ELEME'NTS OF PRIME ORDER [N REFLECTION GROUPS

Stephanie P. Burgoyne Department of Mathematics

Subrnitted in partial fbifWnent of the requirernents for the degree of

Doctor of Philosophy

F a d t y of Graduate Studies The University of Western Ontario

London, Ontario April1998

O Stephanie P. Burgoyne 1998

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Abstract

Let W(A) be a Euclidean reflection group with root system A and fundamental system

X. Carter showed that involutions cm be decomposeci into a product of mut* orthogonal

reflections (involutions). It can be proven that in certain reflectioo groups an eiement cp of prime

order p (p odd) can be decomposecl h o a product of canonid elements of order p. The

decomposition pro- &es 9 = cp, cp2 . . . cp, "th each q+ E W (4, ). The uniqueness of this

decomposition can be determineci up to conjugacy and such elements of order p are shown to be

regular on certain subgroups ofthe reflection group. The decornposition of rp can produce an

elementary Abeiian p-subgroup, but not aii elementary Abelian p-subgroups are generated in this

m e r .

"With God aii things are possible." Matthew 19:26

There are many people without whom the completion of this thesis wodd not have been

possible. 1 am indebted to my parents, whose suppon has k e n unwavering over the years. 1

would especially üke to thank Dr. Richard Kane of the UWO Mathematics Depamnent. whose

guidance and perseverance have enabled me to finish rny shidies. To Anne White, many th-

for chauffeur and courier services. Finaiiy, I would like to thank my typin and fiend, Heather

Robinson, for her cornputer abilities and her sense of humour in the face of particuiarly messfil

situations.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 2 O 1 Euclidean Reflections 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 2 . 2 Root Systems 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 2 . 3 Fundamental Systems 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Section2-4 CoxeterGroupsandGraphs 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 2 5 Further Definitions 12

Chapter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 3 O 1 Paraboüc Subgroups 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 3 œ 2 Isotropy Subgroups 16 . . . . . . . . . . . . . . . . . . . . . Section 3 O 3 Nomabers of Parabolic Subgroups 17

Chapter4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 4 - 1 involutions 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 4 - 2 Elementary Equivalences 20 . . . . . . . . . . . . . . . . . . Section 4 - 3 Mioimal Decomposition of an Element ip 22

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5 The Decomposition Theorem 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 6 Smicture and Uniqueness 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section6- 1 Strucweofh, 28

. . . . . . . . . . . . . . . . . . . . Section 6 - 2 The Uniqueues of the Structure of A, 29

Chapter7 .......................................................... 33 ................................... Section 7 - 1 Regdar Elements 33

Chapter8 .......................................................... 35 . . . . . . . . . . . . . . . . . . . . . . . . Section 8 - 1 Elernentary Abelim p-Subgroups 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 8 - 2 TaMe of Nomahzers 37 ............. Section 8 - 3 Grnenuors of Elementary Abeiian p-Subgroups 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 8 - 4 Specific Cases 44

................................................ Chapter 9 Conclusion 49

1

Chapter 1

In this thesis. let W(A) be a reflection group with root systern b and fundamental

systern C. Let (s,, sdz, , .... s,; denote the fundamental reflections and let cp be an

element of prime order p. A question that can be asked is:

Can elements of order p be decomposed into a product of canonical cornmuthg elements?

For example, every element in the syametric group on n letters can be expressed

as a product of disjoint cycles. Such eiements have as order the least cornmon multiple of

the length of the disjoint cycles. Thus, an element of order p mua be a product of cycles

of length p. We will discover that the symmetric group is a reflection group and we wiil

generaiize this cycle decomposition to other reflection groups.

Other related questions are:

What is the structure of elements in such a decomposition?

1s the decomposition unique?

Do these elernents of prime order behave as regular elemems?

Can analogous decomposition be obtained for elements of any order?

It is necessary to provide some basic definitions and results of the study of

reflection groups. Such introductory material will be covered in Chapters 2 to 4. Chapter

4 will also state and explain Carter's results on the decomposition of elements of order 3

(involutions). In Chapter 5, we wiii prove a decomposition for elernents of prime order, p

odd. which is the mein resuit of this paper. Chapter 6 disasses the uniqueness and

structure of the decomposition and Chapter 7 examines elements of prime order as regular

elements. Chapter 8 studies a generaîization of elements of prime order, nameiy

Cbapter 2

We will work in P dimensional -EE. In other words, E = 18' where

& R has the usual imer product. More abstractly, E is an P-dimensional vector space over

IR provideci with a pairing

C - ) : E x E - B

satisfyuig :

(0 (ax + ~ Y J ) = + MY,^

(ü) (KY) = (YJ)

(üi) ( ~ x ) r O and ( ~ x ) = O if and only if x = O

(In the above Ky,z É E while a,b E B)

Given a hyperplane H c E through the origin there is a rdection associated with

H. Let L = the line through the origin which is orthogonal to H. (So E = H @ L). Then

defhe the iinear transformations SH: E - E sH(x) = x if x E H

s~(x) = -X a x e L.

Actudy, it will be more useful to define reflections with respect to vecton rather thao

with respect to hyperplanes. aven O * a E E let Ha c E be the hyperplane

H,= ( 4 ( w ) = O )

We then defhe the rdection s, : E - E by the des

S=(X) = x if x E Ha

sJa) = -a

m e : Given O * k ~ i R then &=HI, and sa=&

Here are some usefiil properties of s, and Ha.

Proaerties

(P - 1) SJX) = x - (2(ga) / (a,a)) a for ail x e E

(P - 2) s, is onhogond i.e. (s,(x), s,(y)) = (&y) for all &y E E

( P - 3 ) d e t ~ , = - 1

(P - 4) if <p is an orthogonal autornorphism of V then: (i) <pH, = H (3 (P s,TfL = $(a)

Besides reflections we also have the concept of a reflection group. Given W c

GL(E) we say that W is a if W is generated, as a group, by its

reflections,

So a reflection group is just a spetial type of representation - a faiffil

representation p: W - GL(E) where W c GL(E) is a reflection group as defineci above.

n 2 - 2

m: A root s y s t a is a finite set of non-zero vecton A c E sati@np the

properties R - 1, R - 2, and R - 3.

(R - 1) A generates E.

This property is just for convenience. If A does not generate E then replace E by the

subspace E'c E genmed by A. Next, one has the properiy

(R-2) If a e A then A ~ E A ifandoniyif A = * 1.

Finaily, the set A is invariant under the action ofthe reflectioas deteded by A. This

invariance pro- is usually expressed as

(R-3) if a$ E A then sdP)cA.

A root system is, essentiaily, a iinear algebra version of a fimte Euclidean reflection

group. Given a root system A c E, we can convert it into a finite Euclidean refiection

group. Let W(A) denote the reflection group generated by the reflections associated

with the root system A.

A root system is said to be ifit also satisfis

(R - 4) (?(a, B)i(a,a)) E Z for any a, p E A.

Definition: A W e y l is a f i t e subgroup W c GL,(Z) nich that W c GL,(B) is a

Euclidean rdection group.

DefinitiM: The wroot av of a root a is Za/(a,a) , where a i A , a crystaiiographic

root systern. The coroots A' = (av} form a crystdographic system.

In the root systems we wiU study (narnely, inreducible crystaiiographic root

systems), we will discover 2 possible root lengths, thereby giving loaP roou and shPn

LQP~S. As we will see in d o n 2-4, the irreduciile crystdographic root syaems are

labeiied 4, B,, C, D, F,, E, E,, E,, and G, . Only B, C, F,, and have two root

lengths. The ratio of long to short roots is given by

la12 1 !Pi2 = &,a (P,P = 2 (F4, Bi cas=)

= 3 (GZcase).

Definitinn: Let A c E be a crystaüographic root system. We d e h e the Q =

Q(A) to be ai l 2-hear combinations in E of the elements of A. The root lanice

satisfis the following propertîes:

(i) Q = Z a, e .... e Z - a, for any fiindamentai system (a,, ...., a,) of A

(i) Q is invariant under W(A)

The Wrwt lpttiEe is the root lanice ofthe coroot system A'.

5

S=tioa 2 - 3 Fun-

By definition a root syaem A c E contains a basis of E. ln the next few sections

we wül demonstrate that A acîually contains a very special type of basis.

Definition: Given a root system A c E then C c A is a fundamental e s t a of A if

(i) C is Iinearly independent

(ii) every element of A is a hear combination of elements of C where the

coefficients are d non-negative or aü non-positive.

Since the elements of A span E, it follows âom (i) and (ü) that X is a basis of E.

The elements of C are called mou. Other ternis commonly used for C are

"base" or "SimpIe systemn. In the latter case the elements of Z are d e d "simple roots".

