP(Φ) of Φ is defined by
P(Φ):={x∈V∣〈x,α〉∈ℤ for all α∈Φ}.
By [7, VI, 1.9] it is again a lattice containing
L(Φ) and the group P(Φ)/L(Φ) is finite. We call its order the connection index of Φ and denote it by i(Φ).
Note that if Φ is simply laced, the weight lattice is equal to the dual root lattice, namely,
P(Φ)=L*(Φ):={x∈V∣(x∣y)∈ℤ for all y∈L(Φ)}.
Proposition 5.5Let Φ be a simply laced root system and let C be the Cartan matrix of Φ. Then
i(Φ)=det(C).
Proof.
Let Δ={α1,…,αm}⊆Φ be a basis of the root system Φ. Then Δ is a basis of L(Φ). Denote by M the Gram matrix of L(Φ) with respect to Δ. By general lattice theory (see
for instance [17, Section 1.1]) one has
|L*(Φ):L(Φ)|=det(M).
Since Φ is simply laced, we have L*(Φ)=P(Φ) and hence
i(Φ)=|L*(Φ):L(Φ)|.
Again since Φ is simply laced, we have C=M, which concludes the proof.
∎
We list i(Φ) for the irreducible, simply laced root systems. These can be found in [7, Planches I, IV , V, VI, VII].
| Type of Φ | An | Dn | E6 | E7 | E8 |
|---|
| i(Φ) | n+1 | 4 | 3 | 2 | 1 |
As a consequence we obtain the following result.
Proposition 5.6Let Φ be an irreducible, simply laced root system. Then Φ is determined by the pair (rk(Φ),i(Φ)).
The following lemma seems to be folklore, but we could not find a proof in the literature, hence we state it here.
Lemma 5.7Let Φ be a simply laced root system. Then the root lattice determines the
root system, that is,
Φ(L(Φ))=Φ.
Proof.
By the previous proposition, we have to show that the rank and connection indices of Φ and Φ(L(Φ)) coincide. Since
Φ⊆Φ(L(Φ)), we have
rk(Φ)≤rk(Φ(L(Φ))).
On the other hand, the rank of Φ(L(Φ)) is bounded above by the dimension
of the ambient vector space which equals rk(Φ).
In the proof of Proposition 5.5, we used the fact that the Cartan matrix of a root system and the Gram matrix
of the corresponding root lattice coincide. Denote by CΦ the Cartan matrix with respect to Φ. Then
det(CΦ)=vol(L(Φ))=vol(L(Φ(L(Φ))))=det(CΦ(L(Φ))),
which yields i(Φ)=i(Φ(L(Φ))) by Proposition 5.5.
∎
Lemma 5.8Let Φ be as in Lemma 5.7 and let Φ′⊆Φ be a root subsystem. Then
L(Φ′)∩Φ=Φ′.
Proof.
We have the equalities
L(Φ′)∩Φ=L(Φ′)∩{α∈L(Φ)∣(α∣α)=2} by Lemma 5.7
={α∈L(Φ′)∣(α∣α)=2}
=Φ′ by Lemma 5.7.∎
Notation 5.9Let Φ be a finite root system and R⊆Φ a set of roots. We denote by WR the group 〈sr|r∈R〉.
Note that this is consistent with the notation introduced in Definition 3.1 in the case where R=Φ.
Proposition 5.10Let Φ be a crystallographic root system, let
Φ′⊆Φ be a root subsystem, and let R:={β1,…,βk}⊆Φ′ be a set of roots. The following statements are equivalent:
The root subsystem Φ′ is the smallest root subsystem of
Φ
containing R (i.e., the intersection of all root subsystems containing R).
Moreover, if any of the above equivalent conditions is satisfied, then
Proof.
Obviously, WR(R) is a root system with R⊆WR(R). Thus if (a) holds, then Φ′⊆WR(R).
As WR(R)⊆WΦ′(Φ′)=Φ′, it follows that (a) implies (b). The converse direction follows from the definition of a root subsystem.
Statement (b) implies that R⊆Φ′ and therefore we have WR⊆WΦ′.
We show that in this case WΦ′⊆WR. To this end, let α∈Φ′=WR(R). Then
α=w(βi) for some w∈WR and 1≤i≤k. Then
sα=sw(βi)=wsβiw-1∈WR,
which shows the claim. Thus (b) implies (c).
Assume (c) and let Φ′′ be the smallest root subsystem of Φ containing R. Then WΦ′=WR⊆WΦ′′. By the definition of Φ′′ we have
Φ′′⊆Φ′. If Φ′′⊊Φ′, then WΦ′′⊊WΦ, a contradiction. Hence Φ′′=Φ′, which shows (a).
It remains to show that (c) implies (d).
So assume (c) and let ti:=sβi, 1≤i≤k. Let TΦ′ be the set of reflections in WΦ′. By [16, Corollary 3.11 (ii)], we have
TΦ′={wtiw-1∣1≤i≤k,w∈WΦ′}. In particular, any root in Φ′ has the form w(βi) for some w∈WΦ′, 1≤i≤k.
Since WΦ′=〈t1,…,tk〉, we can write w=ti1⋯tim with 1≤ij≤k for each 1≤j≤m. Since Φ′ is
crystallographic, it follows that w(βi)=ti1⋯tim(βi) is an integral linear combination of the roots βj, hence that Φ′⊆L(R).
Since R⊆Φ′, we get L(R)=L(Φ′), which proves (d).
