Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology

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Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology

Vincent Beck

To cite this version:

Vincent Beck. Abelianization of Subgroups of Reflection Group and their Braid Group; an Application

to Cohomology. manuscripta mathematica, Springer Verlag, 2011, 136 (3), pp.273-293. �hal-01198821�

Abelianization of Subgroups of Reflection Groups and their Braid Groups; an Application to Cohomology

Vincent Beck Friday 28

August, 2015

Abstract The final result of this article gives the order of the extension

1 //P/[P, P] ^j //B/[P, P] ^p //W //1

as an element of the cohomology groupH²(W, P/[P, P])(whereB andP stands for the braid group and the pure braid group associated to the complex reflection groupW). To obtain this result, we first refine Stanley-Springer’s theorem on the abelianization of a reflection group to describe the abelianization of the stabilizerNH of a hyperplaneH. The second step is to describe the abelianization of big subgroups of the braid groupB ofW. More precisely, we just need a group homomorphism from the inverse image ofNH by p(wherep:B→W is the canonical morphism) but a slight enhancement gives a complete description of the abelianization ofp⁻¹(W⁰)whereW⁰is a reflection subgroup of W or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection inW.

1 Introduction

Let us start with setting the framework. LetV be a finite dimensional complex vector space; areflectionis a non trivial finite order elementsofGL(V)which pointwise fixes a hyperplane ofV, calledthe hyperplane of s.

Theline ofsis the one dimensional eigenspace ofsassociated to the non trivial eigenvalue ofs. LetW ⊂GL(V) be a(complex) reflection groupthat is to say a finite group generated by reflections. We denote byS the set of reflections ofW andH the set of hyperplanes ofW :

S ={s∈W, codimKer (s−id) = 1} and H ={Ker (s−id), s∈S}.

For a reflection group, we denote byV^reg=V r∪H∈HH the set of regular vectors. According to a classical result of Steinberg [12, Corollary 1.6],V^reg is precisely the set of vectors that no non trivial element ofW fixes.

Thus, the canonical mapπ:V^reg→V^reg/W is a Galois covering. So let us fix a base pointx₀∈V^regand denote byP =π₁(V^reg, x₀)and B=π₁(V^reg/W, π(x₀))the fundamental groups ofV^regand its quotientV^reg/W, we obtain the short exact sequence

1 //P //B ^p //W //1 (1)

The groupsB andP are respectively called the braid groupand thepure braid groupofW. The final result of this article (Corollary 27) gives the order of the extension

1 //P/[P, P] ^j //B/[P, P] ^p //W //1

as an element of the cohomology groupH²(W, P/[P, P]). This order turns out to be the integerκ(W)defined by Marin in [9]. As explained in [9], this integer is linked to the periodicity of the monodromy representation of B associated to the action ofW on its set of hyperplanes.

To obtain this result, we first describe in section 2 the abelianization of some subgroups of complex reflection groups. Specifically, we study the stabilizer of a hyperplaneH which is the same as the centralizer of a reflection of hyperplane H. Contrary to the case of Coxeter groups, this is not a reflection subgroup of the complex reflection groupW in general. The first step is to refine Stanley-Springer’s theorem [13][14] on the abelianization of a reflection group (see Proposition 6 in Section 2).

The rationale also relies on a good description (Section 3) of abelianization of various types of big subgroups of the braid groupB ofW (here “big” stands for “containing the pure braid groupP”). Though we just need to construct a group homomorphism from the inverse image of the stabilizer of H by p with values in Q (see Definition 19 and Proposition 23), we give in fact a complete description of the abelianization ofp⁻¹(W⁰) withW⁰ a reflection subgroup ofW or the stabilizer of a hyperplane (Proposition 15 and Proposition 24). We also suggest a lifting construction for every element of the centralizer of a reflection in W generalizing the construction of the generator of monodromy of [4, p.14] (see Remark 21).

Orbits of hyperplanes and ramification index are gathered in tables in the last section.

1

We finish the introduction with some notations. We fix a W-invariant hermitian product on V denoted byh·,·i: orthogonality will always be relative to this particular hermitian product. We denote byS(V^∗)the symmetric algebra of the dual ofV which is also the polynomial functions onV.

Notation 1 Around a Hyperplane ofW. ForH∈H,

• one choosesα_H∈V^∗ a linear form with kernelH;

• one sets WH =FixW(H) ={g ∈W, ∀x∈H, gx=x}. This is a cyclic subgroup ofW. We denote by eH its order and bysH its generator with determinant ζH = exp(2iπ/eH). Except for identity, the elements ofWH are precisely the reflections ofW whose hyperplane isH. The reflection sH is called the distinguished reflection for H in W.

For na positive integer, we denote byUn the group of then^th root of unity inCand byUthe group of unit complex numbers. For a groupG, we denote by[G, G]the commutator subgroup ofGand byG^ab=G/[G, G]

the abelianization ofG.

2 Abelianization of Subgroups of Reflection Groups

Stanley-Springer’s Theorem (see [13, Theorem 4.3.4][14, Theorem 3.1]) gives an explicit description of the group of linear characters of a reflection group using the conjugacy classes of hyperplanes. Naturally, it applies to all reflection subgroups of a reflection group and in particular to parabolic subgroups thanks to Steinberg’s theorem [12, Theorem 1.5]. But sinceP/[P, P]is theZW permutation module defined by the hyperplanes ofW, we are interested in the stabilizer of a hyperplane which is not in general a reflection subgroup ofW. So we have to go deeper in the study of the stabilizer of a hyperplane.

Before starting our study of the stabilizer of a hyperplane, we write down Stanley-Springer’s Theorem because we will use it many times.

Theorem 2 Stanley-Springer’s Theorem. For every mapn:H →Nconstant on theW-orbits ofH, there exists a linear characterχ:W→C^× such thatχ(sH) = det(sH)^−nH.