One of the more usefid finctions of a fundamemal system is to provide a partition

& = A - U A -

w here

A- = the elements of A which can be expanded in terms of X with

d c i e n t s r O

A- = the elemeats of A which can be expanded in terms of E with

d c i e n t s s O

Fundamental systems exia but are not unique. It can be show that W (A) acts

transitively on the fùndamental systems. For any two systems ZI, C , there exists cp E

W(A) such that NZ3 = ZJ

Definition: For a root a, 1et a = A i q , where 1 sir P. express a as a linear

combination of the hdamental mots cl,, .... 4. Then h(a) = LAi is the of a

with respect to that fundamental system.

Let A be a crystallographic root system. For each fbndamental systern C of A

there is a unique root q of greatest height with respect to 2. Q is characterized by the

faa that #ven any root a, ifwe expand q and a in terms of the fundamental mots

al,a3 .... a,, say % =x hi$, a = x Aiai, then bz Ai fora11 i. It can beshown that

a, is a long root.

Let a, denote the root of greatest height. Let q = hiai , where 1 sis P. Then

it can be shown that 1 W 1 = P! II 4 g where Q is the & the card'iality of the

fundamental system, and g = k + 1, where k is the number of hi quai to 1.

One can a h consider the wroot of a,. One can expand

a$ = &*a,* l r id

where (al*, .... , a,*) are the coroots of ( a l , .... , a& The integers {hlS, .... , b*) are

related to the integers (hl, ... . , h2J by the following nile.

m: Let a, = Ehi ai and let q* = &*aif 1 sisL

n e n hi = hi* (q,q,)4q,ai) -

gPPf 2cg/(eg,tq,) = zhi* 2 a i / ( ~ Q , l s i d byequating thetwo expressions for CC$.

Dividing both sides by 2 and cross-multiplying and replacing q, with 14: 4 , 1 ris4 we

have:

Ehi ai =ch* ( ~ ~ o ) / ( ~ a , ) ] a ~ . lsid

andso 4 =hi*(q,q#(ai,ai). QED.

Also, the foilowiag is mie:

(%vq)l(ai,ai) = 1 (& Da E6, &, &

= I or2 (F,,B,cases)

= l or3 (Gzcase)

Shce we have the formula hi* = hi (a,a,) i (ao.qJ (Steinberg) and since a, is a

long root, we can caldate the hi* using the list of h, for each root system If ai is a long

mot the ratio (q,,q) / (&a is 1. If ai is a short root, the ratio may be 1 1 2 in the B.

and F, cases or it may be 1 1 3 in the G2 case. So h,* = hi or 1/2 hi or 1/3 4

depending on the root length and the root system.

m: A is a connected component of E - (U H 3. Intuitively, the

chambers partition E into pie-shaped regiom. The following picture represents chambers

aod the w d s separathg hem:

There is a one-tosne correspondence between fiindamentai systems and

chambersS Given a fundamentai system (a, ,o; .... a,), its corresponding chamber is C =

( t E E 1 (ai, t) > O, i = 1,2, .... Q}. W (A) permutes both bdamental systems and

chamben. It acts fieely and transitively on both sets and the above one-to-one

correspoadence respects these anions.

8

The closure C* = ( t E El (a , t) 2 O, i = 1.2, .... P J will also play a role inour

work. There is a decomposition C = LI CI, indexed by I c { 1 2, . . . . E 1, where

CI={ t a E l (a , t )=O, i e l a n d ( a i , t ) > O , i e 1).

5-t-4 Coreter~.rwjmd GGueéi

Given any fundamental system Z c A, then W (A) is generated by the set

S={s,l a d j .

D e f k i h : Let cp E W(A), then P(q), the of <p with respect to the set S, where

S = (s,l cr E Z), is the srnailest integer n such that <p can be written as a product of n

eiements tiom S.

The foilowing properties of length are useful.

(i) P (cp sNi, ) = ((q) + 1 if and only if cp ai > O, where ai is a

fundamentai root and cp E W(A)

(ii) P (cp s,,,) =t(cp) - i ifand only if 9 q < 0

De-: A poup W is a w e r ifthere exists S c W wch that

W=cs$SI (si, s ~ ) ~ = l >

where q = 1 and m, E (2,3. ....} u (m) if i+j. The order of si sj is a d y qj .

Every finite reflection group is a Coxeter group. The Coxeter structure of a finite

reflection group can be represented by a graph.

Defimtion: A -ter is a graph with each edge labelleci by an integer 2 3. The

vertices of the graph are the fùndamental roots a,, . . .. a, . if sMj, s,, has order 2, then

no edge connects ai to aj . Othenvise, when sMi, s , has order m r 3, then an edge

connects ori to q- with label m men rn = 3 is left unlabeiled.

The followiq List shows the root system A, fundamental syaem C and Coxeter

graph for the reflection groups used in this thesis.

(Gm) A = {(cos (kx)/m, sin (kn)/m) 1 Os kr 2m- 1 )

m

C = - where a, = (cos d m , sin dm) Ct) % = (COS (~R)/III, sin (2n)/m)

Definition: A- . . is a Coxeter graph in which L44 is replaced

with a double bond - and ~~4 is replaced with a aiple bond

----- O===== O and ~~4 remains as a single bond W. tn the case

of double and triple bonds an arrow points tiom the Longer to the shorter root. Dynkin

* * dagranis are ody used in the crysîaliographic case. An nynkindiagrpm is

formed by adjoining -Q to the Dynkin diagram where Q is the Iongm rom A number

is piaced at each vertex to indicate the coefficient of ülat hdamental mot in the

expansion a,, = 1 4 c+ (1 sise).

. . ended DMikin di-

-: A -ter el- is an element W = % sa, ,. . . .sa(a, where C = (a, , . . . .

a,) is a fundamentai system of A of the reflection group W(A). Different Coxeter

elements can be produced by using different ftndarnentd syaems and by using a Herent

order of the hdamental reflections for a given system. DXerent Coxeter elements of

W(A) ue conjugate to one another. The order of a Coxeter element is (C hi ) + 1 where

q = C hi ai, Isis(. The order also quais 1A)I t.

Definmon: A is an element q with an eigenvector that does not üe on the

reflecting hyperplane ofany reflection. If*) = & x is a reguiar eigenvector, 6 a regular

eigenvdue. Coxeter elements are examples of reguiar elements.

4, B, C,, D,, F,, E, E,, E,, G, are examples of Weyl groups. There exkt other

irreduci'ble groups that are non-Weyl, notably ?& and H, ([4]).

We wiU aiso note the following:

A, = S, -,, the symmetric group on P + 1 letten

B, = S, B. (Z/2zy

D, = S, D< (2/22)i -'

13

C, produces the same group as Bi. C, is obtained @om B, by altering some of

the root Iengths in B,. The in A are replaced with {*2fi) to fom C:.

The following is another result required for this thesis.

S t e b ' s L e m of the Stnilp: Let 2 be an irreducible root system (a root system

with a ~ o ~ e c t e d Coxeter graph) and %, a,, ... the extendeci system of fundamental roots

so labelleci that & a,, ... a, is a minimal string (Le., (q, q,) * O for ali i) co~ecting a,

to a root for which the label n* = rnax q* is achieved. (Here the q* corne f?om the

expression -a,,* = q* a,* where -a,,* is the coroot of -a, .) Then each q (Osiaq) is

a long root and q* = i +i . nie coczfficients q* form a co~ected string of integen

starting with 1 or 2 and increasing.

Chapter 3

Section 3 - I P-lic S-

Let W = W(A) be an Euclidean reflection group with root qstem A c E. Let

C = ( a, , . . . . a, j be a fundamental system for A. So W is generated by the fùndamenta

reflections S = {sq , ,, , . . . . s ,,,, } . Given 1 c ( 1,2,. . . .,@ 1 let

W, = the subgroup of W generated by {sNi,l i E 1).

Such a subgroup is cded a of W. In this case we have defined a

parabolic subgroup with respect to 2. To obtain the complete collection of parabolic

subgroups one considers those defineci with respect to each fiindamental system of A.

Ifwe pick one particuiar bdarnental system Z then every parabolic subgroup is

conjugate to a parabolic subgroup with respect to C. For, given another hdamental

system 2', then by d o n 2-3, cp C' = Z for some <p E W. By property P - 4 of section

2.1 the inner automorphism cp-q' : W - W maps the fùndamental reflections S' = (s,l a

E X' 1 to the fundamental rde*ions S = {sp 1 P E 2 ) and, thus, the parabolic subgroups

of W defined using 2' to those dehed using 2.

There are M e r conjugacy relations between parabolic subgroups. i f w e fk

and look at the Werent parabolic subgroups dehed using 2 then they are often

conjugate.

