Finite complex reflection groups

Finite complex reflection groups

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Cohen, A. M. (1975). Finite complex reflection groups. Utrecht University.

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PROEFSCHRIFT

ter verkrijging van de graad van doctor in de Wiskunde en ~atuu~wetenschappen aan de Rijksuniversiteit te Utrecht, op gezag van de Rector Magnificus Prof.Dr. Sj. Groenman, volgens besluit van het College van Dekanen in het openbaar te verdedigen op rnaandag 24 maart 1975 des narniddags te 4.15 uur

door

ARJEH MARCEL COHEN

geboren op 26 juni 1949 te Haifa

Shirayuri ya

Nikei utsurite

Ike kaoru

Seien

-the elegant reflection

of two white lilies;

PREFACE

In 1954 G.C. Shephard and J.A. Todd published a list of all finite irreducible complex reflection groups (up to conjugacy). In their classification they separately studied the imprimitive groups and the primitive groups. In the latter case they extensively used the classification of finite collineation groups containing homologies as worked out by G. Bagnera (1905), B.F. Blichfeldt

(1905), and H.H. Mitchell (1914). Furthermore, Shephard and Todd determined the degrees of the reflection groups, using the

invariant theory of the corresponding collineation groups in the primitive case. In 1967 H.S.M. Coxeter (cf. [ 6]) presented a number of graphs connected with complex reflection groups in an attempt to systematize the results of Shephard and Todd.

In this thesis we classify the complex reflection-groups by m~~ns of some new methods (without reference to the old literature). Furthermore, we give some new results concerning these groups. Chapter 1 contains some familiar facts about reflection groups and is of a preparatory nature.

Chapter 2 deals with the imprimitive ~ase and contains a study of systems of imprimitivity.

In Chapter 3 .we look for all complex reflection groups among the finite subgro~ps of Gl2(~).

As to Chapter 4, inspired by Coxeter's 'complex' graphs (see

above) and by root systems associated with real reflection groups, we define root graphs and root systems connected with finite

reflection groups. Furthermore, we show how these root graphs may be useful by constructing for a given reflection group G a root graph with corresponding reflection group H in such a way that H is a subgroup of G with properties resembling those of G. In Chapter 5 a number of root graphs are brought together. Using these root graphs, we build root systems and study the associated reflection groups. These groups are all primitive and complex. Thanks to T.A. Springer's work on regular elements of reflection groups (cf. [18] ), we are able to determine the degrees of these groups in a manner analogous to the one in Bourbaki.

In Chapter 6, a theorem of Blichfeldt concerning finite primitive groups is discussed. From this theorem we deduce several necessary

conditions for a primitive finite subgroup of Gln(~) (n ~ 3) to be a reflect~on group. The classification is completed by

manipulating with root graphs.

In Chapter 7 we derive some new properties of finite complex reflection groups. We present the orders of all regular

transformations in the reflection groups, we exhibit some real reflection subgroups of primitive complex reflection groups, and determine all groups normalizing a finite complex primitive

reflection group in dimension

>

2.

To my promoter, Professor dr. T.A. Springer, I am especially

indebted for his encouragement and for many stimulating discussions. I gratefully acknowledge the inspiring help and criticism of

R. van de~ Hout, H. Maazen and

J.

Stienstra.

Thanks are due to :mrs. P. van der Kuilen-Arentsen for carefully typing the manuscript.

Finally, I would like to thank my wife and parents, who contributed to this thesis, all in their own way.

CONTENTS

Symbols and conventions viii

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 References Index

Generalities about reflection groups 1

The imprimitive case 9

The two-dimensional case 17

Root graphs and root systems 22

A construction of primitive reflection groups 35

The primitive case 49

Further properties of reflection groups 72

77 T9 Samenvatting Curriculum vitae 80 80

SYMBOLS AND CONVENTIONS lJJ w µd µ00 arg x k* IR>o n

-Sn (123)

Ix I '

#=X 0 .. l ] V1 ± Vz

±

V3 (CD C (CD+1 S(V) deg(f)

c

I )

standard unitary inner product

= . {x E

a:

11 x I 1} = exp(27fi/3) {x E

_a:

_Ix d 1} (d E IN) = = =

u

µ ~1 d

the argument of the complex number x lying in (-rr,v]

the multiplicative group of the field k = {x E IR Ix

>

O}

=

{1,2, . . . ,n} (n E IN)

the group of all permutations of n (n E IN)

the permutation of 3 that transforms 1 into 2, 2 into 3, and 3 into 1

greatest common divisor of the numbers Pi , P2 , . • . , Pn E IN

least common multiple of the numbers P1 , P2 , · • • , Pn E IN

the number of elements in the set X = \ 1 i f i = j (i,j E IN)

l

0 otherwise

= { v1 -~v2 +v3 , v1 +v2 -v3 , v1 -v2 +v3 , v1 -v2 -v3 } by way of (a1 ,a2 , ... ,an)

=

= (a1 ,a2 ' . . . ,an,O) E a;n+1

=

U a;n

11>1

~ (0,0, . . . ,0,1) (n coOrdinates)

of it)

the algebra over

a:

generated by 1, x1 , x2 , ..• , xn

the algebra of polynomial functions on the complex vector space V

the degree of f E S(V) unitary inner

always linear determined by

product on a vector space, in the first factor

( e: .

I

e: . )

=

o . .

on 0:00

xl Ix! x is unitary

v

l

w

s s s a, i:' a

,m'

a r. J Gl(V) U(V) Un ( ({;)

sun

(er:) Zln(tI) Sln ( ({;)

Dn,T

,Q,

I

c

µ4

I

µ2 ;

D4

I

Cs

> , µ2 •

T, · · ·

0₃ CIF₇ ,q) ,PQ₆ (IF₃ ,q), .•• G G+ K :,.;;; G, K

<

G K ~ G, K <:JG Z(G)

<

g E Gig E L

>,<

L

>

the orthogonal complement of the sub-set X of a complex vector space with respect to a given unitary inner product

=

/Cxjx)

lxl

=

1

Cvlwl

=

0 for all v E V, w E W

reflections, see (1.6) and (5.1)

=

s )' see Chapter 5 ej,w(ej

the group of all nonsingular linear transformations of the complex vector space V

=

Gl(tin). The elements bf Gln(({;) will be identified with their matrix with respect to the standard basis of ({;n. identity element of Gln(({;)

the kernel of the linear transformation~ the determinant of the linear

transformation

~-the group of linear transformations of V, unitary with respect to a given unitary inner product

= U( a:;n)

=

{g E Un(cr:)!det(g)

=

1} center of Gln(Q;)

=

{g E Gln(({;)ldet(g)

=

1} see (3.1) see (3.1) in accordance with [ 8]

finite subgroup of Gl(V) (V is a complex vector space)

G

n

Sl(V)

K is a subgroup of G

K is a normal subgroup of G center of G

the subgroup of G generated by all elements in the subset L of G

[ G, G] XG An,Bn,Cn,Dn,En,F₄ ,Hn,Iz (m) W (An ) , W ( Bn ) , . . .

OG

G(m,p,n) R(m,m,n), R(m,p,n) d(G), d(f)

r

wcr)

~, W(L)

the corrunutator group of G

=

{x E X!Gx

=

{x}} (G acts on X) root graphs, see (4.4)(i)

the corresponding Coxeter groups [ loc. c it. , pp. 2 0 2 , . . . , 2 2 O , 2 31 , 71] considered as subgroups of Un(~),

n . .

leaving IR invariant

complex root graphs, see (4.4) and Chapter 5 see (1.6) see ( 2. 3) see (4.12) see (4.2) see (4.1) see (4.2) see (4.8)

§1., GENERALITIES ABOUT REFLECTION GROUPS

Let V be a complex vector sp~ce of dimension n.

(1.1) DEFINITIONS. A reflection in V is a linear transformation of V

of finite order with exactly n-1 eigenvalues equal to 1. A

reflection group in V is a finite group generated by reflections

in V. From (1.6) on we will assume a reflection (group) to be unitary with respect to a unitary inner product. A reflection

subgroup of a group G of linear transformations of V is a

sub-group of G which is a reflectionsub-group in

V. A

reflection group G in V is called a real group or Coxeter group if there is a G-invariant IR-subspace V0 of V such that the canonical map

~8IRVo ~ V is bijective. If this is not the case, G will be called complex (note that, according to this definition, a real reflection group is not complex). A reflection group G is called

r-dimensional if the dimension of the subspace

v

8 of points fixed

by G is n-r. We will say that G is irreducible in dimension r (also irreducible r-dimensional or merely irreducible if no

confusion is possible) if G is r-dimensional and the restriction of G to a G-invariant complement of VG in V is irreducible. We will use the same convention for other properties than irre-ducibility (e.g. primitivity, see §2).