Moreover, theχ-isotypic component of S(V^∗)is a freeS(V^∗)^W-module of rank 1generated by Qχ= Y

H∈H

αHnH

where thenH are related toχby the formula above and satisfy the relations06nH 6eH−1.

ForH ∈H, we set

N_H={w∈W, wH=H}={w∈W, ws_H =s_Hw}

the stabilizer of H which is also the centralizer of sH. We denote by D = H^⊥ the line of sH (or of every reflection of W with hyperplane H) it is the unique NH-stable line of V such that H ⊕D = V and NH

is also the stabilizer of D (see [3, Proposition 1.19]). We denote the parabolic subgroup associated to D by CH={w∈W, ∀x∈D, wx=x}.

Since Dis a line stable by every element ofNH andWH ⊂NH, there exists an integerfH such thateH |fH

and the following sequence is exact

1 //CH

i //NH

r //Uf_H //1 (2)

whereiis the natural inclusion andrdenote the restriction toD. We define rto be thenatural linear character ofN_H.

Before stating our main result on the abelianization of the stabilizer of a hyperplane, let us start with a straightforward lemma.

Lemma 3 Abelianization of an exact sequence. Let us consider the following exact sequence of groups whereM is an abelian group.

1 //C ⁱ //N ^r //M //1 Then the following sequence is exact

C^{ab i}

ab //N^{ab r}

ab //M //1

Moreover the mapi^abis injective if and only if[N, N] = [C, C]. When C^ab andM are finite, the injectivity ofi^abis equivalent to |C^ab||M|=|N^ab|.

We also have the following criterion : the mapi^abis injective if and only if the canonical restriction map Homgr.(N,C^×)→Homgr.(C,C^×)is surjective.

Proof. The first injectivity criterion is an easy diagram chasing computation. The second one is trivial. Let us focus on the third one. SinceC^× is a commutative group, we have the following commutative square whose vertical arrows are isomorphisms

Homgr.(N,C^×) //

Homgr.(C,C^×)

Homgr.(N^ab,C^×) ^◦i

ab //Homgr.(C^ab,C^×)

Moreover, sinceC^× is a divisible abelian group, i^abis injective if and only if◦i^ab is surjective.

Before applying the preceding lemma to the stabilizer of a hyperplane, let us introduce a definition and a classical linear algebra lemma.

Definition 4 Commuting hyperplanes. Let H, H⁰∈H. We say thatH andH⁰ commuteifsH andsH⁰

commute.

We denote byHH the set of hyperplanes which commute withH andHH⁰ =HHr{H}.

The next lemma on commuting hyperplanes is stated in [3, Lemma 1.7].

Lemma 5 We have the following equivalent characterizations : (i) the hyperplanesH andH⁰ commute

(ii)H =H⁰ orD=H^⊥⊂H⁰

(iii) every reflection ofW with hyperplaneH commutes with every reflection ofW with hyperplaneH⁰ (iv) there exists a reflection ofW with hyperplaneH which commutes with a reflection ofW with hyperplaneH⁰

As a consequence of Lemma 3, we are now able to formulate the following proposition.

Proposition 6 Abelianization of the stabilizer of a hyperplane. For a hyperplaneH ∈H, the following sequence is exact

C_H^{ab iab} //N_H^{ab rab} //U^fH //1

Moreover, we have the following geometric characterization of the injectivity ofi^ab : the mapi^ab is injective if and only if the orbits of the hyperplanes commuting withH under NH andCH are the same.

Proof. Steinberg’s theorem and Lemma 5(ii)ensure us thatC_H is the reflection subgroup ofW generated by thesH⁰ forH⁰ 6=H commuting withH. Thanks to Theorem 2, we are able to describe the linear characters of CH. For every linear characterδofCH, there exists integerse_O forO ∈HH⁰/CH such that the polynomial

Q_δ = Y

O∈H_H⁰/C_H

Y

H⁰∈O

α_H^0eO ∈S(V^∗)

verifiesgQδ =δ(g)Qδ for every g∈CH.

Let us assume that the orbits of the hyperplanes commuting with H underN_H and C_H are the same. For every g∈N_H and everyO ∈HH⁰/C_H, there existsλ_g ∈C^× such that

g Y

H⁰∈O

αH⁰ =λg

Y

H⁰∈O

αH⁰

So we obtain that, for everyg∈N_H, there existsµ_g ∈C^× such that gQ_δ =µ_gQ_δ. Thus every linear character ofC_H extends toN_H and Lemma 3 tells us that the mapi^abis injective.

Let us assume now that every linear character ofCH extends toNH. The orbit ofH underNH andCH is {H}. So let us consider an orbitO ∈HH⁰/CH. We define

Q= Y

H⁰∈O

α_H⁰ ∈S(V^∗).

Thanks to Theorem 2,Qdefine a linear characterχ ofCH: for everyc∈CH, there exists χ(c)∈C^× such that cQ=χ(c)Qfor everyc∈CH. We then consider theNH-submoduleM ofS(V^∗)generated byQ. As a vector space,M is generated by the family(nQ)n∈N_H. But, sinceCH is normal inNH, we obtain forc∈CH,

cnQ=nn⁻¹cnQ=nχ(n⁻¹cn)Q .

Sinceχextends toN_H, we haveχ(n⁻¹cn) =χ(c)and then cnQ=χ(c)nQ. Theorem 2 allows us to conclude thatnQ=λ_nQfor someλ_n∈C^×. SinceS(V^∗)is a UFD, we obtain thatO is still an orbit underN_H.

Remark 7 Commuting Orbits. The orbits of the hyperplanes commuting withH under NH andCH are the same for every hyperplaneH of every complex reflection group except the hyperplanes of the exceptional groupG25 and the hyperplanesHi (16i6r) of the groupG(de, e, r)whenr= 3 andeis even (see section 5 for the notations).