∎
Remark 5.11Notice that condition (d) in Proposition 5.10 is in general not equivalent to conditions (a)–(c). For example, if Φ is of type B2,
then one can choose two orthogonal roots α and β generating a proper root subsystem of type A1×A1 while one has L({α,β})=L(Φ).
Nevertheless, one has the equivalence for simply laced root systems:
Lemma 5.12Let Φ,Φ′,R be as in Proposition 5.10 and assume in addition that Φ is simply laced. Then condition (d) in Proposition 5.10
is equivalent to any of conditions (a)–(c).
Proof.
Since conditions (a)–(c) are equivalent, it suffices to show that condition (d) implies (a). Assume that there exists a root subsystem Φ′′ of Φ with
Φ′′⊆Φ′ and R⊆Φ′′. It follows that
L(Φ′)=L(R)⊆L(Φ′′). One then has
Φ′=L(Φ′)∩Φ⊆L(Φ′′)∩Φ=Φ′′
by Lemma 5.8.
Hence Φ′=Φ′′, which concludes the proof.
∎
The following two results are useful tools for the next section. The following proposition and its proof are borrowed from [37, Proposition 1.5.2].
Proposition 5.13Let Φ be a simply laced root system and let Φ′⊆Φ be a root subsystem with rk(Φ′)=m=rk(Φ). Then:
|L(Φ):L(Φ′)|=|P(Φ′):P(Φ)|,
|L(Φ):L(Φ′)|=i(Φ′)⋅i(Φ)-1.
Proof.
Since Φ and Φ′ have the same rank, there is a lattice isomorphism
ℒ:L(Φ)→∼L(Φ′),
hence |L(Φ):L(Φ′)|=|det(ℒ)|
(see [17, Section 1.1]). Furthermore, we have
P(Φ′)={x∈V∣(x∣y)∈ℤ for all y∈L(Φ′)}
={x∈V∣(x∣ℒ(v))∈ℤ for all v∈L(Φ)}
={x∈V∣(ℒt(x)∣v)∈ℤ for all v∈L(Φ)}
={x∈V∣ℒt(x)∈P(Φ)}
=(ℒt)-1(P(Φ)).
Thus [P(Φ′):P(Φ)]=|det(ℒt)|=|det(ℒ)|, which shows (a).
We have
L(Φ′)⊆L(Φ)⊆P(Φ)⊆P(Φ′).
It follows that
|L(Φ):L(Φ′)||P(Φ):L(Φ)|⏟=i(Φ)|P(Φ′):P(Φ)|⏟=|L(Φ):L(Φ′)|=|P(Φ′):L(Φ′)|⏟=i(Φ′),
which concludes the proof.
∎
For a simply laced root system Φ we extend the definition of connection index to subsets of Φ. For an arbitrary subset R⊆Φ we define i(R):=|L*(R):L(R)|.
Note that i(R) is well-defined by
Proposition 5.10, because i(R)=i(Φ′), where Φ′ is the smallest root subsystem of Φ
in spanℝ(R) containing R.
The following theorem is part of the Diploma thesis of Kluitmann [27].
Theorem 5.14Let Φ be a simply laced root system. Further, let w∈WΦ and let (sα1,…,sαk), (sβ1,…,sβk)∈RedT(w), where
αi,βi∈Φ, for 1≤i≤k.
Then for R:={α1,…,αk} and Q:={β1,…,βk} we have
i(R)=i(Q).
Proof.
Consider L(R) and L*(R) (respectively L(Q) and L*(Q)) as lattices in V′:=spanℝ(R) (respectively in V′′:=spanℝ(Q)). By [12, Lemma 3] the set
R is a basis for L(R). Let
{α1*,…,αk*} be the basis of L*(R) dual to R. In particular, sαj(αi*)=αi* for i≠j and
sαi(αi*)=αi*-αi. Therefore
θi:=(w-1)(αi*)=-sα1⋯sαi-1(αi)∈L(R)∩Φ.
Note that θ1=-α1, θ2∈-α2+spanℤ(α1), and more generally
θi∈-αi+spanℤ(α1,…,αi-1).
It follows that
{θ1,…,θk} is a basis of V′ and that
the map
(w-1)|V′:L*(R)→L(R)
is bijective.
Thus we have i(R)=|det(w-1)|V′|. The same argument with Q instead of R yields that
i(Q)=|det(w-1)|V′′|. By the proof of [1, Theorem 2.4.7] we have that R and Q are both bases for Mov(w), hence
V′=V′′=Mov(w) and hence
i(R)=i(Q).
∎
6 Reflection subgroups related to prefixes of quasi-Coxeter elements
In this section, we prove Theorem 1.2 for simply laced dual
Coxeter systems, see Theorem 6.1.
Further we show that the reflections in a reduced T-decomposition of an element w∈W generate a parabolic subgroup
whenever w≤Tw′ for some quasi-Coxeter element w′. Last but not least we demonstrate that parabolic quasi-Coxeter elements coincide with parabolic Coxeter elements in types An,
Bn and I2(m).
Recall that for w∈W, we denote by Pw the parabolic closure of w (see Definition 4.1) and that Pw=CW(Fix(w)).
Theorem 6.1Let (W,T) be a simply laced dual Coxeter system of rank n. If w∈W is a parabolic quasi-Coxeter element, then the reflections in any reduced T-decomposition of w generate the parabolic subgroup Pw. That is, for each (t1,…,tm)∈RedT(w) we have Pw=〈t1,…,tm〉.
