The Coxeter group Dn

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Vol. 24, No. 10 (2000) 715–720 S0161171200002167 © Hindawi Publishing Corp.

THE COXETER GROUP

D

n

M. A. ALBAR and NORAH AL-SALEH

(Received3 September 1998 andin revisedform 16 November 1998)

Abstract.We show that the Coxeter groupDnis the split extension ofn−1 copies ofZ2

bySnfor a given action ofSndescribedin the paper. We also ﬁndthe centre ofDnand

some of its other important subgroups.

Keywords and phrases. Coxeter group, split extension.

2000 Mathematics Subject Classiﬁcation. Primary 20F05.

1. Introduction. The Coxeter groupDn[5] has the presentation

Dn=x1,x2,...,xn|xi2=e,1≤i≤n;

xixi+13=e,2≤i≤n−1;

xixj2=e,|i−j|≠1 and(i,j)≠(1,3); x1x22=x1x33=e.

(1.1)

andthe graph given in Figure 1.1.

1

2 3 4 n−2 n−1 n

...

Figure1.1.

In [1], we have shown thatD4is solvable with derivedlength 4 andthat its order is 192. In this paper, we explain the algebraic structure ofDnandﬁndits centre. We also ﬁndthe derivedseries; andthe growth series ofDnfor 4≤n≤8.

2. The structure ofDn. In [4], we have shown that the Coxeter groupBn whose graph is given in Figure 2.1

1 2 3 4 n−1 n

...

Figure2.1.

is the wreath product ofZ2bySn, that is,Bnis the split extension ofZ2nbySn. LetZ2n have the presentation

H=Zn 2 =

a1,a2,...,an|ai2=e,1≤i≤n;

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LetKbe the even subgroup ofH, that is, the subgroup consisting of even words in

H. It is easy to ﬁndthe following presentation forK:

K=b2,b3,...,bn|bi2=e,2≤i≤n;

bibj2=e,2≤i < j≤n, (2.2) wherebi=a1ai,2≤i≤n. ThusKisZ2n−1. In the extensionBnZ2nSn, the action of

SnonZ2nis a natural one that can be explainedas follows. LetSnhave the presentation

Sn=x1,x2,...,xn−1|xi2=e,1≤i≤n−1;

xixi+13=e,1≤i≤n−2;

xixj2=e,1≤i < j−1≤n−2,

(2.3)

wherexiis the transposition(ii+1). The action ofSnonHis given by

a1,a2,...,ai−1,ai,ai+1,...,anxi=a1,a2,...,ai−1,ai+1,ai,...,an. (2.4) Using this action we compute the action ofSnonKas follows:

b2,b3,...,bnx1=a1a2,a1a3,...,a1anx1 =a2a1,a2a3,...,a2an =b−1

2 ,b−21b3,...,b−21bn,

b2,b3,...,bi,bi+1,...,bnxi=a1a2,...,a1ai,a1ai+1,...,a1anxi =a1a2,...,a1ai+1,a1ai,...,a1an =b2,...,bi+1,bi,...,bn, 2≤i≤n−1.

(2.5)

We use this action to construct a split extensionEofK=Zn−1

2 bySn. A presentation for this extension is given by the methodin [2],E= generators ofK, generators of

Sn|relations ofK, relations ofSn, the action ofSnonK. We change the action ofSnonKto the following relations:

bx1

2 =b2−1, (2.6)

bx1

i =b−21bi, 3≤i≤n, (2.7)

bxi

i =bi+1, 2≤i≤n−1, (2.8)

bxi

i+1=bi, 2≤i≤n−1, (2.9)

bxi

j =bj, 2≤j≤n,2≤i≤n−1, j≠i, j≠i+1. (2.10)

We will use Tietze transformations to show thatEis isomorphic toDn. But before that we observe the following. The relation (2.8) implies thatbi=b2x2x3···xi−1, 3≤i≤n. Letui=x2x3···xi.