In Section 5, we give tables for the various orbits of hyperplanes for the infinite series G(de, e, r). For the exceptional complex reflection groups, we check the injectivity or non-injectivity ofi^abusing the packageCHEVIE ofGAP [6][8].

For a hyperplane H∈H, the comparison ofeH and fH leads to the following definition.

Definition 8 Ramification at a hyperplane. We defined_H=f_H/e_H to bethe index of ramification of W at the hyperplaneH. We say thatW isunramified at H ifd_H= 1.

We say that an elementw∈N_H such thatr(w) = exp(2iπ/f_H)realizes the ramification.

Remark 9 The Coxeter Case. When H is an unramified hyperplane, we have NH =CH×WH which is generated by reflections thanks to Steinberg’s theorem [12, Theorem 1.5] andsH realizes the ramification.

Moreoveri^abis trivially injective.

In a Coxeter group, every hyperplane is unramified. Indeed, the eigenvalue on the line D of an element of NH is a finite order element of the field of the real numbers.

Remark 10 The 2-dimensional Case. When W is a 2-dimensional reflection group, NH is an abelian group andi=i^abis injective.

In section 5, we give tables for the values of eH, fH anddH for every hyperplane of every complex reflection groups. From these tables, we obtain the following remarks.

Remark 11 UnramifiedG(de, e, r). All the hyperplanes ofG(de, e, r)are unramified only whenr= 1or whenG(de, e, r)is a Coxeter group (that is to say ifd= 2ande= 1andr>2(Coxeter group of typeBr) or if d= 1ande= 2andr>3(Coxeter group of typeDr) or ifd= 1ande= 1 andr>3(Coxeter group of type A_r−1) of ifd= 1andr= 2(Coxeter group of type I2(e)).

Remark 12 Unramified exceptional groups. The only non Coxeter groups for which every hyperplane is unramified areG8, G12andG24.

Remark 13 Generating Set. SinceCH is the parabolic subgroup associated toD, it is generated by the reflections it contains (this is Steinberg’s theorem). Moreover, ifw_H ∈N_H realizes the ramification. Then, the exact sequence (2) tells us thatN_H is generated byw_H and the family ofs_H⁰ such thatH⁰ ∈HH⁰.

3 Abelianization of Subgroups of Braid Groups

In this section, we describe abelianizations of subgroups ofB containingP that is to say of inverse images of subgroupsW⁰ ofW. Explicitly, we are able to give a complete description ofp⁻¹(W⁰)^ab ifW⁰ is a reflection subgroup (Proposition 15) or if W⁰ is the stabilizer of a hyperplane under geometrical assumptions on the hyperplane (Proposition 24). We also construct a particular linear character of p⁻¹(NH) lifting the natural linear characterrofNH which is of importance for the next section (Definition 19).

Our method is similar to the method of [4] for the description of B^ab: we integrate along paths invariants polynomial functions. So, we have first to construct invariant polynomial functions and then verify that we have constructed enough of them.

3.1 Subgroup Generated by Reflections

In this subsection, we fixC a subgroup ofW generated by reflections. We denote byHC⊂H the set of hyperplanes ofC. ForH ∈HC, thenCH ={c∈C, ∀x∈H, cx=x}is a subgroup ofWH and so generated by sHaH with aH | eH. For H ∈ H rHC, we set aH =eH. We then obtain C =hsHaH, H ∈ Hi. For C ∈H/C a C-class of hyperplanes ofW, we denote bya_C the common value ofaH forH ∈ C.

The aim of this subsection is to give a description of the abelianization ofp⁻¹(C)⊂B. For this, we follow the method of [4] and we start to exhibit invariants which will be useful to show the freeness of our generating set ofp⁻¹(C)^ab.

Lemma 14 An invariant. We define, forC ∈H/C, α_C = Y

H∈C

αHeH/aH ∈S(V^∗).

Thenα_C is invariant under the action ofC.

Proof. IfC is a class of hyperplanes ofHC then this is an easy consequence of Theorem 2.

Assume that C is not a class of hyperplanes ofHC. Let us choose a reflectionsofC and letnsbe the order ofs. SinceC is not a class inHC, the hyperplane ofsdoes not belong to C. We then deduce that the orbits of C under the action ofhsiare of two types.

First type : the orbits ofH ∈ C such thatsH andscommute. Since ssHs⁻¹=s_s(H), we then deduce that s(H) =H. And so the orbit of H under hsi is reduced toH. We denote by Hs the hyperplane ofs. Since H 6=Hs, Lemma 5 tells us thatD=H^⊥⊂Hsand sosacts trivially onDwhich is identified to the line spanned byαH through the inner product.

Second type : the orbits of H ∈ C such thatsHs6=ssH. If sⁱH =H thensⁱ andsH commute and thus, Lemma 5 ensures us thatsⁱ is trivial. We then obtain that the orbit ofH underhsihas cardinalityns. So if we denote byQthe following productα_Hα_sH· · ·α_sns−1H=λα_Hsα_H· · ·s^ns−1α_H withλ∈C^×, we havesQ=Q.

We then easily obtain sα_C =α_C for every s∈C and soα_C is invariant under the action ofC.

Before stating the main result of the subsection, we recall the notion of “generator of the monodromy around a hyperplane” as defined in [4, p.14]. ForH∈H, we define a generator of the monodromy around H to be a pathsH,γ inV^regwhich is the composition of three paths. The first path is a pathγgoing fromx0to a point xH which is nearH and far from other hyperplanes. To describe the second path, we writexH=h+dwith h∈H andd∈D=H^⊥, and the second path ist∈[0,1]7→h+ exp(2iπt/eH)dgoing fromxH tosH(xH). The third path issH(γ⁻¹)going from sH(xH)to sH(x0). We can now state our abelianization result.

