Journal of Algebra 319 (2008) 1428–1449 Www.elsevier.com/Locate/Jalgebra

8/14/2019 Journal of Algebra 319 (2008) 14281449 Www.elsevier.com/Locate/Jalgebra

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Journal of Algebra 319 (2008) 14281449www.elsevier.com/locate/jalgebra

Presentation by conjugation for A1-type extended afneWeyl groups

Saeid Azam , Valiollah Shahsanaei

Department of Mathematics, University of Isfahan, PO Box 81745-163, Isfahan, IranReceived 9 October 2006

Available online 21 December 2007

Communicated by Em Zelmanov

Abstract

There is a well-known presentation for nite and afne Weyl groups called the presentation by conjuga-tion . Recently, it has been proved that this presentation holds for certain sub-classes of extended afne Weylgroups, the Weyl groups of extended afne root systems. In particular, it is shown that if nullity is 2, anA1-type extended afne Weyl group has the presentation by conjugation. We set up a general framework for the study of simply laced extended afne Weyl groups. As a result, we obtain certain necessary andsufcient conditions for an A1-type extended afne Weyl group of arbitrary nullity to have the presentationby conjugation. This gives an afrmative answer to a conjecture that there are extended afne Weyl groupswhich are not presented by presentation by conjugation. 2007 Elsevier Inc. All rights reserved.

Keywords: Weyl groups; Root systems; Presentations

0. Introduction

There are several well-known presentations for nite and afne Weyl groups [St,H,MP] in-cluding the Coexter presentation and the so-called the presentation by conjugation [MP]. Theelements of the Coexter presentation can be read from the corresponding (type dependent) Car-tan matrix while the elements of the presentation by conjugation are given uniformly for alltypes (see Denition 4.1). Our main focus in this work is on the presentation by conjugation forA1-type extended afne Weyl groups .

* Corresponding author. E-mail addresses: [email protected] (S. Azam), [email protected] (V. Shahsanaei).

0021-8693/$ see front matter 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2007.10.041


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S. Azam, V. Shahsanaei / Journal of Algebra 319 (2008) 14281449 1429

Extended afne Weyl groups (Denition 2.3) are the Weyl groups of extended afne rootsystems (Denition 2.1). Finite and afne root systems are nullity zero and nullity one extendedafne root systems, respectively. In 1995, Y. Krylyuk [K] showed that simply laced extended

afne Weyl groups of rank > 1 have the presentation by conjugation. He used this to establishcertain classical results for extended afne Lie algebras. In 2000, S. Azam [A3] proved that thepresentation by conjugation holds for a large subclass of extended afne Weyl groups, includingreduced extended afne Weyl groups of nullity 2.

In this work we obtain certain necessary and sufcient conditions for arbitrary nullity ex-tended afne Weyl groups of type A1 to have the presentation by conjugation. This has severalinteresting implications which are recorded in Section 5. In Section 1, we recall the denition of a semilattice from [AABGP] and introduce the notions of support , essential support and integralcollections for semilattices. These notions naturally arise in the study of extended afne Weylgroups and their presentations. In Sections 2 and 3, we record some important results mainly

from [AS1] which are crucial for the up coming sections, and establish a few preliminary resultsregarding extended afne Weyl groups.

In Section 4, we assign a presented group W to the Weyl group W of an extended afneroot system R (Denition 4.1) and study its structure in details. Following [K], we say the Weylgroup W has the presentation by conjugation if W = W . Since our setting has set up for thestudy of general simply laced extended afne root systems, the results of this section give inpart a simple and new proof of [K, Theorem III.1.14], namely simply laced extended afne Weylgroups of rank > 1 have the presentation by conjugation. In Section 5, the main section, wecompute the kernel of a natural epimorphism : W W and show that it is isomorphic to thedirect sum of a nite number of copies of the cyclic group of order 2 (Proposition 5.11). Thisthen allows us to prove that W has the presentation by conjugation if and only if the center of W is free abelian group, if and only if a certain integral condition (see Section 1 for denition)holds for the semilattice involved in the structure of R , if and only if a minimal condition holdson a particular set of generators for the Weyl group (Theorem 5.16). This reduces the problemof deciding which extended afne Weyl group has the presentation by conjugation to a niteproblem which can in principle be solved for any 0. Using this criterion it is then easyto show that if 3, the Weyl groups of all extended afne root systems of type A1 havethe presentation by conjugation except only one root system, namely the one of index 7 whichhappens in nullity 3 (Corollary 4.2). This in particular gives an afrmative answer to a conjecture

due to S. Azam [A3, Remark 3.14] that certain extended afne Weyl groups of type A1 may nothave the presentation by conjugation (see Remark 5.21).