Proof.
By the definition of a parabolic quasi-Coxeter element, there exists an element (t1,…,tm)∈RedT(w) such that P:=〈t1,…,tm〉 is a parabolic subgroup.
By Lemma 4.2, we have P⊆CW(Fix(w))=Pw. Since w∈P, we have by definition of the parabolic closure that P=Pw.
Let (q1,…,qm)∈RedT(w). Then for all 1≤i≤m we have qi≤Tw, which yields that qi is in CW(Fix(w))=Pw. Thus
W′:=〈q1,…,qm〉 is a subgroup of Pw.
Let Φ be the root system of Pw and βi∈Φ be such that qi=sβi, for 1≤i≤m.
Then L(β1,…,βm) is a sublattice of L(Φ) and Φ′=L(β1,…,βm)∩Φ is the smallest root subsystem of
Φ that contains β1,…,βm. Therefore
Theorem 5.14 yields that i(Φ′)=i(Φ).
This implies L(Φ)=L(Φ′) by Proposition 5.13.
Thus W′=Pw by Lemma 5.12.
∎
We will show in Corollary 6.11 that the following property of parabolic quasi-Coxeter elements does in fact characterize them.
Proposition 6.2Let (W,T) be a finite dual Coxeter system. If w∈W is a parabolic quasi-Coxeter element, then there exists a quasi-Coxeter element w′∈W such that w≤Tw′.
Proof.
Let w∈W be a parabolic quasi-Coxeter element.
By definition, there exists a simple system S={s1,…,sn} for W and a T-reduced decomposition w=t1⋯tm such that
〈t1,…,tm〉=〈s1,…,sm〉,
with m≤n. Set w′:=t1⋯tmsm+1⋯sn. It is clear that
〈t1,…,tm,sm+1,…,sn〉=W.
Moreover, we have that ℓT(w′)=n, hence w′ is a quasi-Coxeter element with w≤Tw′.
∎
Lemma 6.3Let (W,T) be a dual Coxeter system of type An. Then each w∈W is a classical parabolic Coxeter element.
Proof.
In type An, the set of (n+1)-cycles forms a single conjugacy class. Hence the set of classical Coxeter elements is exactly the set of (n+1)-cycles (see Remark 2.3 (a)). The assertion follows with
Remark 2.3 (b) as
for every element w∈W, we have w≤Tw′ for at least one (n+1)-cycle w′.
∎
Lemma 6.4Let (W,T) be a dual Coxeter system of type Bn. Then every parabolic quasi-Coxeter element w∈W for (W,T) is a classical parabolic Coxeter element.
Proof.
For the proof we use the combinatorial description of WBn as given in [6, Section 8.1].
Therefore let S-n,n be the group of permutations of
[-n,n]={-n,-n+1,…,-1,1,…,n}
and define
W=WBn:={w∈S-n,n∣w(-i)=-w(i) for all i∈[-n,n]},
also known as the hyperoctahedral group. Then (W,S) is a Coxeter system of type Bn with
S={(1,-1),(1,2)(-1,-2),…,(n-1,n)(-n+1,-n)}.
The set of reflections T for this choice of S is given by
T={(i,-i)∣i∈[n]}∪{(i,j)(-i,-j)∣1≤i<|j|≤n}.
We show that every quasi-Coxeter element for (W,T) is a classical Coxeter element. If w is a parabolic quasi-Coxeter element, then
by Proposition 6.2, there exists a quasi-Coxeter element w′∈W such that w≤Tw′. Hence if w′ is a classical Coxeter element,
then w≤Tw′ implies that w is a classical parabolic Coxeter element.
Let R={r1,…,rn}⊆T be such that 〈R〉=W. It suffices to show that r1r2⋯rn is in fact a classical Coxeter element.
The group W cannot be generated only by reflections of type (i,j)(-i,-j), i≠±j. Therefore there exists i∈[n] with (i,-i)∈R.
If there exists j∈[n], j≠i, with
(j,-j)∈R, then R cannot generate the whole group W.
Since classical Coxeter elements are closed under conjugation, we can conjugate the set R with
(1,i)(-1,-i) (if necessary) and assume (1,-1)∈R.
Since R generates the whole group W, there does not exist j∈[n] which is fixed by each r∈R. Thus for each k∈[n],k≠1,
we can find ik∈[±n] with k≠±ik such that (k,ik)(-k,-ik)∈R. Therefore
R={(1,-1),(2,i2)(-2,-i2),…,(n,in)(-n,-in)}.
Note that some ij has to equal ±1, because otherwise (1,-1) would commute with any element of W. By conjugating R with (j,2)(-j,-2) respectively
(j,-2)(-j,2) (if necessary) we can assume that i2=1. Hence after rearrangement we can assume that R is of the form
R={(1,-1),(2,1)(-2,-1),(3,i3)(-3,-i3),…,(n,in)(-n,-in)}.
Similarly to what we did above, there exists j≥3 with ij∈{±1,±2}.
By conjugating R with (j,3)(-j,-3) respectively (j,-3)(-j,3) (if necessary) we can assume that i3∈{1,2}. Continuing in this manner we obtain
R={(1,-1),(2,i2)(-2,-i2),…,(n,in)(-n,-in)}
with ij∈{1,…,j-1} for each j∈{2,…,n}. A direct computation shows that c:=r1r2⋯rn is a 2n-cycle and
thus a classical Coxeter element. Indeed, there is a single conjugacy class of 2n-cycles in W.