Proposition 15 Abelianization of subgroups of the braid group. LetC be a subgroup ofW generated by reflections. Thenp⁻¹(C)^ab is the free abelian group overH/C theC-classes of hyperplanes ofW.

Explicitly, we have p⁻¹(C) =hsH,γa_H, (H, γ)i(see [4, Theorem 2.18]). ForC ∈H/C, we denote by(s^a_C^C)^ab the common value inp⁻¹(C)^abof thesH,γa_H forH ∈ C. Thenp⁻¹(C)^ab=h(s^a_C^C)^ab, C ∈H/Ci. Moreover, for C ∈H/C, there exists a group homomorphismϕC :p⁻¹(C)→Zsuch thatϕC((s^a_C^C)^ab) = 1andϕC((s^a_C^C00 )^ab) = 0 forC⁰6=C.

Proof. First of all, Lemma 2.14.(2) of [4] shows thats_H,γ^aH =s_H,γ^0aH inp⁻¹(C)^ab. Now, forc∈C, we choose x∈p⁻¹(C)such thatp(x) =c. We havexs_H,γx⁻¹=s_cH,x(cγ). Sos_cH,x(cγ)^acH ands_H,γ^aH are conjugate by an element ofp⁻¹(C). So, we have

s_cH,x(cγ)^acH =s_H,γ^aH ∈p⁻¹(C)^ab. And thenp⁻¹(C)^ab=h(s^a_C^C)^ab, C ∈H/Ci.

Let us now show that the family((s^a_C^C)^ab)_C∈H/C is free overZ. We identifyp⁻¹(C)with

p⁻¹(C) =



 G

c,c⁰∈C

π₁(V^reg, cx₀, c⁰x₀)



/C

whereπ1(V^reg, cx0, c⁰x0)denotes the homotopy classes of paths from c(x0) toc⁰(x0)and the action ofC on paths is simply the composition.

Since α_C : V^reg→C^× is C-invariant (Lemma 14), the functoriality of π1 defines a group homomorphism π1(α_C)from p⁻¹(C)to π1(C^×, α_C(x0)). Moreover, the map

I:γ7−→ 1 2iπ

Z

γ

dz z

realizes a group isomorphism between π1(C^×, α_C(x0)) andZ. The composition of these two maps defines a group homomorphism. We denote it byϕ_C and we now want to show thatϕ_C verifies the condition stated in the Proposition.

ForH ∈ C andC⁰ ∈H/C, let us computeϕC⁰(sH,γa_H). The pathsH,γa_H is the composition of three paths.

The first one isγ, the third one issHa_H(γ⁻¹)and the second one is η:t∈[0,1]7→h+ exp(2iπaHt/eH)d.

Since αC⁰ isC-invariant, when we applyπ1(αC⁰), the first part of the path and the third one are inverse from each other. So when applyingI, they do not appear. We thus obtain

ϕ_C⁰(sH,γa_H) = 1 2iπ

Z

α_C0◦η

dz z .

Using the logarithmic derivative, we obtain ϕ_C⁰(s_H,γ^aH) = 1

2iπ X

H⁰∈C⁰

e_H⁰ aH⁰

Z 1 0

2iπa_H eH

exp(2iπa_Ht/e_H)α_H⁰(d) αH⁰(h+ exp(2iπaHt/eH)d)dt To compute this sum, we regroup the terms according to the orbit ofH⁰ underhsHaHi.

Lemma 5 shows that there are three types of orbits : two types of orbits reduced to one single hyperplane and one other type of orbits corresponding to reflections that do not commute withsH.

Let us first study the orbits reduced to one single hyperplane. The first type corresponds to the hyperplane H whose term of the sum is1and this term appears if and only if H ∈ C⁰. The second type corresponds to hyperplanesH⁰ such thatD=H^⊥⊂H⁰. The corresponding term of the sum is0 sinceαH⁰(d) = 0

Let us now study the non trivial orbits. The orbits of H⁰ under sHaH is {H⁰, . . . , sHaH(e_H/a_H−1)(H⁰)}.

Moreover, since a quotient of the formαH⁰(x)/αH⁰(y)does not depend of the linear form with kernelH⁰, we can replaceα_sHkH⁰ bys^k_HαH⁰ to obtain

exp(2iπaHt/eH)α_s

H−kaHH⁰(d) α_s

H−kaHH⁰(h) + exp(2iπa_Ht/e_H)α_s

H−kaHH⁰(d)

= exp(2iπaHt/eH)sH−aHkαH⁰(d)

sH−aHkαH⁰(h) + exp(2iπaHt/eH)sH−aHkαH⁰(d)

= exp(2iπaH(t+k)/eH)αH⁰(d) αH⁰(h) + exp(2iπaH(t+k)/eH)αH⁰(d). Considering the sum over the orbit underhs_H^aHiofH⁰, we obtain

e_H/a_H−1

X

k=0

Z 1 0

exp(2iπaH(t+k)/eH)αH⁰(d)

αH⁰(h) + exp(2iπaH(t+k)/eH)αH⁰(d)dt= eH

aH

Z 1 0

exp(2iπt)αH⁰(d)

αH⁰(h) + exp(2iπt)αH⁰(d)dt .

Sincex_H is chosen such thatα_H⁰(h)6= 0forH⁰6=H anddis small, the last term is0as the index of the circle of center 0and radius|α_H⁰(d)| relatively to the point−α_H⁰(h).

Remark 16 Extreme cases. The two extreme cases where C = 1 and C = W may be found in [4, Prop. 2.2.(2)] and [4, Theorem 2.17.(2)]. In the first case, p⁻¹(C) = P is the pure braid group whose abelianization is the free abelian group over H. In the second case p⁻¹(C) =B is the braid group whose abelianization is the free abelian group overH/W.

