Internat. J. Math. & Math. Sci. Vol. I0 No. 1 (1987) 147-154 147 ON THE AFFINE WEYL GROUP OF TYPE A n_ MUHAMMAD A. ALBAR Department of Mathematical Sciences University of Petroleum and Minerals Dhahran, Saudi Arabia (Received April 4, 1985 and in revised form March 26, 1986) n-l’ We study in this paper the affine Weyl group of type [I]. Coxeter [I] is a split showed that this group is infinite. We see in Bourbaki [2] that extension of S n, the symmetric group of degree n, by a group of translations and of is one of the crystallographic Coxeter groups considered a lattice of weights. ABSTRACT. n-I #n-I by Maxwell [3], [4]. We prove the following" THEOREM ]. THEOREM 2. The group 4. .ength and so S is a split extension of >3 n-l’ n by the direct product of n (n-l) Z. copies of n-] THEOREM 3. 2 3 is soluble of derived is soluble of derived length 3, coincides with the first n-] the second derived group For n 4, is not soluble for n The center of n-] 1 An_ 4. > n is trivial for > 3. KEY WORDS AND PHRASES. Presentation, Reidemeister-Schreier method, Coxeter group. 1980 AMS SUBJECT CLASSIFICATION CODE. 20F05. 1. INTRODUCTION. Consider the presentation An_ <Yl, 2 YnlYi Y2 yiYi+lYi YiYj Y lYnY where n > e Yi+lYiYi+l YjYi if <_i if <i <n, n-l, if < j-I n and (i,j)# (l,n), Yny zYn > 3. n vertices. Using is infinite [4]. This group is some geometrical methods Coxeter showed that An_ We see in Bourbaki also a Weyl group [I]. It is the affine Weyl group of .type is a split extension of S n, the symmetric group of degree n, by a [2] that This is an irreducible Coxeter group whose graph is a polygon with n-I n-l" M.A. ALBAR 148 group of translations and of a lattice of weights. This group was also considered by Maxwell [3], [4]. is a split extension of S by The purpose of this paper is to prove that n a direct product of (n-l) copies of Z. The method depends on presentations of group is soluble extension [5]. We also find that A3 is soluble of derived length 3, coincides with the first of derived length 4 and that the second derived group An_ n-I if n 4 and hence ,n_ is not soluble in this case. We finally show that the center of n-I is trivial. n-I 4 THE STRUCTURE OF n-l" We show in this section that An_ is a split extension of S n by the direct product of (n-l) copies of Z. We achieve this by using the method in [5] as folS n such that the extension lows. We find an epimorphism e" An_l 2. kero n - (21). S n --+I splits. It will be required to find a presentation for kere. We guess that it will be isomorphic to A z x(n-l) (given by generators and relations). We then construct a new short exact sequence (2.3), where A is embedded as normal subgroup of a group E G. in ,such a way that A is the kernel of an epimorphism o’. E (2.2) G.. kere ’ ’ A E , G , (2.3) Then we use Tietze transformations to identify E with i.e., to find an isomor[ which makes the right-hand square commute. It then follows that phism A kero. A presentation for the symmetric group of degree n >_ 2 is " Sn x n-1 x <X 2 e <i <n-l, if xixi+ix xi+ixixi+ xix j xjx if e. We define the mapping e. n-I e " xi then Yi j-I < n-l>. by Xn_2Xn_iXn_2 is an epimorphism. < n-I if Yi---+ xi Yn---+ xlx2 Then Sn <_ n-2, if x2x1. is the mapping from If <n-l, if e is a homomorphism and ISn. Thus the extension 1-- kere n-I T S splits. n We construct the short exact sequence ---+A---+E---+Sn---+ I. Sn to n-I defined by AFFINE WEYL GROUP A presentation of E Let A E will be generators of <generators of A, relations of A, relations action of Sn on A> [6]. an_ laiak aka <a We define the action of S n xl a a -I x a a -I a a xi k { 149 if n k <_ (2.4) n-l> A as follows" on (2.5) 2<i if S of (2.6) <n-I k+l, k, ak+l if ak_ if