Besides paraôolic subgroups one can also define m r m t -. Given

1 c (1,2 ,...., Q} k t

Z i = ( q ( k I ]

A, = the subroot system generated by LI

Then ZI is a hdamental system for AI. There is an obvious relarion between patabolic

15

subgroups and pamboüc subroot systems. Namely, W, is the refleaion group associateci

to the root system A, . So we can write W = W(A$ The transitivity of the paraboiic

relation (for groups as weU as root systems) follows fiom

-: if A" c A' and A' c A are parabolic subroot syaems then A" c A is a

parabolic subroot system as weil.

PrPnf: Pick inclusions of fiindamentai systems C'* c C ' and C, c C realizing the

parabolic relations A" c A' and A' c A, respectively. Since both C' and LI are

fundamental systems of A' one can choose cp E W' = W(A') such that 2' = cp OZ,.

But then 2', c X' = (Q *2, c cp -C which verifies the parabolic relation A" c A.

Q.ED.

F3: For any subspace V c E, A(V) = A n V is a parabolic subroot syaem of A

with respect to some fundamentai system 2.

Proof: Consider V = V, 3 V, 3 V, 3 ... where dim(ViA,) = dim (Vi) - 1. Ifit is shown

that Vi_, is parabolic in Vi, and Vi is parabolic in Vi+ then by induction and transitivity

(Lemma A), we will have Vi parabolic in V for any i. Since Vi-, is of one dimension

l es than Vi, Vi-, can be obtained fkom Vi by defining a hear fùnctional on Vis So it is

diCient to show that for any t a E, A(t) = (a e A 1 (&a) = O ) is a parabolic nibrwt

system.

Now if A' c A is parabolic then there exist hdamental systems 2' c A' and 2

c A with C' c C. Since ME') c w) are also hdamental systems of A', A

respectively, HA7) c A is aloo parabolic. Since W acts transitiveiy on chambers, there

exists p with Mt)€ C* = {XÉE 1 (cq,x) 2 O for aii i}. So wecanassurnethat t t C*.

Hae Z = (a,, .... 4) is the fiindamentai system.

Choose Ic (1.2. ... PJsothat t e C , = ( x e E 1 (aj,x)=O if j e 1 and

(aj, x)>O if j O 1).

Then A(t) has 8, = (ai 1 i E 1) as a fundamental systemand so A(t) =A,. Q.E.D.

In this section we show how parabolic ~bgroups arise natwally when we consider

the action o f W on E.

m: Given a set r c E the btr- ofr is

W r = { q e WI c p = ~ = ~ f o r d x ~ l ? ) .

-: For any set r c E, Wr c W is a parabolic subgroup.

-: For any set I' c E, Wr c W is a reflection subgroup.

We end this section by discussing coset representatives for Wl . Let

W ' = ( ~ I E W I c p * a i > O f o r i ~ I )

= ( q ~ E WI P(cpsaci9 = P(q) + 1 for i E I]

The quivalence of the two Merem conditions used to descrii W' is provideci by

properties of length. We will prove that the set W' provides coset representatives for W,

Le. w = W' W, .

d: Every Q E W can be decornposed cp = cp, where cp' E W' , cp, E W1.

pro& By induction on Im@. One cm assume cp e w'. So there exins i e 1 such bat

'm* sq$ = WP) - 1

By induction p sMi, = cp' - q+ where cp' E VVf . (PI E WI - SO

This decomposition is of the desired form.

Definitian: Two subroot systems AI, A, or two findamental systems ZI, Cj are

-, iftbere exists cp E W(A) such that @Al) = A, or q(C3 = 2,.

-: Given 1, J c ( 1,. . . . ,P ) t hen the following are equivalent :

(i) W, and W, are conjugate in W

(ü) A, and AJ are W equivalent

(üi) CI and C, are W equivaient.

eLPPf: First of a& (i) and (ii) are equivalent. This foUows corn Property P - 4 (of section

2-1) plus the identities Wl = W(A3 and W, = W(A,). Secondly, (ü) and (Ki) are

equivalent. One uses the fact that ZI and XJ are fundamemal systerns for A, and AJ .

Clearly, (i) irnplies (ii). Conversely, suppose ip *Al = Aj. Then cp Z* is a fundamentai

system of q *A, = A, . Since W, acts transitively on the tiindarnental s y ~ e m s of A ,, it

foilows that cp - XI and 2, are reiated by some cp' E Wj Le. cp7*cp-Cl = Cj . Q.E.D.

The study of conjugation of parabolic subgroups includes the study of nomahzen

of parabolics. Let

N(W3 = (p E W 1 c p - ~ p p - L = W,)

Normalizers have simple decompositions hto semidirect products. Let

N , = t c p ~ w I ~ Z i EC,}

From Howlett, we have the following proposition

m: N(K) = N* WI

P t e Cl-, W, a N(W) and NF N(W& Also, NI n W, = ( 1 }. For W, acts fieely on

18

the fundamental syaems of AI . So (P CI = 2, forces cp = 1. Lastly, N(W3 = Nt W, .

For. given cp E N(W3, then cp-Al = A, and so p.Ci is a fundamental synem of cp-At = A,.

However. since WI acts transitively on the fundamental systems of A, one can pick q+ E

W, rnapping Et to q1-2~ i.e.

'p.2, = #Z,.

If one rewrites this equation as cp ch-'. CI = CI one also has that (p q+-l E Nt. If we

let cp' = cp cpt'' then we have the desired decomposition <p = cpl - q+ . Q.E.D.

Chapter 4

on 4 - 1 In-

Given a root syaem A then for each fiindamental system C c A , there exists an

element mz E W(A) characterireci by the property:

0 , - X = -X

m: m2 is unique for a given fundamental system Z c A.

hoof Suppose there exins JE' such that oz'@) = 4. Then % (%'-'*c) = C . But

the elements of W(A) act M y and transitively on the tùndamemal systems of A, so 9

&?.1 d

= 1 and 9 =+'.

The element y: is an m. For. by linearity, %'*c = 2. The fact that W acts eeely

on the fiindamental systems then forces wZ2 = 1. The fact that -Z is a fundamemal syaern

and the tmsitivity o f W(A) on fundamental systems guarantee the existence of (JE.

The propex-ty agZ = -2 implies that o, agrees with the linear map c = - 1 moddo

a map cz which permutes the hdamentai rwts. In other words,

% = -Tz.

Since (JE is an involution r2 must also be an involution. The involution rz is most

easily understood as a symmeay of the Coxeter graph of W. Clearly, ta is equivaent to

such a symmetry, for X = the vertices of the Coxeter graph.

The key to understanding arbivary invohitions in a Euclidean reflection goup is

the îinear map c = -1. This is m e despite the fàct that -1 does not necessarily belong to

every refldon group. If-1 belongs to W(A) then it d e s the above definition and so,

20

mua be identical to the involution % for any fundamental system Z of A. Analyzing oz

on a case by case basis and determining when % = - 1. one can ver@ that, for an

irreducible reflection group W(A) c GL(E), one has

w: -1 E W(A) if and only i f A = A,, Bk, D,, E,, E,, F,, G2(2k), H,, H,.

Convenely, - 1 e W(A) if and ody if A = 4(k22), DZk- E6 and G2(2k+l). As

memioned before. these are the cases that oz = -q where rz permutes the elements of C

in a non-trivial fashion ( [5 , Chapter 91).

We are interested in a specid case of W equivdences. Its definition requires the

elemems % and rK as d&ed above. Consider the case XI c CK where

&=CI u ( a )

for some a E Z -CI. The element

o(La) = qf q

maps CI to a subset 2, of ZK . We will cd o(I,a) an and also

say that CI and 2, are related by an -. We will adopt the notation

w = s

to indicate the above elementary equivaience.

We can slightly rephrase the above dennition 2, and ZJ are related by an

2, = c, u (a ) = ZJ u {P)

The importance of elementary equivalences Lies in the foiiowing result.

a: C, and 2, are W equivalent then they are related by a senes of elementary

equivaiences.

It is very easy to determine for a given subset CI c X exady what other subsets

of C are related to it via an elementary equivalence. One takes each a E C - 2, . lets

C, = Cl u {a) and determines the effect of TK on Cl. Cleariy, by lemma (4-l ), one need

only concem oneseif with the cases where C, =4(k22), D,,, E, and G2(2k+l ).

a1 a, a3 0% w: Consider the case of A = D, with Coxeter graph u 0%

Then

(i) for 2, = (a,) we have the elementary equivalences

) o r = a , } we have the elementary equivaiences

(Ki) for X, = (a,, a3} we have the elementary equivalences

(iv) for 2, = (a , a,) there are no elernentary equivalences between it and any other

subset of C .

O b w e that the above equivaience classes are concerneci with diEerent types of root

syaems. In (i) we have root systems of type A, . ln (ii) we have root systems of type

A2 . In (iii) and (iv) we have root systems of type A, u A, . The above discussion can

easily be extended to show that al1 subroot systems of type A, , or of type A, . are linked

by elementary equivalences. On the other hand, the mot systems of type A, A, Mi

into two classes with (üi) and (iv) providing representatives.

By M e r extending these arguments we cm show that, except for subroot

systems of type A, u A, , any two nibsets fiom 2 of the same type are linked by a

series of elementary equivaiences.

Elementary equivaleoces can be used to show that al1 parabolic subroot systems of

type A+, within a given root system are W-equivdent (i.e. linked by a series of

elementary equivaiences). By studying graph symmetry on subsets of Coxeter graphs, we

aui Msualize elementary equivalences ([S, Chapter 101).

The minimal decomposition of cp E W(A) is a product of reflenions where the

number of rdections is as s d as possible. Let

EP = the elements of E fixed by cp and

E, = the orthogonal complement ofEq

If p is aa involution, then E, and EP are the 11 eigenspaces.

If cp has orda greater than 2, thm E, decomposes as a direct s u m of

eigenspaces where one cornpldes, Le., passes fiom PL to C. For each E C, we will

use the following notation for eigenspaces.

v(cp,a = { X I q(x) =&;.

Caner showed that, for each cp c W(A), one can h d roots { a ( 1 ), . . . - a (m) 1

giving a basis of E, such that

(9 = sa ( 1 ) .--- Sa (ml

is a minimal decomposition of p . This provides a very us& decomposition of elements

in W(A) . However, it is not always canonid. Moreover, the factors

(s, ( ...., s, ,,,} do not comrnute. This would require that the vaors (a (il, .. . ., a (ml )

be rnutually orthogonal. There is only one case where this occurs. Carter showed that an

involution can be decomposed into a product of reflectioas of muhially orthogonal roots.

Carter achieved his result by reducing to the case of the involution - 1. Suppose

-1 E W and let Q = rank W. then it is shown that -1 has Q reflections in its minimal

decornposition, say - 1 = sa, l , . ... sa, . Carter proves that a, is orthogonal to { %, .. ..,

a, 1 by showing that the subspace orthogonal to a, is spanned by (az, . . .., a: } . By

generaiizing this argument to show that (a,, ...., q} are orthogonal to {aiAI, ...., 6 },

we cm see that (a,, . . .., at ) are a set of rnutudy orthogonai roots ([Il).

Cbapter 5

Tbe Decompos~on Theorerq 1 .

Pro~osition: If W is a Weyl group with crystallographic root system A and cp E W is of

order p, then there exists a subroot system A, u . . . . u A, nich that :

(i) rank Ai = p- l

(ü) (P = q, . . .. q+ , where I * cpi e W(Ai) is of order p.

In other words, cp uui be decomposed into a product of commuting elements of order p.

Pro& Let ip = sMl, s,, .. .. s,, be a minimal decomposition of cp.

= n HMi,, I s i s k .

is the aibspace of vectors ked under the action of cp .

V, = the subspace spanned by {al, ,.... ai ) .

A,=AnV,

W, = the subgroup W(Ap) = the isotropy subgroup for .

Then V is the direct sum of V, and v, where V, and are orthogonal

complements and W acts trivially on V' while it acts as a reflection group on V,.

By restricting attention to V, , A, and Wp = W(AO)% (P has no fixed points in

v=vc .

Let L c V be the mot M c e ( d o n 2-2) on wbich W is considered to be acting.

rp permutes the elements of A. Each orbit of p in h consists of either a single element

or p elements. Let CI = (al , ...,g} be an orbit with p elements. Let L(a) be the

sublaaice of V spanned by o o v a 2. Let V(o) be the subspace of V spanned by o over

R. We have L(o) c V(o).

Since cp acts on Lw) with no k e d points, L(a) is not the trivial representation of

25

Z/(pZ). ï fdim L(o) = p, then a, + ... +a, * O and L(a) contains the fixed point a, + ... - a, and thus a copy of the trivial representation. But since L(a) has no k e d points under

the action of cp, it must be that dim L(a) + p.

Since Z/pZ has two irreducible representations over 2. one of dimension 1 and

the other of dimension p 1, we therefore have dim L(o) = p- 1 and (a,,%,. ..,a,] satisfy

the relation al+. . .+ap = 0.

Let A@) = the subroot system A n V(a), that i s the element of A lying in

V(o). By lemma B in section 3- 1. h(o) is parabolic. So if we choose the appropriate

fundamental system Z of A and the appropriate 1 c { 1,2.. ..,P } where Z = (a,, a,,.. . .a2 1,

we can write V(o) = VI, &(a) = Al and W(A(o)) = W,, and = {ak 1 i e 1). in

particular, the subset 1 mua have size p-1 .

Since cp maps both A and V, to themselves. it aiso maps Al = A n V! to itself.

Thus, cp E N(W9, the nonnalizer of Wl , for as already observed, if <p (a) = P, then

'P sa qi = p.

From Howlett, section 3-3. there is a decomposition N(WI) h o the semi-direct

product of NI and W, where NI = (8 E W 1 8 (ai) E 2,, i E 1). Let Z, corrtained in NI

bethesubgroup Z I = ( + ~ N I 1 $ ( a i ) = û i , i ~ I ) .

We observe that elements of 2, cornmute with elements of W,. Let E ZI and

let sail E Wt . Then, we have

4 s~il d 'L - - % (propaties of rdections)

= s,,, (defiaicion of 2, )

and thus 4 s,,, = sN9 @ . We also observe that Z, n W, = ( 1 } since the ody element in

36

W, which fixes every root in Z1 is the identity element. Hence ZI and W, forrn a direct

produa. We want to show that cp E 2, W, .

Lemma: 'P E ZI W,

Proof. N(WI) = W1 PS NI

Consider the exact sequence:

Kernel of x = Wl . Let N = N(W&

Wecanwrite q = q , cpz where 9, € N I , cp, É W,. Wewant to show that c p , ~

Z I c NI .

Since cp, q1 ciifFer by a factor in WI , they have the same image in N/WI , say 8.

Suppose cp ,' = 1, then either (r,p) = 1 or (r,p) # 1.

If(r,p) = 1, then since cp, cp, both map to 8 and since cpP = 1, we have r = 1 and 0

is the identity element in NAVf . So cp, E W1 . But cp, E NI and WI n NI = ( i ) by

definition of semi-direct product. Hence, pl = 1 and cp = cpl e WI .

If (r,p) t 1, r = tp, t e z ~ . n induces an isomorphism between NI and N/W1 ,

and so the order of pl in N(Wc) is the same as in N/WI. Since OP = 1 in N W , plP E

W, inNw) . S inceWInN,=( l ) , c p l P = 1.

In either case, cplP = 1. Since cpl o NI, pl permutes the eiements of 21.

However, ZI is of rank p 1 aad q, is of order p. Thus, cp, &es each element of Et .

In fact, cp, E Z1 and minmutes with the reflections associateà with the elements of ZI .

Therefore cp = 9, cp2 E ZI WI .

Now 2, is the isotropy group for VI and thus a parabolic subgroup of W (section

3-2). Let Zr = W(A,). A, is contained in the onhogond complement of VI and so we

27

can now repeat the procedure with W(AI) and continue the decomposition of cp uito

orthogonal (commuting) elements of order p. Q.E.D.

It is not always possible to decornpose an efement of order k where k is

composite, into a canonical product of elements of order k. For elernents of order k

any factor of k is a possible orbit length. For example, consider &., , the symmeuic

group on P letters. Let P 2 5 and let cp = (123)(45) an element of order k = 6.

uistead of decomposing into a produa of canonical elements of order 6, cp is expresseci

as a product of an element of order 3 and an element of order 2.

We can also produce coumerexamples to the decomposition process for primes in

the case of non-Weyl groups. For example, the order of H, is divisible by 5 and so

H, contains an element of order 5. However, H3 does not comain a copy of A, since

H, is of rank 3 and A, is of rank 4 (rank is the cardinality of the fiindamental system).

This problern does not occur in the Weyl groups because if p 1 1 W 1, where W is Weyl of

rank n, then p s n+l.

Chapter 6

Strucm= and upi4iipaegg

Section 6 - 1 Structure of' AI

From Chapter 5, we have that A, is parabolic and that q+ has order p on A,.

Suppose A, is discomected, consishg of n segments. So AI = hl U .. . U A, and

W(A) = W(A,) x ... * W(AJ. Then q+ = q, qz ... qn where cpi acts on segment i. The

elements ( c p , ..... cp,) also cornmute with each other. Since = 1 and (pl rp2 ... cp,)P

= cplP ~ p , ~ ...cpnP = 1, it foUows that q? = 1 V i. Since p is prime, q+ has order p

or is the identity on each segment. We can reduce to considering (pi acting on Ai . We

can therefore assume AI is COM-€!~ and consider the possible structure of A, . where

A, has at most p-l vertices and q+ has order p on A,. In the rest of this section we

wiU use these hypotheses to show we can always choose:

a: Ar = %, .