Remark 17 The logarithmic derivative shows that for everyγ∈p⁻¹(C)andn∈Z, we have Z

α_Cⁿ◦γ

dz

z =nϕC(γ).

3.2 Stabilizer of a hyperplane

Let us recall the notation of Section 2; we consider H ∈H a hyperplane of the reflection groupW. We denote byN_H the stabilizer ofH inW andC_H the parabolic subgroup ofW associated to the lineD=H^⊥. The set of hyperplanes commuting withH isHH (see Definition 4).

A group homomorphism

The aim of this paragraph is to construct an “extension” of the natural character ofNHto the groupp⁻¹(NH) which will be useful for the third section. We still follow the method of [4] : we construct an invariant function with values inC^× (Lemma 18) and integrate it (Definition 19). To obtain the “extension” properties of the linear character ofp⁻¹(NH)(Proposition 23), we construct a lifting in the braid group of the elements ofNH

(Remark 21). This lifting is inspired from the construction of the generator of the monodromy.

Lemma 18 An invariant function. The functionα_NH =α_H^fH ∈S(V^∗)is invariant underN_H. Proof. This is clear since the line spanned byαH is identified to D through the inner product.

Definition 19 The group homomorphism. As in the proof of Proposition 15, we write

p⁻¹(NH) =



 G

n,n⁰∈NH

π1(V^reg, nx0, n⁰x0)



/NH.

Since α_NH : V^reg→C^× is N_H-invariant (Lemma 18), the functoriality of π₁ allows us to define a group homomorphismπ₁(α_NH)fromp⁻¹(N_H)toπ₁(C^×, α_NH(x₀)). Moreover, the map

I:γ7−→ 1 2iπ

Z

γ

dz z

realizes a group isomorphism between π1(C^×, αN_H(x0)) andZ. The composition of this two maps defines a group homomorphismρ⁰:p⁻¹(NH)→Z. We also defineρ=fH−1ρ⁰ :p⁻¹(NH)→Q.

Remark 20 Center of the braid group ofG31. In [4, Theorem 2.24], it is shown that the center of the braid groupB of an irreducible reflection groupW is an infinite cyclic group generated byβ:t7→exp(2iπt/|Z(W)|)x0

(where x0∈V^reg is a base point) for all but six exceptional reflection groups. In his articles [1][2], Bessis proves that the result holds for all reflection groups but the exceptional oneG31.

This remark is a first step toward the case of G31 : we show that if ZB is an infinite cyclic group, it is generated byβ. For this, let us considerH ∈H a hyperplane ofG31 andρ⁰ the group homomorphism defined above. SinceZB⊂p⁻¹(NH),ρ⁰ restricts to a group homomorphism fromZB toZsuch thatρ⁰(β) = 1. So if ZBis an infinite cyclic group, it is generated byβ.

Remark 21 The lifting construction. Let us considerw∈N_H. We now construct a pathweinV^regstarting from x₀ and ending atw(x₀): p(w) =e w. We use the notations of the description of the generators of the monodromy aroundH : we writex_H=h+dwithh∈H andα_H⁰(h)6= 0forH⁰6=H andd∈D=H^⊥. Since w∈N_H, we havew(x_H) =h⁰+ exp(2ikπ/f_H)dwithh⁰∈H and06k < f_H.

The path we consists into four parts. As in the case of the generators of the monodromy, the first part is a pathγfromx0 toxH and the fourth path isw(γ⁻¹)fromw(xH)tow(x0). Let us now describe the second part and the third part. The second part ofwe is the path

t∈[0,1]7→h+ exp(2ikπt/fH)d∈V^reg.

The third part is of the formt∈[0,1]7→θ(t) + exp(2ikπ/fH)dwhereθ(t)is a path in the complex affine lineD generated byh⁰ andh. It is easy to force the third part ofweto stay inV^reg since its image is contained in the affine lineexp(2ikπ/fH)d+D which is parallel to the hyperplane H and meets each of the other hyperplanes in a single point : so we just have to avoid a finite number of points inC.

Remark 22 Generating set. We have seen in Remark 13 that NH=hwH, sH⁰, H⁰∈HHi

wherewH ∈NH is a once and for all fixed element realizing the ramification. It is now an easy consequence of Theorem 2.18 of [4] that

p⁻¹(NH) =hgwH, sH⁰,γ, sH⁰⁰,γ⁰e_H00, H⁰∈HH, H⁰⁰∈H rHH, γ, γ⁰i.

It remains to show that the constructed group homomorphismρis an “extension” of the natural character of NH. More precisely, we have the following proposition.

Proposition 23 The “extension” property. We have the following commutative square p⁻¹(N_H) ^ρ //

p

Q

π⁰

N_H ^r //UfH

whereπ⁰:x∈Q7→exp(2iπx).

Proof. Using the generating set ofNH given in Remark 22, we only need to show that (i)ρ⁰(wgH) = 1

(ii)ρ⁰(sH,γ) =fH/eH

(iii)ρ⁰(sH⁰,γ) = 0forH⁰∈HH⁰ =HHr{H}

(iv)ρ⁰(sH⁰,γe_H0) = 0forH⁰∈H rHH.