1. Semilattices

In this section, we briey recall the denition of a semilattice from [AABGP] and recordcertain properties of semilattices which will be important for our work. We also introduce anotion of integral collections for semilattices which turn out to be crucial in the study of Weylgroups of extended afne root systems. For the details on semilattices the reader is referred to[AABGP, Chapter II.1]. In this section, we x several sets which will be used throughout thepaper.

Denition 1.1. A semilattice is a subset S of a nite dimensional real vector space V 0 suchthat 0 S , S 2S S , S spans V 0 and S is discrete in V 0 . The rank of S is dened to be the


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1430 S. Azam, V. Shahsanaei / Journal of Algebra 319 (2008) 14281449

dimension of V 0 . Note that the replacement of S 2S S by S S S in the denition givesone of the equivalent denitions for a lattice in V 0 .

Let S be a semilattice in V 0 . TheZ

-span of S in V 0 is a lattice in V 0 with a basis Bconsisting of elements of S . Namely

B = { 1 , . . . , } S with =

i = 1

Z i . (1.2)

We x this basis B . For a set J {1, . . . , } we put

J :=j J

j .

(If J = we have by convention j J j = 0.) By [AS1, 1], there is a unique subset, denotedsupp (S) , consisting of subsets of {1, . . . , } such that

supp (S) and S =J supp (S)

( J + 2). (1.3)

Note that since B S , we have {r } supp (S) for all r J . The collection supp (S) is called thesupporting class of S (with respect to B ). We also introduce a notion of essential support for S ,denoted Esupp (S) , namely

Esupp (S) = J supp (S) : |J | 3 .

Clearly we have

Esupp (S) 2 1 2 .

Following [A4], we call the integer ind (S) := | supp (S) | 1, index of S .We assign to each semilattice S certain collections of integers, called integral collections . This

notion will play a crucial role in the sequel.Let S be a semilattice of rank . For 1 r < s and J supp (S) we set

(r,s) =1, if {r, s } supp (S),2, if {r, s } /supp (S),

and (J ,r,s) =1, if {r, s } J,0, otherwise .

We call a collection = { J }J Esupp (S) , J {0, 1}, an integral collection for S if

1(r,s)

J Esupp (S)

(J,r,s) J Z, for all 1 r < s .

If Esupp (S) = , we interpret to be the zero collection and the sum on an empty set to be zero.If J = 0 for all J Esupp (S) , we call = { 0}J Esupp (S) the trivial collection. Clearly the trivialcollection is an integral collection and there are at most 2 |Esupp (S) | integral collections for S . Anyintegral collection which is not trivial is called non-trivial .


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Example 1.4.

(i) If Esupp (S) = or J = 0 for all J Esupp (S) , then it is clear from denition that the only

integral collection for S is the trivial collection.(ii) Let S be such that {r, s } supp (S) for all 1 r < s . Then (r,s) = 1 for all such r < sand so any collection { J }J Esupp (S) , J {0, 1} is an integral collection. Therefore thereare 2 |Esupp (S) | integral collections for S . In particular, if S is a lattice of rank 3, thenthere is at least one non-trivial integral collection for S , as |Esupp (S) | 1.

(iii) Suppose Esupp (S) contains a set J such that {r, s } supp (S) for all r, s J . Set

J =1, if J = J ,0, otherwise .

Now for 1 r < s we have (J ,r,s) = 0 if {r, s } J . It follows that { J } is a non-trivial integral collection for S . In particular, if > 3 and ind(S) = 2 2, then there isat least one non-trivial integral collection for S , as in this case there is always a set J Esupp (S) with {r, s } supp (S) for all r, s J .

(iv) Let S and S be semilattices with the same Z-span and with S S . Then supp (S) supp (S ) . Now if is a non-trivial integral collection for S then the collection ={ J }J Esupp (S ) dened by

J =J , if J Esupp (S),

0, otherwise ,

is a non-trivial integral collection for S .

Lemma 1.5. If ind(S) 3 , then the only integral collection for S is the trivial collection.