The rest of this section will be devoted to the proof of this lemma

From section 2-3, we have 1 W(A3 1 = P ! II hi g, 1 r i d , where Q is the rank of

the group, where (4 1 1 a i d ) are the coefficients in <b = Chi ai 1 s i s P, the expression

of the highest mot in terms offundamentai roots and where g = k + 1 where k is the

number of î+ equal to 1.

Since k pl, p does not divide P!. Since cpl is of order p and since the order

of q+ mua divide the order of W(A3, p 1 I[ 4 , 1 i i d or p 1 g.

If p 1 g, there mua be p vertices on the extended Dyakin diagram with label 1.

' ' h e m - . . is formed by adding -a, to the Coxeter graph (section 2-4)

and labehg each vefiex with its coefficient hi in a,, = xb ai 1 ria@ (and IabeUins 0%

39

with 1). From the diagrams (section 2-4), the only such diagram with p vertices wit h

label 1 is %, .

Consider the case p 1 Il h; , 1 r is P. and let a$= 2ao /(a ,a,) be the coroot of

cq, where cc,, is the highest root of W(A,).

As explained in section 2-3. we have the expansion q* = h,* q * where

4 = 4' (%%)/(%a- So if p l II hi , l si&, p divides 4 for some i and nom the above formula

relating hi and h,* , we mua have p 1 hi * or p 1 (%%)/(ai, ai) .

If pl hi*, then from Steinberg's lemma of the string (section 2-5). the structure of

the coroot system must include a singly-bonded string of p vertices with labels 1,2.. .,p.

in the extended Dynkin diagram. From the diagrams and labels (section 2-4), there is no

such case.

If pl(&cq$(ai7ai) , then from the possible ratios tisted, we have that if p is odd,

the only possibilityis p=3 in the G2 case. Now 1G21 = 12 and (A,I = 6. Since W(G)

is the semidirect product of 2/22 and W(A2), an element of order 3 in G2 is actudy

contained in A*. Therefore, in all instances, we have shown that the structure of Al is 4,.

Every element q of order p is of the form cp = cp, cp2 ... cp, where cpi E W(Ai)

and Ai has structure %,. So cp E W ( h L ) W(+I) ... W(Apl) (t thes). We

consider two decompositions of cp to be equivalent if they are conjugate to each other.

M: Any two copies of 4, in the A, D, E6, Ç, &, B, cases are W-equivdent and

are related by a series of elementary equivalences (section 4-2).

30

Proof: Fint we will show that any copy of al is parabolic. Consider a copy of +, in

h and choose cp F W ( 4 , ) = S,c W(A), such that cp is of order p. cp acts on the

roots 4,. Choose an orbit (P l , P2, ...., P,) in Let V, =the subspace spanned by

( P i , P2, ...., Ppj and let A, = A n V, . By the sameargument as in the proofofthe

decomposition theorem (chapter 5 ) , A, is parabolic of rank p- 1 (Le. dim V, = p 1 ) .

Now the roots of Apl span a (p-1 )dimensional space V. A h , the roots ( P , , Pz, . ..., P,) span a (p-1)-dimensional space V, . By definition, (P ,, P2, . . .., Pp)c %I and so Vq c V. Since the dimension of V, equds the dimension of V, we must have

V, = V. Since %, c V, we have 4, c V, . The fact that +, c A and 4, c V,

gives that %, c Aq or A, c G2 = Av . Thus, any copy of SI is paraboiic in A,, D;,

E6, E,, E, . From the results of section 4-2, any two copies of

%, are W-quivalent and are thus related by a series of elementary equivdences.

Q.E.D.

We can therefore move the &st copy of Ap, obtained in the decomposition

process to begin at a, (in the extended diagram of the original group W(A)). ( R d that

this copy of %, is parabolic in the original group.) Then we restrict to the orthogonal

complement of the first copy of GI , and as the decomposition process yields tùrther

copies of %, , we can repeat the process of moving each copy to begin at the highest

root of the orthogonal complement.

This algorithm produces a cawnical product W ( 4 , ) W(%,) ... W(&).

For each W(+,), any two eiements of order p are conjugate since they are Coxeter

elements ofW(+,) (section 2-5). Thus the choice of (pi in the decomposition of cp is

unique up to conjugacy.

In the case of F,, there is a copy of A? (long roots) and a copy of À2 (short roots).

Using Carter's list of conjugacy classes. we have the following possibilities for

elements cp of odd prime order:

Note that not al of the above are parabolic.

In the cases of A,, BI, Dg, we will provide a method for caidating the maximal

number of copies of 4, for a given p. Suppose A, contains a copy of (beginniag

at highest root) for some odd prime p. Once this copy is rernoved f?om the extended

diagram of Aj , we have (Q+l) - ($4) - 2 = P - p vertices remainimg in the orthogonal

cornplment. We proceed by induction on this cornplmeutent Let k = P - p. We have

Np1 copies of ql in the complement. ïhus the amximai number of copies of in A

32

is [Wp] + 1 = [klp] + plp = [(k+p)lp] = [Plpl.

Similady, in the Di case, we have (P+l) - (p-1) - 2 = t - p vertices rernaining in the

orthogonal complement after removhg a copy of Apt (that begins at highea root ).

Again, we can have nom i to [Plpl copies of %, . In the B, case, P - (p-1) - 2 = Q - p -

1 vertices remain in the cornplement and [(P- 1 )/pl is the maximal number o f copies

possible.

Cbapter 7

w o n 7 - 1 Elencm

In the decomposition of ip = cp, cp2 ... cp, . each cpi acts as a Coxeter element

(section 2-5) permuting the roots of Ai (a copy of a,). Since all copies of %, are

orthogonal (nom the decomposition process) and since Coxeter elements are regular

(section 2-5), cp is also a regular element. if vi is a regular eigenvector of cpi with

rpi (vi) = vi , then cp(vi) = cp, q2 .. . cpt (vi) = cpi (vi) = cvi since cpj (vi) = vi , i j . Thus,

the regular eigenvectors of cpi , 1 sist are also regular eigenvectors of cp , on the

subgroup W(%,) x W ( 4 , ) ... W(Apl) (t times).

The question remains whether cp is regular on the original group W(A). We know

that the fundamentai reflections generate W(A) and that an eigenvector x of <p lies on

the hyperplane of a reflection s, if(xa) = O. Suppose there is a fundamental root P

orthogonal to the roots in %, Y ... U 4, (determineci by the decomposition of 9).

n e n (xP) = ( cp(x),

= (km

= €(SPI

and this implies that (x$) = O and x lies on the hyperplane b. Thus, ifthere are

fundamemal roots ofW(A) in the orthogonal component of the copies of %, associateci

with 9, cp is not regular on the original group. Conversely, if there are no fùndamentai

roots of W(A) in the orthogonal cornpiement, then cp is regular on W(A).

We can also ask whether an element cp of order k where k is not prime, is

regular on some subgroup. In the h-, = S, case (symmetric group ou Q letters), cp is a

product of disjoint cycles of length k,, k2, k,, . . ., 4, where the least common mdtiple of

34

(k,, 1 sir n} is k. Each cycle of length ici is conjugate to a Coxeter element of a

subgroup hi, ., and thus regular on that subgroup. Since the cycles are disjoint, any

regular eigenvector of a cycle ici is also a regular eigenvector of cp.

In the B: = S: DC (2/22)' case, an element cp of order k is a product of disjoint

cycles involving sign changes. Let cp = cp, cp2 ... cp, with cycles of length k,, k,, ..., .

Each pi is conjugate to a Coxeter element of a subgroup hi, ., (Even number of sign

changes in cpi) or a Coxeter element of a uibgroup Bui, (Odd number of sign changes in

cp). So each cpi is regular on a subgroup and again, since the cycles are disjoint, any

regular eigenvector of a cycle cpi is also a regular eigenvector of cp.

in the D, = S. D. (2/22y -' case, we fhd the following counterexarnple to the

question of bding a subgroup on which an elemerrt cp of order k is regular. Consider

(p = (123)(456) with one sign change in eacb cycle. Now cp is regular on W(B3)

W(B,) (by the previous discussion of the B, case). However, B3 is not comained in any

D, group since âom the Coxeter graphs, B, has a double bond while D2 is C O M - ~ ~

ody with single bonds.

Therefore, for an eiement of composite order, it is not aiways possible to h d a

wbgroup on which it is regder.