∎
Remark 6.5Notice that by Remark 2.3 (c), it is already known that classical Coxeter elements and Coxeter elements must coincide in type Bn.
Moreover, it follows from [12, Lemma 8, Theorem A] that every quasi-Coxeter element is actually a Coxeter element. Hence one can derive
Lemma 6.4 from these two observations. However since both of them rely on sophisticated methods, we preferred to give here a direct proof
using the combinatorics of the hyperoctahedral group.
Remark 6.6In type An, we even have that every element w such that ℓT(w)=n is a classical Coxeter element (thus quasi-Coxeter),
because such an element is necessarily an (n+1)-cycle. Notice that this fails in type Bn. For instance, the product
(1,-1)(2,-2)⋯(n,-n) in WBn has reflection length equal to n, but it is not a quasi-Coxeter element.
The following is well known (see [7, IV, 1.2, Proposition 2]):
Proposition 6.7A group W is a dihedral group if and only if it is generated by two elements s, t of order 2, in which case {s,t} is a simple system for W.
Corollary 6.8Let (W,T) be a dual Coxeter system of type I2(m), m≥3. Then w is a quasi-Coxeter element in W if and only if w is a Coxeter element in W.
It follows that w∈W is a parabolic quasi-Coxeter element if and only if w is a parabolic Coxeter element.
Note that Coxeter elements and classical Coxeter elements do not coincide in general in dihedral type (see Remark 2.3 (c)).
Theorem 1.5Let w be a quasi-Coxeter element in a finite dual Coxeter system (W,T) of rank n and (t1,…,tn)∈RedT(w) such that
W=〈t1,…,tn〉. Then the reflection subgroup W′:=〈t1,…,tn-1〉 is parabolic.
Proof.
The reduction to the case where W is irreducible is immediate. The proof is uniform for the simply laced types and case-by-case for the non-simply laced types.
Dihedral type. The claim is obvious in that case.
Simply laced types. Let Φ be a root system for (W,T) with ambient vector space V.
Let PW′ be the parabolic closure of W′. For 1≤i≤n, let βi∈Φ be a root corresponding to ti and let
Φ′ be the smallest root subsystem of Φ containing R:={β1,…,βn-1} so that WR=WΦ′=W′
(see Proposition 5.10). Let Ψ⊆Φ be the root subsystem of Φ associated to PW′.
By [12, Lemma 3] the set R∪{βn} is a basis of V.
Let U be the ambient vector space for Ψ. As the linear independent set R is a subset of Ψ,
the dimension of U is at least n-1. Since PW′ is the parabolic closure of t1⋯tn-1 it
has to be the centralizer of a line in V and therefore dimU=n-1. It follows that U=spanℝ(β1,…,βn-1).
By Proposition 5.10 we have that
L(Φ′)=L({β1,…,βn-1}) and L(Φ)=L({β1,…,βn}).
Since V=U⊕ℝβn, we have that
L(Φ)∩U=L({β1,…,βn-1})=L(Φ′). As L(Ψ)⊆U, it follows that L(Ψ)⊆L(Φ′).
But since Φ′⊆Ψ, we get that
L(Φ′)⊆L(Ψ) and therefore L(Φ′)=L(Ψ). Thus
W′=PW′ by Lemma 5.12.
Type Bn. By Lemma 6.4 the element w is a classical Coxeter element. It follows that wtn is a classical parabolic Coxeter element,
hence a parabolic Coxeter element (see Remark 2.3 (c)). It follows that W′ is parabolic.
Type F4. By [15, Table 6], the group WF4 cannot be generated by just adding one reflection to one of its non-parabolic rank 3 reflection subgroups.
Types H3 and H4. We refer
to [15, Tables 8 and 9], where the reflection subgroups of WH3 and WH4 and their parabolic closures are determined.
Each rank 2 reflection subgroup of the group WH3 is already parabolic.
The only rank 3 reflection subgroup of WH4 that is not parabolic has type A1×A1×A1. Taking a set of three reflections generating such a reflection subgroup, we checked using [38] that this set cannot be completed to obtain a generating set for WH4 by adding a single reflection.∎
Remark 6.9Theorem 1.5 is not true in general. It can even fail if w is a Coxeter element, as the following example borrowed from [20, Example 5.7] shows: Let W=〈s,t,u〉 be of affine type A~2, and let w=stu. Then we have s(tut)≤Tc, but W′=〈s,tut〉 is an infinite dihedral group, hence it is not a parabolic subgroup in the classical sense since proper parabolic subgroups of W are finite. We mentioned in Section 4 that for affine Coxeter groups parabolic subgroups in the classical sense also coincide with parabolic subgroups as defined in Section 2.1.
Corollary 6.10Let (W,T) be a finite dual Coxeter system of rank n and let W′ be a reflection subgroup of rank n-1. Then W′ is a parabolic subgroup if and only if there exists t∈T such that 〈W′,t〉=W.
Proof.
The necessary condition is clear by the definition of parabolic subgroup. The sufficient condition is a direct consequence of Theorem 1.5.
∎
We also derive a characterization of parabolic quasi-Coxeter elements analogous to that of parabolic Coxeter elements (see Remark 2.3 (b)).
Corollary 6.11Let (W,T) be a finite dual Coxeter system and w∈W. Then w is a parabolic quasi-Coxeter element if and only if there
exists a quasi-Coxeter element w′∈W such that w≤Tw′.