Lemma2.1. The following identities hold in the groupSn−1: (i) ukxi=xi+1uk,if2≤i≤k,

(ii) ukxi=uk−1,ifi=k, (iii) ukxi=uk+1,ifi=k+1, (iv) ukxi=xiuk,ifi > k+1,

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Proof. We will make use of the relations ofSn−1.

(i)

ukxi=x2x3···xixi+1···xkxi =x2x3···xixi+1xi···xk =x2x3···xi+1xixi+1···xk =xi+1x2x3···xixk=xi+1uk.

(2.11)

(ii), (iii), and(iv) are clear, while (v) and(vi) are applications of (i) to (iv). We reduce relation (2.6) to (2.10) as follows. Relation (2.6) easily becomes

x1b22=e. (2.12)

Using Lemma 2.1, (2.7) becomes

b2x1x23=e. (2.13)

Usingbi=bx22x3···xi−1, (2.8) becomes redundant. Relation (2.9), using Lemma 2.1, be-comes redundant. Using Lemma 2.1, (2.10) bebe-comes

xib22=e, 3≤i≤n. (2.14)

The relationb2

i =e=(bibj)2become redundant fori≥3. Letc=x1b2. Then (2.12) be-comes c2 ₌ _e_{. Relation (2.13) becomes} _(cx₂₎3 ₌ _e_{. Relation (2.14) becomes}

(cxi)2=efori≥3. The relationb22=ebecomes(cx1)2=e. Therefore a presentation forEis

E=x1,x2,...,xn−1,c|xi2=e,1≤i≤n−1; c2=e;

xixj2=e,1≤ i < j−1≤n−2;xixi+13=e,1≤i≤n−2;

cx23=e;cxi2=e,1≤i≤n−1 andi≠2.

(2.15)

Consequently, we have provedthe following theorem.

Theorem2.2. The groupDnis the split extension ofn−1copies ofZ2bySn. Remark2.3. Let us describe the relation between the Coxeter groupsBnandDn. It is easy to show thatDn is the subgroup ofBn consisting of all elements of even length inBn. ThusDn is a subgroup ofBn of index 2. On the other hand, letSn be the symmetric group of degreengeneratedbyyi,1≤i≤n−1. We consider the map

θ : Dn→Sndeﬁned byθ(x1)=y2, θ(x2)=y2, andθ(xi)=yi−1, 3≤i≤n. Using the Reidemeister-Schreier process, it is possible to show thatθ−1_(S

n−1)Bn−1. Hence Bn−1is a subgroup ofDnof indexn. We observe the graph given in Figure 2.2.

The orders ofBnandDnare|Bn| =2nn! and|Dn| =2n−1n!, respectively. It is also easy to see thatB

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Bn

2

Dn

n

Bn−1

2

Dn−1

.. .

B4

2

D4

4

B3

Figure2.2.

3. The derived series of Dn. In [1], we showedthat D4 is solvable of derived length 4. For n >4 we use the Reidemeister-Shreier process to ﬁnd the following presentation forD

n:

D

n=b2,b3,...,bn|b22=b33=b2i=e,4≤i≤n;

bibi+13=e, 2≤i≤n−1; bibj2=e,2≤i < j−1≤n−1.

(3.1)

The groupD

n/Dnis trivial. HenceDnis a complete group. We thus have the following theorem.

Theorem3.1. Dnis solvable of derived length 4 ifn=4. Ifn >4, thenDn is not solvable.

4. The centre ofDn. We use the structure ofDn explainedin Section 2 to prove the following theorem.

Theorem4.1. The centre ofDnisZ2ifnis even and trivial ifnis odd.