Chapter 8

w o n 8 - 1 A m n-Subgrp(lpg

An elementary Abelian psubgroup is a subgroup in which every eiement has order

apowerofapnme p. Let9 = < p l cp2. . . c p , ~ W ( + , ) x W ( % , ) x ... x W(+,) b e a

decomposition of cp . where cp is of order p. Since each cpi is of order p, it generates a

group isomorphic to UpZ. So fiom the decomposition cp = cp, <p2 . . . cp, , we obtain an

elementary Abelian p-subgroup (z/~z)'. One question we can ask is whether al1

elementary Abeiian p-subgroups are generated in this manner.

C o ~ e r e a : In the case @, the Klein4 subgroup provides a negative answer to

this question. It is generated by (12)(34) and (1 3)(24), and also contains ( l4)(23) and

(1). Since it does not contain any single 2-cycle, the Klein4 group is not the redt of a

decomposition of an involution (p=2) h o a product of reflections. We can generalize this

Ktein-4 situation to other primes p. For instance, the 3-subgroup generated by

(123)(456)(789) and (147)(258)(369), which again does not contain any single 3-cycle.

These Klein4 type subgroups are contained in symmevic groups (4 -,).

To see the pattern of these counterexamples, consider the Sylow p-subgroup Z/pZ

b< (Z/PZ)~ c S, where ZlpZ is generaîed by

2 2 0 = (1, p+l, 2p+l, ..... p - pl)(?, p+2, ...., p - pt2) ... (p,2p, ....,

while (ZlpZ)P is generated by {(1,2, .. .., p),(p+l, .. . .. 2p), ... ., (p2- p+ 1, . .. .. p2) }

and in particdar contains the element

2 0' = (1.2, .S.., p)(p+l, ...-, 2p) ... (p - pl, ....,

The psubgroup generafed by 8 and 8' does not contain any single pcycle.

Ifwe want to produce an elementary Abelian psubgroup using an element p of

36

order p, we kst need to determine another element 8 of order p. such that cp and 0

cornmute. This will give ( w ~ z ) ~ by taking the cross product of the groups generated by

cp and 8. We cm continue to h d more commuting generators of order p to create

more copies of U p Z . Thus, by midying the centralizers of elements of order p, we can

better understand how to produce eiementary Abetian psubgroups.

a: Let cp be an element of order p ui the symmetric group on P letters

S; = W(4 -,) (cp is a produa of k p-cycles). Then the centraker of cp is

sk .< (Z/~Z)' sg -kp .

Pro* (i) S, &es the possible permutations of the order of the k cycles.

(ii) (~~2)' produces ali the powers of each cycle.

(iii) S, .,, gives the symmetric group on the remaining P - kp letten that are not in

the cycle expression of 9. Q.E.D.

In the B, case, S, w ( 2 / 2 ~ 1 , we have the extension (2l22)' a S, o- (2/22)' - S: . In the Dg case, S, (z/@ " , we have the extension (z/zz)~ -' Q S, w (z/zz)' -' - S, . So, in the case of an element of odd prime order, the cmtralizer in the B, and Di

cases has the same p-prîmary part as in the 4 case.

In the F,, Es, Ç, ,E, cases, the structure of the centralizer is more complicated.

We will use the foiiowing proposition.

-: Let q = s,, , sq2, .. . sa,, be a minimal decomposition of p and let

V, =<a,, %, ..-, a& = (v[Nv) =v} . Let AD = A n V, and let W, = W (Ad.

The centraliter Z(q) is containeci in the normalizer N(WD).

Pro& From Carter's work on minimal decomposition, we have that (ai 1 1 sise) are

lineariy independent and that the nxed space V' is the orthogonal cornplment. The

37

generating roots of V, depend on the decomposition, but the resulting space is the same

regardless of the decomposition So we can say V = V, @ where V, = VD .

Suppose x E and s E Z(@. Since cpr(x) = q ( x ) = s(x), it foliows that

r(x) E V<P and that Z(q) maps to itsetE Since AD = A n V, and since Z(q) maps

A to A (by definition of root system) and V, to V, , Z ( q ) maps AD to AD.

Suppose a$ E AD and s e Z(cp), with ~ ( a ) = P . From properties of reflections,

we have that r s, r" = sp . Thus r WD f t = WD and Z(q) c N (MiD) . Q.E.D.

Since we need elements of order p in the centralizer Z(q) and since Z(q) is

containeci in the nomaiizer, we can use Howlett's tables of normalizers and isolate the

Sylow-p subgroup to discover the maximum possible number or copies of (Z/pZ) in the

centralizer Z(cp).

p-S ylow

The notation under cp of order p represents the Coxeter eiement of the given root

systern. The list of root systems comprises ail copies of (&,)GP to W-equivalence.

In the situations where the nomalizer is not provided in Howlett's paper, Le. those

in which we are dealing with a non-parabolic subroot system, we have used the p-Sylow

subgroup of W(A) together with the structure of the subroot system correspondhg to cp

within A. We w d i explain each of these situations where the normaiizer is not given.

1. Consider A,X& in F, . A ~ À , contains 2/32 x n32. The 3-Sylow

nibgroup of the cenaalirer can be no larger than this since the 3-Sylow subgroup of F,

(order = ~ ~ 3 ~ ) is of order 9.

2. E6 has order 273'~. The 3-Sylow subgroup is P = 2/32 EX ( ~ 3 2 ) ~ . The fim

copy of (Z/3Z) is obtained by w,w,' where wo, w i act as foiiows:

a , q, to -a3 . Thus w, w,' "rotates" the above diagram by acting on {a, al- --- -a6} in

the foiiowins fashion. wo w,' maps al to (., , % to a,, a3 ta q. a4 to a6, as to

The other three copies of (U3Z) are produced by the product of the two

reflections correspondhg to the roots in each of the three sets ( a,&, { a3,a4 1, ( u6,a() 1

Now (A,)~ h E6 represents the diagonal element of P (that is, it commutes with

ali elements of P) and so the 3-Sylow subgroup of its centralizer is P.

3. Since E, c E, and since the order of E, is 21°3"5 7, P is also the 3-Sylow

subgroup of 5 . Again, (A*)~ represents the diagonal element of P and its centralizer

has aii of P as its 3-Sylow subgroup.

We have E, x A, c E8. This gives P 2/32 as its 3-Syiow subgroup. Since

(A$ in E8 is the diagonal element, the 3-Sylow subgroup of its centralizer is P x 2/32.

Since any two copies of (A*)) are conjugate, we cm choose (A$ in E6 , h m

we 2 its centralizer in E6 is P = (ZnZ) oc (2/32)3 . Also the additional copy of 2/32

in the other copy of A, in E, wiii form part of the centralizer. Thus, the p-Sylow

obtained for ( ~ 3 ' in E8 is P x 2/32.

13 5 7 the cemralizer can be no larger than tbis since the 5-Sylow subgroup ofEs (order 2 3 5'

7) is of ords 52 .

We wil i now use our list of centralizers to discover the possible generators of

elementary Abelian p-subgroups.

40

in F,. the table of centralken reveais that we have three possible 3-subgroups

and that p=3 is the ody odd prime for p-subgroups in F,. For a given fundamental

system of F,, there exists a copy of A? (long roots), À2 (short roots) and their produa

A, x À,. An element of order 3 in A2 or À2 generates a subgroup (2 1 32). while

using an element of order 3 from each of A2 and À, we have (2 1 32)'.

For E,, E,, E, , we wi.ü need fùrther detail to understand the 3-subgroups.

However, before pursuing this. we will deal with p=5 in E,, E,, E, and p=7 in 5. For a

&en fundamental system of E,, its diagram contains a copy of AS and an element of

order 5 in A, generates a subgroup (ZfSZ). We h d copies of A, and & in E, and these

contain elements of order 5 and 7 respectively, thereby generating wbgroups (2152) and

(Zf72). In E8, we have two copies of A4 and one copy of &. By choosing an element of

order 5 in each copy of A, and an element of order 7 in 4, we can produce (USZ).

( w ~ z ) ~ , or (2.72). In all these instances, the choice of generators is unique in the sense

that any two copies of %, are related by a series of elementary equivalences and any

two elements of order p in +I are wnjugate (Coxeter elements).

We now hm to the case of possible 3-subgroups in E6, &, Es. We will begh

with IWO lemmas.

-: Let 0 = 8 e2 ... et E W(hl) x ..- x W(Apl) be an element of order p.

Then

(i) The multiplicities of non-trivial eigenvalues are the same.

(ü) Decomposition length k can be characterized as the mdtiplicity of the

eigenvaiues.

eLQPf: Each Bi can be represented as a p-cycie in the symmetric group on p letters, S,

4 1

= %, . We have S, c GLPl (Z)c GLPl (R) c GLpl (C). The characteristic poIynomial

for Bi is xP- 1 = (x- 1 )(x + x + .- - X+ 1) = (x- 1) II (x-< '), where 6 is a pm root of

unity (1 5 i i p- 1). Hence Bi has eigenvalues f. e2. .. ., 5 with multipticity 1. Thus, 8

= 8, 8, ... et has eigenvalues 1 , c, p. . . ., e each of multiplicity k. Q.E.D.