Proof.
The forward direction is given by Proposition 6.2.
Now let w≤Tw′ where w′∈W is a quasi-Coxeter element. Using Theorem 1.5 inductively we get that w is a parabolic quasi-Coxeter element in W.
∎
Remark 6.12Corollary 6.11 does not hold for infinite Coxeter groups as it fails for the Coxeter element given in Remark 6.9.
7 Intersection of maximal parabolic subgroups in type Dn
The aim of this section is to show the following result which will
be needed in the next section in the proof of Theorem 1.1.
Proposition 7.1Let (W,S) be a Coxeter system of type Dn (n≥6). Then the intersection of two maximal parabolic subgroups is non-trivial.
Remark 7.2This statement is not true in general, not even in the simply laced case. In particular, it fails in types D4,D5,E7 and E8. For example:
Let (W,S) be of type D4 where S={s0,s1,s2,s3} with s2 commuting with no other simple reflection, then both
P:=〈s0,s1,s3〉 and s2Ps2 are maximal parabolic subgroups of type A1×A1×A1 and have trivial intersection.
Let (W,S) be of type E7 where S={s1,…,s7} labelled as in [7, Planche VI].
Let I={s1,s2,s3,s4,s6,s7} and J={s1,s2,s3,s5,s6,s7}. Then the non-conjugate parabolic subgroups WJ
and wWIw-1 intersect trivially, where
w=s6s2s4s5s3s4s1s3s2s4s5s6s7s4s5s6s4s5s3s4s1s3s2s4s5s4s3s2s4s1.
This was checked using [38].
For the rest of this section we work with the combinatorial realization of W as a subgroup (which we denote by WDn) of the hyperoctahedral group
WBn (see Section 6). To this end, set
s0=(1,-2)(-1,2),
si=(i,i+1)(-i,-(i+1)) for i∈[n-1].
Then {s0,s1,…,sn-1} is a simple system for a Coxeter group WDn of type Dn. The set of reflections is given by
T={(i,j)(-i,-j)∣i,j∈[-n,n],i≠±j}. Notice that W is a subgroup of the group WBn of type Bn;
indeed, the above generators are clearly contained in WBn. Given A⊂[-n,n], write Stab(A) for the subgroup of WDn
of elements preserving the set A. Notice that since WDn⊆WBn, we have Stab(A)=Stab(-A). The maximal standard
parabolic subgroups of WDn are then described as follows (see [6, Proposition 8.2.4]). Let i∈{0,1,…,n-1} and
I=S∖{si}. Then WI=Stab(AI), where
AI:={[i+1,n]if i≠1,{-1,2,3,…,n}if i=1.
Since WI stabilizes both AI and -AI, it stabilizes also the complement AI0 of AI∪(-AI) in [-n,n]. Notice that AI0=-AI0.
From this description we can easily achieve a description of maximal parabolic subgroups:
Lemma 7.3If WJ=Stab(AJ) is a maximal standard parabolic subgroup and w∈W, then wWJw-1=Stab(w(AJ)).
Proof of Proposition 7.1.
It is enough to show that WI∩wWJw-1≠{id} for two subsets I,J⊆S with |I|=|J|=n-1 and w∈W. We therefore assume that
WI∩wWJw-1={id} for some I,J⊆S with |I|=|J|=n-1 and w∈W and show that this implies that n≤5.
Consider the intersections AI∩w(AJ) and AI∩(-w(AJ)). If one of these intersections contains at least two elements, say k and l,
then (k,l)(-k,-l)∈WI∩wWJw-1 since
AI∩(-AI)=∅=w(AJ)∩(-w(AJ)).
Therefore we can assume that |AI∩w(AJ)|≤1 and |AI∩-w(AJ)|≤1.
Now if |AI|≥4, then |AI∩w(AJ)0|≥2, and since AI∩(-AI)=∅, it follows that there exist k,ℓ∈AI∩w(AJ)0 with k≠±ℓ,
and we then have that
(k,ℓ)(-k,-ℓ)∈WI∩wWJw-1.
Hence we can furthermore assume that |AI|<4. It follows that |AI0|≥2n-6.
But arguing similarly we can also assume that |AI0∩w(AJ)0|<4, hence we have |AI0∩w(AJ)0|≤2 since it has to be even and |AI0∩w(AJ)|≤1.
It follows that |AI0|≤4. Together with the inequality above we get 2n-6≤|AI0|≤4, hence n≤5.
∎
8 The proof of Theorem 1.1
The aim of this section is to prove the sufficient condition for the transitive Hurwitz action as stated in Theorem 1.1.
The proof is a case-by-case analysis.
Let (W,T) be a finite dual Coxeter system and let w∈W. For two elements (t1,…,tm),(r1,…,rm)∈RedT(w) we write
(t1,…,tm)∼(r1,…,rm) if both factorizations lie in the same Hurwitz orbit. Furthermore, note that if w is a
parabolic quasi-Coxeter element in (W,T), then each conjugate of w is also a parabolic quasi-Coxeter element in (W,T).
Since the Hurwitz operation commutes with conjugation, we can restrict ourselves to check transitivity for one representative of each
conjugacy class of parabolic quasi-Coxeter elements of W. The proof of the following is easy:
Lemma 8.1Let (Wi,Ti), i=1,2, be dual Coxeter systems and let wi∈Wi, i=1,2. Then (W1×W2,T:=(T1×{1})∪({1}×T2))
is a dual Coxeter system. Furthermore, if the Hurwitz action is transitive
on RedTi(wi), i=1,2, then the Hurwitz action is transitive on
RedT((w1,w2)).
8.1 Types An, Bn and I2(m)
In all these cases every parabolic quasi-Coxeter element is already a parabolic Coxeter element
by Lemmas 6.3, 6.4 and Corollary 6.8.
Therefore the assertion follows with [2, Theorem 1.3].
Remark 8.2Notice that in types An and Bn, parabolic quasi-Coxeter elements, classical parabolic Coxeter elements and parabolic Coxeter elements coincide. In type I2(m),
parabolic quasi-Coxeter elements and parabolic Coxeter elements coincide, but parabolic Coxeter elements and classical parabolic Coxeter elements do not
(see Remark 2.3(c)).
8.2 The simply laced types
We now treat the parabolic quasi-Coxeter elements in an irreducible, finite, simply laced dual Coxeter system (W,T) of rank n. As we already dealt with the type An,
it remains to consider the types Dn, n≥4 and E6,E7,E8.
We only need to show the assertion for quasi-Coxeter elements. Indeed, let w be a parabolic quasi-Coxeter element in (W,T) and let (t1,…,tm)∈RedT(w). Then
W′=〈t1,…,tm〉 is by Theorem 6.1 a parabolic subgroup of (W,T), in fact W′=Pw.
Therefore, it follows from Lemma 4.2 that all the reflections in any reduced factorization
of w are in W′. The latter group is a direct product of irreducible Coxeter groups of simply laced type.
If we know that the Hurwitz action is transitive on RedT(w~) for all the quasi-Coxeter elements w~ in the irreducible Coxeter groups of simply laced type,
then the Hurwitz action on RedT(w) is transitive as well by Lemma 8.1.
The strategy to prove the theorem is as follows: we first show by induction on the rank n (with n≥4) that the Hurwitz action is transitive on the set of reduced
decompositions of quasi-Coxeter elements of type Dn; for this we will need to use the result for parabolic subgroups, but since they are (products) of groups of type A with groups of type D of smaller rank, the result holds for groups of type A by Section 8.1 and they hold for groups of type Dk, k<n by induction.
Using the fact that it holds for type Dn, n≥4, we then prove the result for the groups E6, E7 and E8. Similarly as for type Dn, parabolic subgroups of
type E are of type A, D or E. It was previously shown to hold for types A and D and holds for type E by induction.
We then prove by computer that the result holds for the remaining exceptional groups.
Let w be a quasi-Coxeter element and let (t1,…,tn)∈RedT(w).
8.2.1 Type Dn
For types D4 and D5 the assertion is checked directly using [38]. Therefore assume n≥6. Let (r1,…,rn)∈RedT(w) be a second reduced
factorization of w. By Theorem 6.1 and Theorem 1.5 the groups 〈t1,…,tn-1〉 and
〈r1,…,rn-1〉 are maximal parabolic subgroups and since ℓT(wtn)=n-1=ℓT(wrn), it follows that CW(Vwtn)=Pwtn=〈t1,…,tn-1〉, CW(Vwrn)=Pwrn=〈r1,…,rn-1〉.
By Proposition 7.1 there exists a reflection t in their intersection. It follows by Lemma 4.2 that t≤Twtn,wrn. Hence there exists t2′,…,tn-1′,r2′,…,rn-1′∈T such that (t,t2′,…,tn-1′)∈RedT(wtn) and (t,r2′,…,rn-1′)∈RedT(wrn). In particular, we get
(t2′,…,tn-1′,tn),(r2′,…,rn-1′,rn)∈RedT(tw).
By Theorem 1.5 the element tw is
quasi-Coxeter in the parabolic subgroup
Ptw=〈t2′,…,tn-1′,tn〉.
It follows from Lemma 4.2 that the reflections
r2′,…,rn-1′,rn are in Ptw since ri′≤Ttw for each i. As Ptw is a direct product of irreducible Coxeter groups of type
A and D of smaller rank, we have by induction together with Lemma 8.1 that
(t2′,…,tn-1′,tn)∼(r2′,…,rn-1′,rn),
as well as
(t,t2′,…,tn-1′)∼(t1,…,tn-1) and (t,r2′,…,rn-1′)∼(r1,…,rn-1).
This implies
(t1,…,tn)∼(t,t2′,…,tn-1′,tn)
∼(t,r2′,…,rn-1′,rn)∼(r1,…,rn)∈RedT(w),
which concludes the proof.
8.2.2 Types E6,E7 and E8
First we calculated representatives of the conjugacy classes of quasi-Coxeter elements using
[38], see also Remark 8.3 (b) below. Then given a
quasi-Coxeter element w we checked, again using [38], that there is a reduced factorization (t1,…,tn) of w such that for every reflection t in T
there exists an element (t1′,…,tn-1′,t)∈RedT(w) with (t1,…,tn)∼(t1′,…,tn-1′,t).
Let (r1,…,rn)∈RedT(w). By our computations in GAP there exists an element
(t1′,…,tn-1′,rn)∈RedT(w) with
(t1,…,tn)∼(t1′,…,tn-1′,rn).