Proof. By Example 1.4(i), it is enough to show that if { J } is an integral collection for S with Esupp (S) = then J = 0 for all J Esupp (S) . Now let Esupp (S) = { J 1 , . . . , J n } with|J i | |J j | for all 1 i < j n . We may choose {r j , s j } J j such that {r j , s j } /supp (S) and{r j , s j } J i for all 1 i < j n . Then from the denition of an integral collection we get

J i / 2Z for all i . This forces J i = 0 for all i . 2

2. Extended afne Weyl groups

In this section, we briey recall the denition and some basic facts about extended afneroot systems and their corresponding Weyl groups. For a more detailed exposition, we refer thereader to [AABGP] and [AS1], in particular we will use the notation and concepts introducedthere without further explanations.

All the groups we consider in this work will be subgroups of the orthogonal group O (V , I )where V is a nite dimensional vector space equipped with a non-degenerate symmetric bilin-ear form I = ( , ) . For a group G , Z(G) denotes the center of G , and for x, y G we denotethe commutator x 1y 1xy by [x, y ]. An element V is called non-isotropic (isotropic ) if (,) = 0 ((,) = 0). We denote the set of non-isotropic elements of a subset T with T andthe set of isotropic elements of T with T 0 .


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Denition 2.1. A subset R of a non-trivial nite dimensional real vector space V , equipped witha positive semi-denite symmetric bilinear form ( , ) , is called an extended afne root systembelonging to (V , ( , )) if R satises the following 8 axioms:

(R1) 0 R , (R2) R = R , (R3) R spans V , (R4) R 2 /R , (R5) R is discrete in V , (R6) For R and R , there exist non-negative integers d, u such that + n R ,

n Z, if and only if d n u , moreover 2 (,)/(,) = d u . (R7) If R = R1 R 2 , where (R 1 , R 2) = 0, then either R1 = or R2 = , (R8) For any R 0 , there exists R such that + R .

The dimension of the radical V 0 of the form is called the nullity of R , and the dimensionof V := V / V 0 is called the rank of R . It turns out that the image of R in V is a nite root systemwhose type is called the type of R . Corresponding to the integers and , we set

J = { 1, . . . , } and J = { 1, . . . , }.

Let R be a -extended afne root system belonging to (V , , ) . It follows that the form re-stricted to V is positive denite and that R , the image of R in V , is an irreducible nite rootsystem (including zero) in V [AABGP, II.2.9]. The type of R is dened to be the type of R .

In this work we always assume that R is an extended afne root system of simply laced type ,that is it has one of the types X = A , D , E 6 , E 7 or E 8 . We x a complement V of V 0 in V such that

R := V + R for some V 0

is a nite root system in V , isometrically isomorphic to R , and that

R = R(X , S) = (S + S) (R + S) (2.2)

where S is a semilattice in V 0. Recall that we have xed a basis B of V

0satisfying (1.2). Here

X denotes the type of R . Throughout this work we x a fundamental basis

= { 1 , . . . , }

of R . It is known that if > 1, then S is a lattice in V 0 . To introduce the extended afne Weylgroup of R , we need to consider the so-called a hyperbolic extension V of V as follows. We xa basis { 1 , . . . , } of (V 0) and we set V := V (V 0), where (V 0) is the dual space of V 0 .Now we extend the from ( , ) on V to a non-degenerate form I , denoted again by ( , ) , on V asfollows:

( , ) | V V := ( , ),

V , V 0 = V 0 , V 0 := 0, ( i , j ) := i,j , i, j J .


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Denition 2.3. The extended afne Weyl group W of R is the subgroup of the orthogonal groupO(V , I ) generated by reections w , R , dened by w (u) = u (u,) , u V . Clearly

W FO(

V , I ) := w O(

V , I ) w() = for all

V 0

. (2.4)The subgroup

H := w + w R , V 0 , + R

of W is called the Heisenberg-like group of R and plays a crucial role in the study of structureand presentations of W . Using the fact that R R , we identify the nite Weyl group W of Rwith a subgroup of W . Then for any R , using (2.4), (2.2), (1.3) and the fact that R = W we have

R =J supp(S)

W () + J + 2 = W J supp (S)

( + J + 2) . (2.5)

We also note that the following important conjugation relation holds in W :

ww w 1 = ww ( R , w W ). (2.6)