ak otherwise k (2.7) (2.8) (2.9) n-l n-l 2 yxy and We let A x i. We also denote the relations xyx x2x by (x,y) and [a,b] respectively. To reduce the relations of E to a manageable form we consider the following NOTATION. ab ba lemma and proposition. LEMMA I. In the group the following identities hold" (i) AkX xi+iA k (ii) &kXi &kXi Ak_ if k &k+l xiA k if k+l if k+l (iii) (iv) (v) (vi) PROOF. Sn (i) AkX AkA 2 Ai AkX if 2 <_ X3... Xi+iAk Xi Ai -I X3 x2x x2 if k 2 <_ xi_ixixi+ xi_ixixi+ix x2 k XkX xk xi_ixi+ixixi+ xk Xi+l& k (ii) to (iv) obvious. (v) and (vi) application of (i). PROPOSITION I. (i) In the group E, Relation (2.5) relations (2.4) to (2.9) become the following- is equivalent to (alXl) 2 e. A (ii) (iii) (iv) Relation (2.7) is equivalent to Relation (2.6) is equivalent to 2 <_ a <_ n-l. (alxl,x2). Relation (2.8) follows from (ii). (v) Relation (2.9) is equivalent to (vi) a Relation [a,xi] (2.4) is equivalent to (x2al) for 2 3 <_i <_n-l. (alx2) 2. M.A. ALBAR 150 (i) PROOF. (ii) (iii) Obvious Easy by induction on i. Using part (ii) relation (2.6) becomes XlAilaIAiX aIAilaiA Using relation (2.9) it reduces to (iv) (alx l, x2). Obvious by using part (ii). (v) (2.9) becomes Using part (ii) relation -I AkXiA k If > -I al then by Lemma k+l, [x i, a] for 3 < [xi+ I, a l] for 2 <_ (iv) we get n-l. (i), we get then by Lemma k If # k+l. k, azAkXiA k <n-l. Therefore relation (2.9) is equivalent to 3<i<n-l. (vi) Using part (ii) relation -I a AkAilaiAiAk (2.4) becomes l<i azka aliA n-I Sn and so Then n- is a split extension E has the following generators" S n, b2=e, (2.o) (b, x2) . [bx I, x i] for 3 <_ (xbx) k _<_ n-lo (alx2) 2. 2 THEOREM I. The group E is isomorphic to of 3. by A where n PROOF. In Proposition I, we let alx b. xl, x2 Xn_l, b. Relations of E are" Relations of for (v) and relation (2.9), we get Using Lemma (x2al) [a l, x i] (bxx) (2.11) (2.12) <_ n-I (2.13) We change relation (2.13) to the form (b, XlX2Xl). (2.14) We change relation (2.12) to [b, x i] We let c -! 2 n iban_ I. Then c Using relation (2.11) and Lemma a Xn-I CXn-I An-! IbAn-I for 3 <_ n-I e. (i), we get (c, x). (2.15) AFFINE WEYL GROUP 151 (ii) and (v) and (2.15) Using Lemma Using Lemma _ -2 n_IbAn_ CXn_iC (vi) -_ 2bn_2 an CXn_lC Xn_lCXn_ 1. (c, Xn_l )" Using Lemma (i) and (2.15) we get Therefore [c, x i] for 2 <_ E Thus n-l. has the following presentation E <x Xn_ c Ix 2 C2 e for =e, (x i, xi+ I) [x i, x k] (Xn_ I, <_i <n-2 for x n. c REMARK I. Then it is clear that c), (x I, c), E S3- 2 a(3, 3, 3) <n-l>. n-I is the same as 0 We notice the special cases i n-l, k-I for [x i, c] for 2 Let <n-l, < Z2. $2 (3, 3, 3) the triangle group and the theorem is proved. [6]. REMARK 2. We used the Reidemeister-Schreier process to find A kere for n=3, 4. From the computations involved we found the action of S n on A. For n >_ 5, we guessed that A Z x(n-l) and the action is a generalization for the case when n =3,4. We then proved this guess by the method in [6]. 3. THE DERIVED SERIES OF n-l" We prove in this section the following theorem" 3 4 is soluble of derived The group is soluble of derived length 3, ~ii with the first An_ coincides 4, the second derived group An_ length 4. For n is not soluble for n > 4. and so Io prove the theorem we consider the derived series of An_ I. We notice that THEOREM 2. n-I n-l A’n-l <ylly>. Hence An_ is a transversal for {e, yl} in n-l" Reidemeister-Shreir process we find the following presentation for #’n-I <b z, b 2 bn_ Ib] b 2. -I (bibi+ I) (bibl)2 b e e n-I if if if <_ <_ < < n-2 <_n-2, <j-I < n-l>. Using the ’ n-l M. A. ALBAR 152 We now consider the following A2 i) n If b21b <bl, 3, cases" A2 [bl, b2] b2 e>. A Using the Reidemeister-Schreier process we find that is soluble of derived length 3. Therefore z. z 2 ii) n If A3 4 <bllb A We use the Reidemeister-Schreier process e>. 4 to find the following presentation for tlx 2 <x,y z A4 y2 xytz xtzy <x,y z t lx 2 y2 " e e>o t2 z2 [y,t] [x,z] t2 z2 [x z] [x,y] [z,t] [y,t] [y,x] [x t] e>. We use the Reidemeister-Schreier process to find that is soluble at derived length 4. 4 iii) n If > 4 An-I A" So the second derived group is trivial A" An_ I. with first derived group 4. Hence An_ z. z " Therefore coincides n-I 4. n is not soluble for THE CENTER OF n-l" We prove in this section that the center of LEMMA 2. The identity of An_ is trivial for n A is the only element fixed by S n- PROOF. We let w be an element of A. We can write w in the fore x m ml m2 w for !j n-l. Let w where mj E Z for a a2 We therefore get the equation annI_, [ mi m2 a a2 xi mI m a I_m2 ann, a2 on A Using the action of S n 2m2+...+ mn_ a mn-I an_ (4 I) [in Section 2] equation (4.1) for implies :e. A is free abel ian this equation gives + 2m + m 2 + Using the action of m -mi_ ai_ Since !i !n-l. n-l. for Since 3. a mn_ S n on A, m -mi_ (4.2) O. equation (4.1) for 2 <i <n-I implies A is free abelian this gives 0 for 2 <i <n- I. mi_ From (4.2) and (4.3) we get ml m2 (4.3) m mn_ O. Therefore w e as required. AFFINE WEYL GROUP THEOREM 3. The center of is trivial for n-I n > 153 3. We know that PROOF. ---+A---+/n_l --+Sn x Z(n_ I) so x as I. where a A and s S n. We let x be a als the Hence xx xlx implies asazs typical element of Applying alssas. ={e}. we Hence epimorphism get e(s)e(sl) e(s)e(s) and so e(s) E Z(S e. Therefore x s s a coutes elementwise with kere A n S n Using {e}. Lemma 3, a e and so We let n-l" n) Sn. Z(n_ I) REMARK 3. From Remark we notice that n-I A REMARK 4. We notice that Sn Z( o) from Theorem I. 2 of length 3 and 4 respectively, we get that 4 respectively. S n is not soluble for n > 4 is not soluble for n 4. n-I Z() Z and 3 and {e}. S4 are soluble are soluble of length 3 and is soluble, it follows that n-I REMARK 5. S Since and and A Z(S 3) One way to view is as a subgroup of the wreath product Z S n deLet Z xn be the free abelian group with base Po on which S acts the basis, x by (i-I, exchanges and pemuting i), n Pi Pi-I k and fixes the others. The subgroup {P0 O} H is Sn-invariant, fined as follows" Pn-I j=okn and has basis {ai and An_ is just this split extension n-I Pll of S n by H. Therefore is the subgroup of the natural wreath product of Z S S n consisting of those elements in which the component from the base group has exponent sum zero. Pi n-I n-I REMARK 6. The motivation behind studying this group about the circular braid group [7]. We see that responding group B n. Consider the diagram to,he n Ain $ IYF n-I was to get some infomation is the Coxeter group cor- Zx(n-l) 111 nn-I n n U B gn s Artqn’s brad Sn rou [6], n the unermuted brad rou, a free x( 1) as described n ths aer. non of countably nfnte rank [7] and Z Z x(n-1) dd not hel us to descrqbe the structure of hch as described n a dfferent ay [7]. e are stql] unable t fnd the rous X and 8ere gn ould ]ke to thank r. avd k. dohnson for hs helpful suggesalso thank the nversty f Petro]aum and Mnerals for suort et for conductn research. CNOkgg[N[NI" tions. 154 M.A. ALBAR REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. CARTER, R.W. Simple Groups of Lie Type, Wiley, London-New York, 1972. BOURBAKI, N. Groupes et algebres de Lie Type,. Chap. IV-VI, Herman, Paris, 1968. MAXWELL, G. The Crystallography of Coxeter Groups, J. Algebra 3__5(1975), 159-177. MAXWELL, G. The Crystallography of Coxeter Groups II, J.. Algebra 44(1977),290-318. ALBAR, MUHAMMAD. 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