Next we tum to the case of elements of order 3 in WG). We prove a technical

lemma about such elements. Every element of order 3 in W(E,) cm be found in a

Sylow 3-subgroup. The Sylow 3-subgroups of W(E,) are of the fom

P = 2/32 DK (2132)~ . To locate nich a subgroup, consider the extended Dynkin

diagram:

Then P = Q C-R isdetenninedasfoUows. Let V1= (a1+& V2= (axa4), V,

= (a,%). Then P =Z/3Z acts by rotatiag V,, V,, V, (i-e. it rotates the amis of the

diagram) whiie Q = 2/32 x 2/32 x 2/32 is @en by the three copies of 2/32 c h,

where Sg = W(A3 is given by the three copies of the mot system A2 hahg V,, V,, V3

as hdamental systems.

w: Let P = Q w R If 8 E P, 0 6 R, then 0 has decomposition length at most

Pro& Since 8 B El, 0 = el û2, where €4 E Q, B2 E 0, + identity. So 8, E Q acts

on a given root a E VI by permuting it with a root E V2 and a root y E V3. Shce a,

P, y are h d y independent, a + P + y * O. The root a + P + y is fixed under the

action of 0 and so it is an eigenvector with eigenvalue 1. Now the other possible

42

eigenvalues of 0 are 6.6' since 0 is of order 3. Shce V, u V, u V, span a 6-

dimensional space and since the eigenvalues C and c2 have the same multiplicity (lemma

A). their multiplicity cannot exceed 2. Thus, by lemrna A, the decomposition length of 0

is at most 2. Q.E.D.

The above lemmas together with an examination of the centraken of elements of

order 3 WU help us to understand the possible 3-subgroups in E,, &, 5. We will simple

to subgroups of rank 2, (2/3212. We will represent the three copies of (ZDZ) in E, and

E, using vector notation where (1,O.O). (0,l ,O), (0,O. 1) correspond to the three

generaton. We will represent the four copies of (ZBZ) in Es ushg four-dimensional

vecton w here (1,0.0,0), (0,1,0,0), (0,O. 1 ,O), (0,0,O. 1 ) correspond to the four generators.

ï h e goup operation is vector addition. Given an elementary Abelian 3-subgroup of rank

2, we can choose an element cp of minimal decomposition length (minimal number of

non-zero elements in vector represematioa) within the subgroup. Then we will consider

the possible choices for a second generator cp' of order 3 which is necessarily in the

cenualizer of cp (as already observeci).

AU the 3-cycles in S, are conjugate to each other. In other words, the non-trivial

elements of ZnZ c S, are conjugate to each other. Switching to vector notation, there is

an element in the oormalizer of (ZDZ)~ or ( z /~z )~ which conjugates each (a,b,c) or

(a,b,c.d) to an element with all coordinates O or 1. Even more e~piicitly~ we can shply

replace each non-zero coordinete by 1. For example, (202) can be replaced by (10 1). So

when we consider generators cp and cp' of the group 2/32 x 2/32 we can reduce to cp

and p' of this special type.

1. Let cp E6 or E, with length 1. So Z(q) has p-Sylow (2/3213 and we can

choose <p = (1.0.0) (corresponding to one copy of A? in E, or E, ). We cm choose <p '

to be either (O, 1.0) or (O, 1.1). We can also combine (1.0,O) with either of these to fonn

cp' = ( 1.1 .O) or 9' = ( 1.1.1). The subgroup generated by ( 1 .O,O) and (0,l ,O) contains

(1,0,0) and (1,1,0) and the subgroup generated by (1.0.0) and (1.1.0) contains ( 1.0.0) and

0 O ) So these two subgroups are equivalent. Similarly, the subgroup generated by

(1.0.0) and (O, 1.1) is the same as that generated by ( 1 ,O$) and (1.1. L ).

1. Let <p E E6 or E7 with length 2. So Z(q) has p-Sylow (2132)' and we cm

choose cp = (1,l.O) (corresponding to two copies of A2 in E, or E, ). if cp' = (1.1, 1).

then 2 9 + cp' = (0,0, l), which contradicts the minimality of decornposition length of <p.

However, we can choose cp' = ( 1 .O, 1) or 9' = (O, 1.1) to fonn a group with minimal

decomposition length 2.

3. Let <p E E6 or Ç with length 3. So Z(q) has p-Sylow 2/32 C- (~32)'

and we can choose cp = (1,1,1) (corresponding to three copies of A, in E, or E, ).

Frorn lemma B. we must choose cp' E R = ( z / ~ z ) ~ in P to have decornposition length 3 .

However. the only such choices are (1, 1,l) or (2.2.2). Thus there is not a second

senerator 9'.

4. Let <p e E, with length 1. So Z(q) has p-Sylow ( U 3 Z P< ( Z / ~ Z ) ~ ) x 2/32

and we can choose p = (1,0,0,0) (corresponding to one copy of A2 in Es). We can

choose p' to be (0,1,0,0), (0,1, 1.0) or (0.1,l.l). We can also combine ( 1,0.0,0) with

each ofthese, thereby giMng (1.1,O.O) or (1.1,l.O) or (1,1,1,1). As in Situation 1. the

subgroup generated by (1,0.0,0) and (O, 1,O.O) is the same as that generated by (1,0,0,0)

and (1,1,0,0); the subgroup generated by (1 ,O,O,O) and (O, 1,l ,O) is the same as that

generated by (1 ,O,O,O) and ( 1, 1, 1,O); the subgroup generated by (1 .O,O,O) and (û, 1,l. 1) is

the same as that generated by (l,O,O,O) and (1.1, 1.1).

S. Let cp E E, with length 2. So Z(q) has pSylow (~32)~ and we can

choose cp = ( 1,l.O.O) (correspondhg to two copies of A2 in E,). If cp' = ( 1.1.1 .O) then

2 9 + rp' =(O,O,l,O) and if p' = (1.1,0,1) then 2rp + cp' =(0.0,0,1). ùi bothcases, the

minimal decomposition length is contradicted. However. we can select cp' = ( 1 ,O. 1 ,O),

rp' = (1,0,0.1). or cp' = (0,0,1,1) (which produces the same group as cp' = (1,1,1,1)).

6. Let cp É E, with length 3. So Z(p) has pSylow (2/32 DC (zBz).') x 2/32

and we can choose cp = (1,1,1 ,ù) (comesponding to three copies of A2 in %). If

cp' = (1,l. 1,l). then 2q1+ cp' = (0,0,0, l), and if cp' = (1,1,0.1), then 29 + cp' =

(0,0,2,1). Both possibüities contradict the minimal decomposition length. The c e n t r e r

of cp is P x (Z3) = [(213) D-= (2/3)'] x (ZB) = (Q w R) x (2/3). From lemma B. we

must choose rp' outside of Q. However, we have jus observeci that choices of rp' of

length 3 or 4 stiU produce a contradiction in composition length. Thus, there is not a

possible second generator 9'.

7. Let (9 E Eg with length 4. So Z(q) has p-Sylow (2/32 B- (2/3a3) x U32

and we can choose cp = (1,1,1.1) (correspondhg to four copies of A, in Es). From

lemma B, we have (1,1,1,1), (2,2,2,2) as the only vectoa of decomposition length 4.

Hence. there is no possibüity for a second generator 9'.

C a

It still remains to midy the choice of Q' in the &-,, B, D, cases using the

formula for the centralizer of cp of odd prime order (section û-1). We will work with

elementary Abelian psubgroups of rank 2; (Z/pZy in the symmdc group on P letters,

45

S,. We will choose an element cp of order p with a decomposition cp = cp, cp2 . . . Uito

pcycles where the decornposition length k (the nurnber of pcycles) is a minimum. We

wili examine possible choices for a second generator cp' of (ZlpZ)' . As previously

observeci, cp' is contained in the centralizer of cp. Aiso, since cp' is of order p, it is a

produa of p-cycles and cp' must have at lest k cycles to satis@ the minimality condition.

R e d that the centraürer of cp, Z(cp), is (SI: D.. (Z/~Z)') x S:. i, (Section 8-1).

1. Suppose cp' E (z/~z)'. Then cp' is the produa of k p-cycles, with each

cycle of length p, and with each cycle a power of a cycle of <p. When we calculate cpcp'.

we multiply each cycle of cp by its correspondhg cycle in cp' to produce a new power of

the same cycle. By ushg a power of cp, say ip: we have cprcp' = [ (cycle I r (cycle 2)' ...

(cycle kr ] [ (cycle 1)' (cycle 2)b ... ] and by choosing r = -a, 9'9' cornains at moa

(k-1) cycles. Since this contradicts the minimal decomposition length of k cycles we

c a ~ o t choose ip' containecl entirely within (ZlpZ)L.