In [AS1, 3] we have studied the structure of W and have obtained a particular nite set of generators for W and its center Z( W ) , using linear maps dened as follows. For V and V 0 , we dene T End(V ) by

T (u) := u (,u) + (,u) (,)

2(,u) (u V ). (2.7)

One can check easily that for V , V , , V 0 and w O(V , I ) ,

T = w + w , T

, T

= T (,)

and wT

w 1 = T w() . (2.8)

For V and , V 0 , we have from (2.8) that

T FO(V , I ) and T

Z FO(V , I ) . (2.9)

For r, s J , we set

cr,s = T s r and C := cr,s : 1 r < s . (2.10)

Then by (2.9) for all r, r , s , s J and V , we have

cr,s

() = and

cr,r

= cr,s

cs,r

= [ cr,s

, cr ,s

] =1

.(2.11)

By [AS1, Lemma 2.16(ii)],

C is a free abelian group of rank ( 1)/ 2. (2.12)


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Also for (i,r) J J , we set

t i,r := T r i = w i + r w i . (2.13)

From (2.8) for all r, s J , i, j J and V , one can see easily that

[t i,r , t j,s ] = c( i , j )r,s and w j t i,r w j = t i,r t

( i , j )j,r . (2.14)

To describe the center Z( W ) of W , we set

Z := zJ | J J Z FO(V , I ) ,

where

zJ :=

{r,s J | r


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3. Preliminary results

We keep all the notations as in Section 2. In particular R is an extended afne root system of

simply laced type andW

is its extended afne Weyl group. We have xed a basis B of V 0

anda nite root system R such that R is of the form (2.2), where the semilattice S is given by (1.3).Also recall from Section 2 that we have xed a basis = { 1 , . . . , } of R .

For a subset J = { i1 , . . . , i n } of J with i1 < i 2 < < i n and a group G we make the con-vention

iJ

a i = a i1 a i2 a in (a i G).

Using this notation, (2.7) and (2.13), we have

rJ

t m ri,r ( j ) = j +rJ

( i , j )m r r ( i , j , m r Z). (3.1)

Lemma 3.2. Let J J and i J .

(i) If J supp (S) then

zJ = w i w i + J rJ

w i + r w i .

(ii) If J = { r, s } /supp (S) , r < s , then

zJ = [ t i,r , t i,s ] = w i w i + r w i w i + s w i + r w i w i + s w i .

Proof. (i) From (2.8), [AS1, Lemma 2.8] and the way zJ is dened we have

w i + J w i = T J

i = T rJ r

i =rJ

t i,r{r,s J | r 1,

(3.3)


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and

W = w | .

It is clear that W W .

Lemma 3.4. t i,r W , (i,r) J J .

Proof. Let w W be such that i = w( 1) . Since W W and {r } supp (S) for all r J ,we have from (2.6)

t i,r = w i + r w i = ww( 1 )+ r ww( 1 ) = ww 1+ r w 1 w 1W . 2

The following result is proved in [A4, Proposition 4.26]. We present a new proof for this resulthere.

Lemma 3.5. W = R and W = W .

Proof. It is clear that W R . So to prove the rst equality, we must show that R W . Using (2.5) and the fact that W W , it is enough to show that for J supp (S) andm r Z, r J ,

:= 1 + J +

r = 1

2m r r W .

By (3.1), (2.4) and the fact that J = iJ i , we have

=

r = 1

t m r1,r ( 1 + J ),

which if = 1 is an element of W , by Lemma 3.4. If > 1, then there exists j suchthat ( 1 , j ) = 1. Thus

=

r = 1

t m r1,r ( 1 + J ) =

r = 1

t m r1,rsJ

t 1j,s ( 1 + s ),

which is again an element of W , by Lemma 3.4. Thus the rst equality in the statement holds.Using (2.6) and the rst equality, one can see that the second equality holds. 2

4. Presentation by conjugation

We keep all the notations and assumptions as in the previous sections. In particular, R is anullity simply laced extended afne root system of the form (2.2), where the semilattice S isgiven by (1.3) and W is its extended afne Weyl group.


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Proof. (i) Let be as in (3.3) and R . From Lemma 3.5 we have = w() for somew W and . So w = w 1 w n , for some i . Then from ( II ) we have

w = ww() = w1

wnw w

n w

1 w | .

(ii)(iii) Considering (4.6) and part (i) we only need to show that for any , w is in theright-hand side of the equality in the statement. Clearly this holds if = i for some i J . If = 1 + J for some J supp (S) , then from the way zJ is dened it is clear that (ii) holds.Therefore to see (iii), it is enough to show that zJ is in the right-hand side of the equality in (ii)for all J J . But this clearly holds if J /supp (S) . Now let J supp (S) . Since > 1, thereexists i J such that ( 1 , i ) = 1. Let w = rJ t 1i,r and w = rJ t

1i,r . We have from (4.4),

(4.7) and (3.1) that

w 1+ J = ww( 1) = w( w)( 1 ) = w w 1 w 1 =rJ

t 1i,r w1rJ

t 1i,r

1

. (4.9)

Therefore zJ is in the right-hand side and we are done. 2

Lemma 4.10.

(i) [t i,r , t j,s ] Z( W ) , for (i,r),(j,s) J J .(ii) zJ Z( W ) , for J J .

Proof. Let R . From (4.4), (4.7), (2.11), (2.14) and the denition of zJ we have

[t i,r , t j,s ] w [t i,r , t j,s ] 1 = w [t i,r ,t j,s ]() = w

and

zJ w z 1J = wzJ () = w .

Thus [t i,r , t j,s ] and zJ commute with all elements of W . 2

Lemma 4.11. Let i J . Then

(i) [t 1,r , t 1,s ] = [ t i,r , t i,s ] , for all r, s J ,(ii) zJ = w i w i + J rJ t i,r for all J supp (S) . In particular, if > 1 , this holds for all

J J .

Proof. Let w W be such that i = w( 1) and x a preimage w W of w , under . Then by(4.4), w t 1,r w 1 = t i,r for r J . Thus from Lemma 4.10 we have

[t 1,r , t 1,s ] = w [t 1,r , t 1,s ] w 1 = w t 1,r w 1 , w t 1,s w 1 = [ t i,r , t i,s ].

This gives (i). Next let J supp (S) . Then we have again from Lemma 4.10 and (4.4) that

zJ = w zJ w 1 = w w1 w 1+ J rJ

t 1,r w 1 = w i w i + J rJ

t i,r .


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[t i,r , t j,s ] = t 1i,r t 1j,s t i,r t j,s

= w i w i + r w j w j + s w i + r w i w j + s w j

= w i + r w i w j + s (w i w i )w j w i w i + r w j + s w j

= w i + r w wi ( j )+ s wwi ( j ) w i + r w j + s w j

= (w i + r w wi ( j )+ s w i + r )( w i + r wwi ( j ) w i + r )w j + s w j

= wwi r (w i ( j )+ s ) wwi r (w i ( j )) w j + s w j

= w j + ( j , i ) r + s w j ( j , i ) r w j + s w j

= w j w j + ( j , i ) r + s w j w j ( j , i ) r w j w j w j + s w j

= w j

w j + ( j , i ) r + s

w j + ( j , i ) r

w j

w j + s

w j

= w j + ( j , i ) r + s w j + ( j , i ) r w j w j + s .

But if ( i , j ) = 1, then by Lemma 4.12 the expression appearing in the right of the lastequality is z 1{r,s } and is clearly 1, if ( i , j ) = 0. This completes the proof. 2

Lemma 4.15. w i t j,r w i = t j,r t ( i , j )i,r , i, j J , r J .

Proof. First, let i = j . Then from ( I) and the fact that ( i , i ) = 2 we have

w i t i,r w i = w i w i + r w i w i = w i w i + r = t 1i,r = t i,r t

( i , i )i,r .

Next let i = j . If ( i , j ) = 0, then using ( I) and (II ) we have

w i t j,r w i = w i w j + r w j w i

= wwi ( j )+ r wwi ( j )

= wj

+ rw

j= t j,r = t j,r t

( i , j )

i,r.

Finally, if ( i , j ) = 1, then using ( I) and (II ), we have

w i t j,r w i = w i w j + r w j w i t 1i,r t

1j,r t j,r t i,r

= w i w j + r w j w i + r w j w j + r t j,r t i,r

= w i w j + r w i + j + r w j + r t j,r t i,r

= wiw

it j,r t i,r = t j,r t i,r = t j,r t

( i , j )

i,r.

This completes the proof. 2

Lemma 4.16. w 1+ 2 rJ r = ( rJ t 1,r )w 1 ( rJ t 1,r ) 1 , J J .