Then
wrn=t1′⋯tn-1′=r1⋯rn-1
are reduced factorizations. Further, wrn is a quasi-Coxeter element in (W′,T′) where
W′:=〈t1′,…,tn-1′〉 and T′:=T∩W′. By Theorem 1.5 we have that
W′ is a equal to the parabolic closure Pt1′⋯tn-1′ of t1′⋯tn-1′
and therefore r1,…,rn-1 are elements of W′ by Lemma 4.2. Thus
(t1′,…,tn-1′) and (r1,…,rn-1) are reduced factorizations of a quasi-Coxeter element in a dual, simply laced Coxeter system of rank n-1.
By induction and by Lemma 8.1 we get
(t1′,…,tn-1′)∼(r1,…,rn-1),
thus
(t1,…,tn)∼(t1′,…,tn-1′,rn)∼(r1,…,rn).
8.3 The types F4, H3 and H4
For these cases we again calculated representatives of the conjugacy classes of quasi-Coxeter elements using
[38] and then we checked Theorem 1.1 directly for each representative using [38].
Remark 8.3(a) The computer programs that we used can be found at
www.math.uni-bielefeld.de/~baumeist/Dual-Coxeter/dual-Coxeter.html.
(b)
For the convenience of the reader we briefly describe Carter’s classification of the conjugacy classes in finite Weyl groups by means of so-called
admissible diagrams [12]. Due to [12, Lemma 8, Theorem A], we obtain the following description of conjugacy classes of quasi-Coxeter
elements (in the notation of [12]):
For the infinite families the conjugacy classes correspond to the admissible diagrams
An,Bn,Dn,Dn(a1),Dn(a2),…,Dn(a⌊12n-1⌋).
In particular, in types An and Bn the conjugacy class of the Coxeter element is the only quasi-Coxeter class (see Lemmata 6.3 and 6.4).
For the exceptional types the conjugacy classes correspond to the admissible diagrams
E6,E6(a1),E6(a2),E7,E7(a1),…,E7(a4),E8,E8(a1),…,E8(a8),F4,F4(a1),G2.
For the remaining non-crystallographic types we found by computer:
For the type H3 respectively H4 there are three respectively eleven conjugacy classes of quasi-Coxeter elements.
Note that there might be more than one admissible diagram for the same conjugacy class (e.g., the class
E7(a2) might also be parametrized by the diagram E7(b2)). For (W,T) irreducible and crystallographic,
the conjugacy classes of quasi-Coxeter elements are precisely described by the connected admissible diagrams with number of nodes equal to the rank of (W,T).
In [13] such a class is called semi-Coxeter class.
Acknowledgements
We thank Joel Lewis as well as the referee for helpful comments and Christian Stump for fruitful discussions.
References
[1]
Armstrong D.,
Generalized noncrossing partitions and combinatorics of Coxeter groups,
Mem. Amer. Math. Soc. 202 (2009), no. 949, 1–159.
10.1090/S0065-9266-09-00565-1Search in Google Scholar
[2]
Baumeister B., Dyer M., Stump C. and Wegener P.,
A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements,
Proc. Amer. Math. Soc. Ser. B 1 (2014), 149–154.
10.1090/S2330-1511-2014-00017-1Search in Google Scholar
[3]
Bessis D.,
The dual braid monoid,
Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 647–683.
10.1016/j.ansens.2003.01.001Search in Google Scholar
[4]
Bessis D.,
Finite complex reflection arrangements are K(π,1),
Ann of Math. (2) 181 (2015), no. 3, 809–904.
10.4007/annals.2015.181.3.1Search in Google Scholar
[5]
Bessis D., Digne F. and Michel J.,
Springer theory in braid groups and the Birman–Ko–Lee monoid,
Pacific J. Math. 205 (2002), 287–309.
10.2140/pjm.2002.205.287Search in Google Scholar
[6]
Björner A. and Brenti F.,
Combinatorics of Coxeter Groups,
Grad. Texts in Math. 231,
Springer, New York, 2005.
Search in Google Scholar
[7]
Bourbaki N.,
Groupes et Algèbres de Lie, Chapitres 4–6,
Hermann, Paris, 1968.
Search in Google Scholar
[8]
Brady T.,
A partial order on the symmetric group and new K(π,1)’s for the braid groups,
Adv. Math. 161 (2001), no. 1, 20–40.
10.1006/aima.2001.1986Search in Google Scholar
[9]
Brady N., McCammond J. P., Mühlherr B. and Neumann W. D.,
Rigidity of Coxeter groups and Artin groups,
Geom. Dedicata 94 (2002), no. 1, 91–109.
10.1023/A:1020948811381Search in Google Scholar
[10]
Brady T. and Watt C.,
K(π,1)’s for Artin groups of finite type,
Geom. Dedicata 94 (2002), 225–250.
10.1023/A:1020902610809Search in Google Scholar
[11]
Brieskorn E.,
Autmomorphic sets and braids and singularities,
Braids (Santa Cruz 1986),
Contemp. Math. 78,
American Mathematical Society, Providence (1988), 45–115.
10.1090/conm/078/975077Search in Google Scholar
[12]
Carter R. W.,
Conjugacy classes in the Weyl group,
Compos. Math. 25 (1972), 1–59.
10.1007/BFb0081548Search in Google Scholar
[13]
Carter R. W. and Elkington G. B.,
A note on the parametrization of conjugacy classes,
J. Algebra 20 (1972), 350–354.
10.1016/0021-8693(72)90062-2Search in Google Scholar
[14]
Digne F. and Gobet T.,
Dual braid monoids, Mikado braids and positivity in Hecke algebras,
Math. Z., to appear.