2. Suppose q' e (Sk PG ( ~ ~ 2 ) ~ ) where cp' is not within ( ~ ~ 2 ) ' . To bener

understand this case, we can represent cp as a series of k wlumns where each column is

a cycle of cp. Now cp' in this case is a combination of pemtations from (z/~z)' which

pennute elements within columos and of permutations fiom Sk which permute columns

(while preserying the order withm columas). For example, let Q = (123)(456)(789)

which can be represented as:

The permutation (1 32)(465)(798) acts within each column producing:

The permutation (147) acts by pemuting the three columns to give:

Observe that ( 1 47) is necessarily accompanied by (25 8) and (3 69) to preserve the

order within the coIumas of cp. Thus, such a permutation involves p columns (Le. p

cycles nom 9).

Suppose that cp' has x permutations fiom ( ~ 1 ~ 2 ) ~ and y permutations h m

S,. Then cp' acts on x columns (cycles) of cp by pemuting the elements within each

colwnn and cp' acts on yxp columns of cp by pennuting each group of p wiumns.

Note that cp' must act on all k columns of q, because otherwise q' would have

decomposition length less than k, thereby contradicting the mhhdity condition.

First assume that x * O. Then cp' cornains a power of at least one cycle fkom ip.

From the same argument as in case 1, we can choose an appropriate power of rp, say ipr,

such that cpr cp' contains at most (k-1) cycles. This contradias the minimal

decomposition length of cp and so we mua have x = 0.

Since x = O and since cp' must act on al i cycles of q, the ody possibility is that

cp consists of yxp cycles and haî cp' consists of y permutations, each of which

permutes a distinct group of p cycles of 9.

Therefore, the number ofpossibilities for this case is determined by the choice of y

and p. Once y and p are determhed for cp t h m exists a unique choice of cp' up to

conjugacy.

3. Suppose cp' E S, . kp . cp' must hae at least k cycles, each of length p.

The Mirent possible choices of cp' are determined by the number of cycles in q', since

any two choices of cp' with the same nurnber of cycles are conjusate to each other.

4. Suppose cp' r (S, B.- (~~2)') S.. . kp, where the part of cp' contained in

each of Sb ( u ~ z ) ~ , and S: . k, is non-trivial. We observe the following:

(a) We can write the centralwr Z(q) as P x S where P = Se B- ( z / ~ z ) ~

and S=Sl+,

@) The cycles of any element in P are powen of those appearing in

cp so that each is a remangement of the numben nom one of the internai cycles of cp.

The external p-cycles appearing in 9' occur in patterns of p, for example, cp' =

(147)(258)(369), and permute groups of cycles of 9. Moreover the cycles can be

arranged in a mavix so that the rows give external cycles of cp' and the columns give

interna1 cycles of cp. For example 1 4 7 .

[: : il (c) nie elernents nom S are made up entireIy of intemal cycles.

(d) We can decompose each elernent of <çp, q'> hto intemal pcycles and

external p-cycles. If an exterd cycle appears in an element then it is a power of an

e x t e d pcycle appearing in q7. This is tnie since every elernent of ccp, <pl> can be

written (cp)i (pl)i-

(e) Each number appearing in an extemal cycle of some element also appears in an

imemai cycle of another element. AU other numbers only appear in a unique intemal

cycle. For example, given B = (1 47)(258)(369), then 1 dso appears in A =

(1 23)(456)(789).

Now, given a second choice of generator cp" E Z(q) of a cornplesnent to p,

suppose t h one can conjugate 9, <p'> into <cp, qf"'. One can choose the conjugdon

so that the following is me.

48

p r o ~ a : Extemal (intemal) cycles in <<p, q'> are conjugated to extemal (internai)

cycles in <<p, cp"?

pro& As the above ma& iilustrates, each number appearing in an extemal cycle of <cp.

cp'> also appean in a unique intemal cycle. AU other numbers appear in a unique intemal

cycle of <cp, p'>. If an externai cycle of <<p, cp'> is conjugated to an internai cycle of

<q, q">, then it occurs in p cases and the entries in a p x p m a h are mapped to a p

p matrix so that rows go to colums and vice versa. But there is a conjugation which

transposes any of these matrices (Le. interdianges the intemal and extemal cycles).

Combining this with the given conjugation and we have a conjugation sending extemais to

extemals. A s d a r argument (involving a matrix transpose conjugation) enables one to

send intemal cycles to interna1 cycles. For example.

via (20 22)( 2 1 25)(24 26). Q-ED.

Corom: Both cp' and cp" have the same number of extemal cycles.

Pro-: If extemai cycles appear in q', then the conjugation maps intemal cycles of

P to those of P and intanal cycles of S to those of S.

ELPPÎ: Othenuise one of the intenial cycles of P is mapped to an intemal cycle of S.

Now some element of <cp, cp7> mua be conjugated to cp E <p, cp"? Since cp' contains

externai factors it foilows that every element of < , cp7> contains externai factors except

for powers of p. By the above proposition oniy these powers of cp could be conjugated

to cp. But if an internai cycle o fq is mapped to S, this cannot happen We now know

that p is mapped to for some i QED.

Cbapter 9 C o q & n i o ~

The main purpose of this thesis was to extend Carter's results on the

decomposition of elements of order 2 to the decomposition of elements of order odd

prime. Carter proved that an element of order 2 (involution) can be expressed as a

product of mutudy orthogonal refledons (involutions). For Weyl groups (4, B,, D,, F,,

E6, E,, E, G, ) , we have show that an element cp of order p can be decomposeci into a

canonical product cp = cp, (R ..- q1 of C O ~ ~ U M ~ elements of order p (p prime). We

produced counterexamples for such a decomposition in non-Weyl groups and in the case

of elements of composite order. These conclusions were achieved ushg properties of

parabolic subgroups and normaijzers in reflection groups.

Each cp, in the decomposition cp = ip, cp2 ... cp, was seen to be conjugate to a

Coxeter element on a subgroup h., . This gave a structure, cp = pl q2 ... cp, E W(&,)

W(+,) x ... x W(Api), unique up to conjugacy. These arguments were based on a

formula for the order of a group W(A), on Steinberg's lemma of the string, on elementary

equivalaices and on Carter's list of mjugacy classes. The knowledge that each cpi is

conjugate to a Coxeter elemem of a copy of 4, was used to show that cp is regular on

a certain subgroup. We do aot aiways have 9 regular on the whole group W(A) and we

produced a counterexample to the question ofwh*her an eiement of composite order

couid be conjugate to a rcgular el- on a certain subgroup.

Since each cpi in the decomposition q = 9, ... cp, is of order p, it generates a

subgroup isomorpbic to ZlpZ. An element 0 of order p can thus produce an

eiementary M a o psubgroup ( z lp~)~ - This led to the question of whether all

elementary Abelian psubgroups can be generated in this manna. Given an elementary

50

Abelian p-subgroup of rank 2 (ZpZ) (Z/pZ). an element cp of minimal decomposition

length was chosen. The possible choices of a second generator ip' were detennineû from

an examination of the centraiizer Z(9) in the case of &, B , D? and fiom an examinacion

of the p-Syiow subgroups of the centraüzer Z(q) in the case of F,, E6, E,, E,. It was

also shown that the ceatralizer Z(9) of cp is contained in its nomaiizer N(W,). It was

discovered that there are elernentary A b e h p-subgroups that cannot be produced by the

decomposition of an elernent cp of prime order. For instance, for p = 3, the generators

can be pairs of 3-cycles, or triples of 3-cycles and the group may not contain any single 3-

cycles. Such instances were considered genefalizations of the Klein-4 group.

1. Carter, R W. Conjugacy classes in the Weyl group. Cornpositio Mathematica 25 (1972) Pp. 1-59.

7. Deodhar. V.V. On the rwt system of a Coxeter group. C~ommunicanons in Algebra 10(6) (1982) Pp. 611-630.

3. Howlett, RB. Normalizers of parabolic subgroups of reflection groups. Journi of L o ~ h n Mdkmat ia I Society (2) 2 1 ( 1980) 4.62-80.

4. Hmphreys, J.E. w o n Ch- Coxeter G r o u . Cambridge: Cambridge University Press, 1990.

5. Kane, Richard. Lecture notes, "Rdection Groups". 1995-6.

6. Richardson, R W . Conjugacy classes of involutions in Coxeter groups. Bulleti~ of AtlmaIian Mahematîc4I&wiey 26 (1982) Pp. 1-15.

7. Springer, T. A. Some remarks on involutions in Coxeter groups. Commtinz~om in Algebm 1q6) (1982) Pp. 63 1-636.

8. Steinbers, R Torsion in reductive groups. Ahunces in Mizthernatïcs 15 (1975) Pp. 63-92.

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