As in the proof of Proposition 15, theγ-part ofs_H,γ(resp. s_H0,γforH⁰∈HH⁰ ands_H0,γe_H0 forH⁰∈H rHH) does not appear in the computation ofρ⁰. We thus obtain

ρ⁰(s_H,γ) = 1 2iπ

Z 1 0

f_H2iπ eH

α_H(exp(2iπt/e_H)d)

αH(h+dexp(2iπt/eH))dt=f_H/e_H. ForH⁰ ∈HH⁰, we setxH⁰ =h⁰+d⁰ withh⁰∈H⁰ andd⁰∈D⁰=H^0⊥. We then obtain

ρ⁰(sH⁰,γ) = 1 2iπ

Z 1 0

fH

2iπ eH⁰

αH(exp(2iπt/eH⁰)d⁰)

αH(h⁰+d⁰exp(2iπt/eH⁰))dt= 0 sinceαH(d⁰) = 0 forH ∈HH⁰. With the same arguments, we obtain forH⁰∈H rHH

ρ⁰(sH⁰,γeH) = 1 2iπ

Z 1 0

2iπfH

exp(2iπt)αH(d⁰)

αH(h⁰) + exp(2iπt)αH(d⁰)dt= 0 sinced⁰ is small andαH(h⁰)6= 0.

For gw_H, neither the first and fourth part are involved in the computation nor the third one. Moreover, as in the computation ofρ⁰(s_H,γ)the second part ofgw_H gives1.

The Stabilizer Case

In this paragraph, we extend the results of Section 2 to the braid group. Namely, sincep:B/[P, P]→W is a surjective homomorphism, the classical isomorphism theorems give the following short exact sequence

1 //p⁻¹(CH) ^j //p⁻¹(NH) ^rp //Uf_H //1 which gives rise to the following exact sequence (Lemma 3)

p⁻¹(C_H)^{ab j}

ab //p⁻¹(N_H)^ab(rp)

ab//UfH //1

and Proposition 6 extends to the braid group in the following way.

Proposition 24 Abelianization in the braid group. If the orbits of the hyperplanes ofH underN_H and C_H are the same, the mapj^ab is injective.

Moreover under this hypothesis,p⁻¹(NH)^abis the free abelian group with basiswgH,(s_C)^abforC ∈HH⁰/CH

and(s^e_C^C)^abforC ∈(H rHH)/CH.

Proof. From Lemma 3, it is enough to show that every linear character ofp⁻¹(C_H)with values inC^× extends top⁻¹(NH). But the group of linear characters ofp⁻¹(CH)is generated by theexp(zϕ_C)forz∈CandC an orbit ofH underCH. So it suffices to show thatϕ_C extends top⁻¹(NH).

Since the orbits ofH underCH andNH are the same, then for everyC ∈H/CH, there existsn∈N^∗ such thatα_Cⁿ is invariant underNH (see Lemma 14 for the definition ofα_C). Then Remark 17 shows that

ψ_C :γ∈p⁻¹(N_H)7−→ 1 n

Z

αCn◦γ

dz z ∈Q is a well defined linear character ofp⁻¹(NH)extendingϕC.

Proposition 15 applied toCH ensures us thatp⁻¹(CH)^ab is the free abelian group generated by(s^e_{H}^{H})^ab, (s_C)^abforC ∈HH⁰/C_H and (s^e_C^C)^abforC ∈(H rHH)/C_H. Moreover, we havegw_H^fH ∈p⁻¹(C_H)and, thanks

to Remark 17,

ϕ_{H}(wgH f_H

) = 1 fH

ρ⁰(wgH f_H

) = 1

We then deduce that the family gw_H^fH,(s_C)^abforC ∈HH⁰/C_H and(s^e_C^C)^abforC ∈(H rHH)/C_H is a basis for p⁻¹(C_H)^ab. The short exact sequence

1 //p⁻¹(C_H)^{ab j}

ab //p⁻¹(N_H)^ab(rp)

ab//UfH //1

gives the result.

Remark 25 Comparison of orbits. In this remark, we give a list of the hyperplanes for which the orbits of hyperplanes underNH andCH are not the same. Of course, we find again in this list the hyperplanes of Remark 7 but we have to add some others.

Let us consider the infinite series (see Section 5 for notations). WhenH =Hi, the orbits underNH andCH

are always the same except whenr= 3andeis even and when r= 2ande>3. IfH =Hi,j,ζ, the orbits under NH andCH are the same whendeis even andr6= 3or whenr= 3ande∈ {1,3} or whenr= 2andd=e= 1.

For the exceptional types,G25is the only case where the commuting orbits under NH andCH are not the same. The only exceptional types where the non commuting orbits underNH andCH are not the same areG4, the second (named afterGAP) class of hyperplanes ofG₆, the first (named afterGAP) class of hyperplanes of G₁₃and the third (named after GAP) class of hyperplanes ofG₁₅.

4 An Application to Cohomology

In this section, we apply the preceding constructions and results to obtain a group cohomology result.

Specifically, the derived subgroup ofP is normal inB, so we obtain the following short exact sequence

1 //P/[P, P] ^j //B/[P, P] ^p //W //1 (3)

which induces a structure ofW-module onP^ab. By a classical result on hyperplanes arrangements (see [10] for example), theW-moduleP^abis nothing else that the permutation moduleZH and this section describes the extension (3) as an element ofH²(W,ZH)using methods of low-dimensional cohomology.

The rationale breaks down into three steps and each step consists of a translation of a standard isomorphism between cohomology groups in terms of group extensions.

(i) We decompose H into orbits underW : H =t C and uses the isomorphism H²(W,ZH) = M

C∈H/W

H²(W,ZC)

(ii) In each orbit, we set a hyperplaneH_C and thenZC= Ind^W_NC(Z)whereN_C is the stabilizer ofH_C. Shapiro’s lemma (see [5, Proposition III.6.2]) then gives us

H²(W,ZH) = M

C∈H/W

H²(N_C,Z)

(iii) The short exact sequence 0→Z→Q→Q/Z→0 ofN_C-modules gives a long exact sequence in cohomology.

Since|N_C| is invertible inQ, we have H¹(N_C,Q) =H²(N_C,Q) = 0and so we obtain the isomorphism H²(N_C,Z) =H¹(N_C,Q/Z)and

H²(W,ZH) = M

C∈H/W

H¹(N_C,Q/Z) = M

C∈H/W

Hom_gr.(N_C,Q/Z). The results of this section are the following proposition and corollary.

Proposition 26 Description. Under the isomorphism H²(W,ZH) = M

C∈H/W

Hom_gr.(N_C,Q/Z)

the extension (3) corresponds to the family(r_C :N_C→Q/Z)_C∈H/W wherer_C is the natural linear character of N_C (we identifyUf_HC with a subgroup of Q/Zvia the exponential map).

The next corollary is a trivial consequence of Proposition 26 and generalizes a result of Digne [7, 5.1] for the case of Coxeter groups.

Corollary 27 Order in H²(W,ZH). Since the order of rC is fHC, we deduce that the order of the extension (3) isκ(W) =lcm(fHC,C ∈H/W)(this integer κ(W)was first introduced in [9]).

The rest of the section is devoted to the proof of Proposition 26 : one subsection for each of the three steps.

4.1 First step : splitting into orbits

The isomorphism

H²(W,ZH) = M

C∈H/W

H²(W,ZC)

is simply given by applying the various projectionsp_C :ZH →ZC to a2-cocycle with values inZH where p_C : X

H∈H

λHH 7−→ X

H∈C

λHH .

To give a nice expression of the corresponding extensions, we need the following lemma.

Lemma 28 Extension and direct sum. LetGbe a group,X =Y ⊕Z a direct sum of G-modules and 0 //X ^u //E ^v //G //1

an extension ofGby X. We denote by q: X→Y the first projection andϕthe class of the extension E in H²(G, X). The extension associated toq(ϕ)is

0 //Y //E/Z //G //1

Proof. Let us denote byθ:E→E/Z the natural surjection andi:Y→Y ⊕Z the natural map. Let us first remark that Z is normal in E sinceZ is stable by the action ofG. Sincev is trivial onZ, then it induces a group homomorphismev:E/Z→Gwhose kernel isX/Z=Y. Thus the sequence

0 //Y ^θui //E/Z ^ev //G //1 (4)

is an exact one.

Ifs:G→Eis a set-theoretic section ofv, thenθsis a set-theoretic section ofev. The expression of a2-cocycle associated to an extension in terms of a set-theoretic section gives the result.

ForC ∈H/W, we denote byB_C the quotient group BC =B/h[P, P], sH,γe_H

, H /∈ Ci

Lemma 28 tells us that the extension (3) is equivalent to the family of extensions

0 //ZC ^jC //B_C ^pC //W //1 (5)

forC ∈H/W.

4.2 Second step : the induction argument

In each orbit C ∈ H/W, we choose a hyperplane H_C ∈ C and write ZC = Ind^W_NC(Z) where N_C ⊂W is the stabilizer ofH_C. Shapiro’s isomorphism lemma [5, Proposition III.6.2] shows thatH²(W,ZC) =H²(N_C,Z).

Exercise III.8.2 of [5] tells us that in term of2-cocycles Shapiro’s isomorphism is described as follow S: (ϕ:G²→ZC)7−→(fC◦ϕ:NC2

→Z) wheref_C :ZC →Zis the projection onto theH_C-component.

Decomposing Shapiro’s isomorphism into the following two steps

(ϕ:G²→ZC)7−→(ϕ:N_C²→ZC)7−→(f_C◦ϕ:N_C²→Z),

allows us to interpret it in terms of group extensions. Exercice IV.3.1.(a) of [5] gives a description of the first step : the corresponding extension is given by

0 //ZC //pC−1(NC) //NC //1

sincep_C⁻¹(N_C)is the fiber product ofB_C andN_C overW. Moreover, sincef_C is a split surjection as aN_C-module map, Lemma 28 gives us the following extension

0 //Z //p_C⁻¹(N_C)/hsH,γeH, H ∈ Cr{H_C}i //N_C //1 Finally, the extension (3) is equivalent to the family of extensions

0 //Z //B_C^{0 pC} //NC //1 (6)

whereB_C⁰ =p⁻¹(N_C)/h[P, P], sH,γe_H, H 6=H_Ciand C ∈H/W.

4.3 Third step : linear character

For the third step, we use results and notations of Section 2 and Section 3. Let us consider the group homomorphismρC :p⁻¹(NC)→Qof Definition 19. Since it is trivial onh[P, P], sH,γe_H, H6=HCi, it induces a group homomorphism fromB_C⁰ toQstill denoted byρC. Moreover, sinceρC(sHC,γe_HC) = 1, Proposition 23 gives the following commutative diagram

0 //Z //Q //Q/Z //0

0 //Z //B⁰_C ^pC //

ρC

OO

N_C //

rC

OO

1

Exercises IV.3.2 and IV.3.3 of [5] tell us precisely that the group homomorphism corresponding to (6) isr_C. So the extension (3) is equivalent to the family(r_C)_C∈H/W. This concludes the proof of Proposition 26.

5 Tables

5.1 The infinite series

In this subsection, we bring together tables for the orbits of the hyperplanes ofG(de, e, r)under the centralizer of a reflection and under the parabolic subgroup associated to the line of the reflection and tables for the values of fH and the index of ramification. So let us consider the complex reflection groupG(de, e, r)acting onC^r with canonical basis(e1, . . . , er). The standard point ofC^ris denoted by(z1, . . . , zr).

The hyperplanes of G(de, e, r) are Hi ={zi = 0} fori∈ {1, . . . , r} (whend >1) andHi,j,ζ ={zi =ζzj} fori < j andζ∈Ude(whenr>2). They split in general into two conjugacy classes underG(de, e, r)whose representant may be chosen as followH1 andH1,2,1.