10.1007/s00209-016-1704-zSearch in Google Scholar
[15]
Douglass J. M., Pfeiffer G. and Röhrle G.,
On reflection subgroups of finite Coxeter groups,
Comm. Algebra 41 (2013), no. 7, 2574–2592.
10.1080/00927872.2012.661005Search in Google Scholar
[16]
Dyer M. J.,
Reflection subgroups of Coxeter systems,
J. Algebra 135 (1990), no. 1, 57–73.
10.1016/0021-8693(90)90149-ISearch in Google Scholar
[17]
Ebeling W.,
Lattices and Codes,
Vieweg, Wiesbaden, 2002.
10.1007/978-3-322-90014-2Search in Google Scholar
[18]
Franzsen W. N., Howlett R. B. and Mühlherr B.,
Reflections in abstract Coxeter groups,
Comment. Math. Helv. 81 (2006), no. 3, 665–697.
10.4171/CMH/69Search in Google Scholar
[19]
Hou X.,
Hurwitz equivalence in tuples of generalized quaternion groups and dihedral groups,
Electron. J. Combin. 15 (2008), no. 1, Paper No. R80.
10.37236/804Search in Google Scholar
[20]
Hubery A. and Krause H.,
A categorification of non-crossing partitions,
preprint 2013, http://arxiv.org/abs/1310.1907;
to appear in J. Eur. Math. Soc.
10.4171/JEMS/641Search in Google Scholar
[21]
Humphreys J. E.,
Reflection Groups and Coxeter Groups,
Cambridge Stud. Adv. Math. 29,
Cambridge University Press, Cambridge, 1990.
10.1017/CBO9780511623646Search in Google Scholar
[22]
Humphries S. P.,
Finite Hurwitz braid group actions on sequences of Euclidean reflections,
J. Algebra 269 (2003), no. 2, 556–588.
10.1016/S0021-8693(03)00377-6Search in Google Scholar
[23]
Hurwitz A.,
Über Riemannsche Flächen mit gegebenen Verzweigungspunkten,
Math. Ann. 39 (1891), 1–61.
10.1007/BF01199469Search in Google Scholar
[24]
Igusa K. and Schiffler R.,
Exceptional sequences and clusters,
J. Algebra 323 (2010), 2183–2202.
10.1016/j.jalgebra.2010.02.003Search in Google Scholar
[25]
James A., Magaard K. and Shpectorov S.,
The lift invariant distinguishes components of Hurwitz spaces for A5,
Proc. Amer. Math. Soc. 143 (2015), no. 4, 1377–1390.
10.1090/S0002-9939-2014-12185-XSearch in Google Scholar
[26]
Kane R.,
Reflection Groups and Invariant Theory,
CMS Books Math./Ouvrages Math. SMC 5,
Springer, New York, 2001.
10.1007/978-1-4757-3542-0Search in Google Scholar
[27]
Kluitmann P.,
Geometrische Basen des Milnorgitters einer einfach elliptischen Singularität,
Diploma thesis, Universität Bonn, Bonn, 1983.
Search in Google Scholar
[28]
Kulikov S. V. and Teicher M.,
Braid monodromy factorizations and diffeomorphism types,
Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 89–120.
10.1070/IM2000v064n02ABEH000285Search in Google Scholar
[29]
Liberman E. and Teicher M.,
The Hurwitz equivalence problem is undecidable,
preprint 2004, http://arxiv.org/abs/math/0511153.
Search in Google Scholar
[30]
Matsumoto H.,
Générateurs et relations des groupes de Weyl généralisés,
C. R. Math. Acad. Sci. Paris 258 (1964), 3419–3422.
Search in Google Scholar
[31]
Michel J.,
Hurwitz action on tuples of Euclidean reflections,
J. Algebra 295 (2006), no. 1, 289–292.
10.1016/j.jalgebra.2004.12.020Search in Google Scholar
[32]
Nuida K.,
On the direct indecomposability of infinite irreducible Coxeter groups and the isomorphism problem of Coxeter groups,
Comm. Algebra 34 (2006), no. 7, 2559–2595.
10.1080/00927870600651281Search in Google Scholar
[33]
Qi D.,
A note on parabolic subgroups of a Coxeter group,
Expo. Math. 25 (2007), no. 1, 77–81.
10.1016/j.exmath.2006.05.001Search in Google Scholar
[34]
Reiner V., Ripoll V. and Stump C.,
On non-conjugate Coxeter elements in well-generated reflection groups,
preprint 2014, http://arxiv.org/abs/1404.5522.
10.46298/dmtcs.2466Search in Google Scholar
[35]
Sia C.,
Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups,
Electron. J. Combin. 16 (2009), no. 1, Paper No. R95.
10.37236/184Search in Google Scholar
[36]
Tits J.,
Buildings of Spherical Type and Finite Bn-Pairs,
Lecture Notes in Math. 386,
Springer, Berlin, 1974.
Search in Google Scholar
[37]
Voigt E.,
Ausgezeichnete Basen von Milnorgittern einfacher Singularitäten,
Bonner Math. Schriften 160 (1985).
10.1007/BF02941496Search in Google Scholar
[38]
The GAP Group,
GAP-Groups, Algorithms, and Programming, Version 4.7.7, 2015, http://www.gap-system.org.
Search in Google Scholar
,
Thomas Gobet