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Proof. Let w = rJ t 1,r . From (3.1) we have w( 1) = 1 + rJ 2 r , and so by (II ) and (4.4)we get

w 1+ 2 rJ r = ww( 1) =rJ

t 1,r w1rJ

t 1,r 1

. 2

Lemma 4.17. z2J z{r,s } | r, s J , r < s , J J .

Proof. If J supp (S) and J = rJ r , then using relations ( I), (II ) and Lemmas 4.16 and4.15 we get

z2J

= zJ zJ = w1w

1+

J rJ

t 1,r zJ

= w1 w 1+ J zJ rJ

t 1,r

= w1 (w 1+ J w 1 w1+ J )rJ

t 1,rrJ

t 1,r

= w1 w 1+ 2 J rJ

t 1,r2

= w1rJ

t 1,r w 1rJ

t 1,r 1

rJ

t 1,r2

=rJ

t 11,rrJ

t 1,r .

But this latter belongs to z{r,s } | r, s J , r < s , by Lemma 4.13. If J /supp (S) , then the resultis clear by the way zJ is dened and Lemma 4.13. 2

Lemma 4.18. z{r,s } | 1 r < s is a free abelian group of rank ( 1)/ 2.

Proof. Let 1 r


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Proof. By (4.7), Lemmas 4.18 and 2.17, the restriction to z{r,s } | 1 r < s induces theisomorphism

z{r,s } | 1 r < s

= z{r,s } | 1 r < s . (4.20)

From the way zJ is dened and (2.16), it follows that

z2J ={r,s J | r


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Therefore from (4.3) and Proposition 2.19, it follows that w = 1 and n i,r = 0, (i,r) J J and m r,s = 0 for all r, s J and so w = 1. 2

5. Presentation by conjugation for type A 1

This section contains the main results of our work. In particular, we give the necessary andsufcient conditions for an extended afne Weyl group of type A1 to have the presentation byconjugation (Theorem 5.16). Throughout this section R is of type A1 , that is

R = (S + S) ( 1 + S).

As in the previous sections let W be the extended afne Weyl group of R and W be the group

dened in Denition 4.1. We recall from (3.3) and Lemma 3.5 that the set

= 1 + J J supp (S)

satises W = w | and W = R .

Lemma 5.1. { w | } is a minimal set of generators for W .

Proof. By Lemma 4.8(i), the set in the statement generates W . To show that it is minimal x

= 1 + J 0 , J 0 supp (S) . We have W = F /N , where F is the free group on the set{r | R } and N is the normal closure of the set {r 2 , r r r rw () | , R

} in F . Thenw = r N , R . From (2.5), it follows that

R =J supp (S)

R J , where R J := 1 + J + 2.

Since (,) = 2 for R , we have

W R J R J J supp (S) . (5.2)

Now let : F Z2 be the epimorphism induced by the assignment

(r ) =1, if R J 0 ,0, if R \ R J 0 .

From (5.2), it follows that (r 2 ) = 0 and (r r r rw () ) = 0, for any , R and so (N) =

{0}. Thus the epimorphism induces a unique epimorphism : W Z2 so that ( w ) = (r ) , R . So w = w for any w w | \ , as ( w ) = 1 and ( w) = 0. 2

Lemma 5.3. If 1 r


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Proof. By Lemma 4.18, it is enough to show that J = 0 for all J Esupp (S) . Suppose to thecontrary that J 0 = 0 for some J 0 Esupp (S) . Then

z J 0J 0 =

1 r


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Set

Z := zJ | J J Z( W ). (5.9)

The following corollary follows immediately from Lemma 5.7.

Corollary 5.10. Each element u of Z has a unique expression of the form

u = u( m, ),

where (m, ) is of the form (5.4) .

Proposition 5.11.

(i) ker() Z .(ii) If u Z , then the following statements are equivalent :

(a) u ker() ,(b) u 2 = 1 ,(c) | u | < ,(d) u = u( m, ) , where is an integral collection satisfying

m r,s = 1

(r,s)J Esupp (S)

( J ,r,s) J (1 r < s ). (5.12)

(iii) The assignment = { J } u( m, ) , where m = { m r,s } satises (5.12) , is a one to onecorrespondence from the set of integral collections for S onto ker() .