Proof. In Section 2, we showedthatDnis the split extension ofZ2n−1bySn. This means the existence of an epimorphismθ:Dn→Sn, where kerθ=Z2n−1. It follows thatθ(Z(Dn))⊆Z(θ(Dn))=Z(Sn)= {e}. HenceZ(Dn)⊆kerθ=zn2−1. We use the previous notation whereSn = x1,x2,...,xn−1 and Z2n−1= b2,b3,...,bn andthe previous action. We letw∈Z(Dn)⇒w∈Z2n−1andwxi =wfor 1≤i≤n−1. Also

w=b2

2b33···bnn, wherei=0 or 1 sinceb2i =e. Using the action ofSn onZ2n−1, we get

b2

2 b33···biibi+i+11···bnn xi

=b2

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Letting 2≤i≤n−1, we get

b2

2b33···bi+i1bii+1···bnn=b22b33···biibi+i+11···bnn, (4.2) andsobi

i+1bii+1=biibi+i+11. This implies i=i+1andso2=3= ··· =n. Hence

w=b2b3···bnorw=b20b30···b0n=e. Now, we consider the action ofx1onwin the following two cases:

(a) Ifnis even, we get(b2b3···bn)x1=b2n−2b2b3···bn=b2b3···bnsinceb22=e. Henceb2b3···bnis in the centre ofDn. Since(b2b3···bn)2=e, we getZ(Dn)=Z2.

(b) Ifnis odd,(b2b3···bn)x1=b3b4···bnandb2b3···bndoes not commute with

x1. Thuswisb0

2b03···b0n=eandZ(Dn)= {e}.

5. The growth series. The growth series, in the sense of Milnor andGromov, ofDn for 4≤n≤8 were computedas follows [3]:

γD4=(1+t)41+t221−t+t21+t+t2, (5.1)

γD5=(1+t)41+t221+t41−t+t21+t+t21+t+t2+t3+t4, (5.2)

γD6=(1+t)61+t221+t41−t+t221+t+t22

×1−t+t2₋_t3₊_t4₁₊_t₊_t2₊_t3₊_t4_, (5.3)

γD7=(1+t)61+t231+t41−t+t221+t+t221−t2+t4

×1−t+t2₋_t3₊_t4₁₊_t₊_t2₊_t3₊_t4₁₊_t₊_t2₊_t3₊_t4₊_t5₊_t6_, (5.4)

γD8=(1+t)81+t241+t421−t+t221+t+t22 ×1−t2₊_t4₁₋_t₊_t2₋_t3₊_t4₁₊_t₊_t2₊_t3₊_t4 ×1−t+t2₋_t3₊_t4₋_t5₊_t6₁₊_t₊_t2₊_t3₊_t4₊_t5₊_t6_.

(5.5)

We make two observations about these growth polynomials. First, each growth series is a product of cyclotomic polynomials. Second, the value of the series at 1 is the order of the corresponding group and the degree of the growth series equals the length of the element of maximal length.

We have not yet succeeded in ﬁnding the growth series ofDnfor generaln. Acknowledgement. The ﬁrst author (MAA) acknowledges KFUPM’s support.

References

[1] M. A. Albar,Analogues of the braid group and their corresponding Coxeter groups, Comm. Algebra12(1984), no. 23-24, 2977–2984. MR 86g:20041. Zbl 591.20046.

[2] ,On presentation of group extensions, Comm. Algebra12(1984), no. 23-24, 2967– 2975. MR 86g:20040. Zbl 551.20017.

[3] M. A. Albar, M. A. Al-Hamed, and N. A. Al-Saleh,The growth of Coxeter groups, Math. Japon.

47(1998), no. 3, 417–428. MR 99f:20066. Zbl 912.20032.

[4] M. A. Albar andN. A. Al-Saleh,On the aﬃne Weyl group of typeBn, Math. Japon.35(1990),

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[5] H. S. M. Coxeter andW. O. J. Moser,Generators and Relations for Discrete Groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and RelatedAreas], vol. 14, Springer-Verlag, Berlin, 1980. MR 81a:20001. Zbl 422.20001.

M. A. Albar: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia