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
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I.
2.
3.
4.
5.
6.
7.
8.
9.
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