Let us continue with more notations. For every triple of integersd, e, r, we denote byπ:G(de, e, r)→Uthe following group morphism : forg∈G(de, e, r),π(g)is the product of the nonzero coefficients of the monomial matrixg. Wheneis even, we denote bye⁰=e/2. We denote bye⁰⁰=e/gcd(e,3)and byP the set of elements ofUdewith strictly positive imaginary part.

The case of the hyperplane H

= {z

= 0}

We then have d >1. The stabilizerN ofH₁ is described by

N ={(α, g), g∈G(de,1, r−1), α∈U^de, (π(g)α)^d= 1}

and the pointwise stabilizer C of D1 =H1⊥ = Ce1 is C = G(de, e, r−1). Table 1 gives the orbits of the hyperplanes underN and C. In Table 1, C.O. stands commuting orbits and N.C.O. stand for non commuting orbits.

The hyperplane H

with r = 2

We set ζ= exp(2iπ/de). The reflection of G(de, e,2) with hyperplaneH1,2,ζ is s=

ζ ζ⁻¹

The line ofsisD=C(e₂−ζe₁). The centralizer ofsis given by N=

dλ=

λ λ

, tλ=

λζ

λζ⁻¹

, λ^2d= 1

.

The eigenvalue oftλ onDisλwhereas the eigenvalue ofdλonD is−λ. So the parabolic subgroupCassociated toD isC={id,−s}. The orbits of hyperplane underC andN are the same. The commuting ones are{H1,2,ζ}

and{H_1,2,−ζ}. The non commuting ones are{H1, H2} and{H1,2,µ, H_1,2,ζ2µ⁻¹}forµ∈Uder{±ζ}.

The hyperplane H

= {z

= z

}

We then haver>2. The reflection ofG(de, e, r)with hyperplaneH1,2,1is the transposition τ12 swapping1 and2. Since the elements ofG(de, e, r)are monomial matrices, an element of G(de, e, r)commuting withτ12

stabilizes the subspace spanned bye1 ande2. Thus the stabilizerN ofH1,2,1 is given by

N =

d_λ,g =

"λ λ

g

#

, t_λ,g =

" λ λ

g

# ,

λ∈Ude, g∈G(de,1, r−2), (π(g)λ²)^d= 1

The line ofτ₁₂ is C(e₁−e₂). So the eigenvalue of d_λ,g on C(e₁−e₂) is λwhereas the eigenvalue of t_λ,g on C(e₁−e₂)is−λ. Thus, whendeis odd, the parabolic subgroup associated to the lineC(e₁−e₂)is given by

C= ("1

1 g

#

, g∈G(de, e, r−2) )

and whendeis even, the parabolic subgroup associated to the lineC(e1−e2)is given by

C= ("

1 1

g

# ,

"

−1

−1 g

#

, g∈G(de, e, r−2) )

Table 2 gives the orbits the hyperplanes ofG(de, e, r)underN andC. In Table 2, C.O. stands commuting orbits and N.C.O. stand for non commuting orbits.

Value for f

and the index of ramification

The computations of the preceding paragraphs also lead to Table 3 which brings together the values ofeH, fH

anddH for every class of hyperplanes. In Table 3, we setζ= exp(2iπ/de).

We obtain the following errata for the proposition 6.1 of [9]. Let us consider r>2. ForW =G(de, e, r), we haveκ(W) = 2deifdeis odd andr>3. We haveκ(W) =deif (d6= 1andr= 2anddeeven) or (r>3 andde even). We haveκ(W) = 2deif (d6= 1andr= 2 anddeodd). We have κ(W) = 2ifd= 1andr= 2¹.

5.2 Exceptional types

With the package CHEVIE of GAP [8][6], we obtain Table 4 for the values of e_H, f_H andf_H/e_H for the hyperplanes of the exceptional reflection groups. In particular, the only non Coxeter groups with only unramified hyperplanes areG8, G12andG24. Table 4 can also easily be obtained from the table of [9] for the value ofκ(W).

The first and fifth columns stand for the number of the group in the Shephard and Todd classification. We also write instructions which determines, for a given hyperplaneH, the orbits of commuting and non commuting hyperplanes underNH andCH which is used to obtain the results of Remark 7 and Remark 25.

1Comparing to the preceding versions, we correct here the value ofκ(W)fordeodd andr= 2. This was pointed out by Ivan Marin.

NH1CH1 r>4C.O.H1H1 {Hi,i>2}{Hi,i>2} {Hi,j,ζ,i6=j>2,ζ∈Ude}{Hi,j,ζ,i6=j>2,ζ∈Ude} N.C.O.{H1,j,ζ,j>2,ζ∈Ude}{H1,j,ζ,j>2,ζ∈Ude} r=3andeoddC.O.H1H1 {H2,H3}{H2,H3} {H2,3,ζ,ζ∈Ude}{H2,3,ζ,ζ∈Ude} N.C.O.{H1,i,ζ,i=2,3,ζ∈Ude}{H1,i,ζ,i=2,3,ζ∈Ude} r=3andeevenC.O.H1H1 {H2,H3}{H2,H3} {H2,3,ζ,ζ∈Ude}{H2,3,ζ,ζ∈c}forc∈Ude/Ude0 N.C.O.{H1,i,ζ,i=2,3,ζ∈Ude}{H1,i,ζ,i=2,3,ζ∈Ude} r=2andeoddC.O.H1H1 H2H2 N.C.O{H1,2,ζ,ζ∈Ude}{H1,2,ζ,ζ∈c}forc∈Ude/Ud r=2andeevenC.O.H1H1 H2H2 N.C.O.{H1,2,ζ,ζ∈c}forc∈Ude/Ude0{H1,2,ζ,ζ∈c}forc∈Ude/Ud r=1C.O.H1H1 Table1:OrbitsofhyperplanesunderNH1andCH1