Proof. (i) Let w ker() . By Lemma 5.7, w has a unique expression of the form (5.8). Thenapplying the homomorphism , we get

1 = ( w) = w n1

r = 1

t n r1,r u( m, ) = wn 1

r = 1

t n r1,r z,

where z = ( u( m, )) Z( W ) . Then from Proposition 2.19 we obtain n = 0 = n r for all r J and so w = u( m, )Z .

(ii) Let u Z . By Corollary 5.10, u = u( m, ) for a pair (m, ) of the form (5.4). Let n Z 0 .By (4.7), (2.15), (2.12) and (2.16) we have

u n =1 r


19/22


=1 r


20/22


implies (vi). So it remains to show that (vi) (i). Let u ker() . By Proposition 5.11, u hasthe form u( m, ), where is an integral collection satisfying (5.12). Now we claim that J = 0for all J Esupp (S) and so u = 1. Suppose to the contrary that J 0 = 0 for some J 0 Esupp (S) .

Then from (4.7) we have

1 = ( u) =1 r


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(i) m + 3 ,(ii) 3 and m = 7 ,

(iii) Esupp (S) = .

In particular, if 3 , then W has the presentation by conjugation if and only if m = 7.

Proof. By Theorem 5.16, we only need to show that the only integral collection for S is thetrivial collection. To see this, use Lemma 1.5 for part (i). Part (ii) is a consequence of (i) asm 7 when 3. For part (iii) see Example 1.4. Next, note that if 2, then Esupp (S) = and so the last statement holds by part (iii). Finally, if = 3 then the result holds by part (ii) andCorollary 5.18. 2

Corollary 5.20. Let R = (S + S) (R + S) and R = (S + S ) (R + S ) be two extended afneroot systems of type A1 , where S and S have the same Z-span and S S . Then if the extended afne Weyl group of R has the presentation by conjugation, then also has the extended afneWeyl group of R .

Proof. It follows immediately from Theorem 5.16 and Example 1.4(iv). 2

Remark 5.21.

(i) In [A3, Remark 2.14] the author, by suggesting an example, has conjectured that certain

extended afne Weyl groups may not have the presentation by conjugation. The Corollar-ies 5.17 and 5.18, give an afrmative answer to this conjecture. However, as it is shown inCorollary 5.19, the suggested example in [A3] is not an appropriate example. In fact, theroot system in the mentioned example is an A1-type extended afne root system of nullity 3and index 6, and so by Corollary 5.19, its Weyl group has the presentation by conjugation.

(ii) One can easily see that Corollary 5.20 is false if we remove the assumption S = S .

Acknowledgments

The authors would like to thank the Center of Excellence for Mathematics, University of

Isfahan.For the study of extended afne root systems and extended afne Weyl groups we refer

the reader to [MS,Sa,AABGP,A1,A2,A3,A4,SaT,T,AS1,AS2]. The authors would like to thank A. Abdollahi for a fruitful discussion which led to the proof of crucial Lemma 5.1. The authorshope that the results appearing in this work prepare the ground for the study of the presentationby conjugation for non-simply laced extended afne Weyl groups.

References

[AABGP] B. Allison, S. Azam, S. Berman, Y. Gao, A. Pianzola, Extended afne Lie algebras and their root systems,Mem. Amer. Math. Soc. 603 (1997) 1122.

[A1] S. Azam, Nonreduced extended afne Weyl groups, J. Algebra 269 (2003) 508527.[A2] S. Azam, Extended afne root systems, J. Lie Theory 12 (2) (2002) 515527.[A3] S. Azam, A presentation for reduced extended afne Weyl groups, Comm. Algebra 28 (2000) 465488.[A4] S. Azam, Extended afne Weyl groups, J. Algebra 214 (1999) 571624.


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[AS1] S. Azam, V. Shahsanaei, Simply laced extended afne Weyl groups (A nite presentation), Publ. Res. Inst.Math. Sci. 43 (2007) 403424.

[AS2] S. Azam, V. Shahsanaei, On the presentations of extended afne Weyl groups, Publ. Res. Inst. Math. Sci.,in press.

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versity of Saskatchewan, 1995.[MP] R.V. Moody, A. Pianzola, Lie Algebra with Triangular Decomposition, Wiley, New York, 1995.[MS] R.V. Moody, Z. Shi, Toroidal Weyl groups, Nova J. Algebra Geom. 11 (1992) 317337.[Sa] K. Saito, Extended afne root systems 1 (Coxeter transformations), Publ. Res. Inst. Math. Sci. 21 (1985)

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