(1.2) Let G be a finite group of linear transformations of V and let S

=

S(V) be the algebra of polynomial functions on V with G-action defined by (g.f)(v)

=

f(g-1v) for any v E V; f ES, g E G.

G The subalgebra of G-invariant polynomials will be denoted by S . The theorem below is a well-known characterization of reflection groups (cf. [ 18] ) •

THEOREM. The following three statements are equivalent:

(i) G is a reflection group in V;

(ii) there are n algebraically independent homogeneous polynomials f1 ,f2 ' . . . ,fn E

s

8 with IGI = deg(f1) .deg(f2). . . . deg(fn); (iii) there are n algebraically independent homogeneous polynomials

G · G

f 1 ,f2 , ... ,fn ES which generate S as an algebra over f

Furthermore, let f1 ,f2 , • • • ,fn be a family of algebraically

independent homogeneous polynomials in

s

8 such that di

~

di+l

(i E n-1) where· d. is the degree of f. (j En); then f 1 ,f2 , • • • ,fn

- J J

-satisfy (ii) i f and only i f they -satisfy (iii). In this situation, the sequence d1 ,d2 , • • • ,dn is independent of the particular choice

of such a family fi ,f2 , ... ,fn.

DEFINITION. d1 ,d2 , ... ,dn are called the degrees of G.

( 1. 3) Let G be a reflection group with degrees d₁ ~d₂ , • • • ,d . Suppose

G - n

( 1 )

( 2 )

f 1 ,f2 , ... ,f _n ES satisfy (1.2) (ii) and d.::: deg(f.). Let I be

l i

the ideal in S generated by f 1 ~f2 , • • • , f . Then I is a G-invariant n

graded ideal, so S/I is a graded G-module. According to

[4], G

acts on S/I by the regular representation. Let i4J be an irreducible character of G; denot~ by a.(~) the multiplicity of~ in the i-th

l

homogeneous component (S/I). of S/I (i ~ 0). Adopting an idea of

l [ 18] , we define 00 • p,₁,(T)

=

L a.(i[J)T1 • 't' • 0 l i=

Note that pi[J(T) is a polynomial specifying for which i the

representation corresponding to i4J occurs in (S/I)i. Writing out left- and right-hand side as a formal power series in T, one obtains the identity

jG[-

1 _{L i[J(g).det(1-gT)-}1

gEG

n d·

=

P,₁,(T). IT (1-T 1 ) -1 •

't' i=1

The following result is due to R. Steinberg (cf. [ 3] ,p.127): if

G is irreducible, then, for each j E ~' the action of G on the j-th exterior power of V is also irreducible. Write XJ for the

a .

corresponding character (j

=

1,2, ... ,n). The coefficient of yJ in Solomon's formula (see [ 3] ,p.136)

1 L

TGT

gEG n det(1+Yg)

=

IT det(1-gT) i=l equals p j (T) .

xa

n d· II ( 1-T l) - i i=1 d·-1 (1+YT i ) (1-Tdi)

(1.4) We mention a few corollaries of these statements.

n

Notations as before. Put N

=

k (d.-1),and Zet

x

he the character

i=1 l

of the given representation of G in V. Suppose G is iPreducibZe.

Then ( i ) Pv ( T)

=

/\. n d·-1 _l -2: T i=1

gcd(d1 ,d2 , • • • ,dn);

TNp~(T-1

) , where o(g)

=

det(g) (g E G);

( i i ) IZ(G)I

=

(iii) Pijj

0(T)

=

(iv) there is a homogeneous polynomial J of degree N such that

g(J) = det(g).J for any g E G;

(v) N is the number of refleetions in G, in fact

n · n

~ h.T1 _{= IT (d.-1+T),}

i=O 1 i=1 1

where hi is the number of elements in G with exactly i

eigenvalues equal to 1;

(vi) d1 ~ 2; furthermore, the following three statements are

equivalent

(a) G is complex, (b)

x

has complex values, (c) di

>

2.

Since (i), (ii), ... ,(v) appear elsewhere in the litterature (cf. [ 3]., [ 18] ) , i t is left as an exercise to the reader to deduce them from (1.3).

For a character ~ of an irreducible representation p of G we define

v ( ~) =

I

G

1-·

1: ~ ( g2 ) •

gEG

It is a known fact that

{

1 if p is a real representation

,v(~) = -1 if p is not real, but conjugate ~o p 0 otherwise, i.e. if

w

has complex values. For further details we refer to [ 11] .

We will now prove the essential part of (vi). Suppose x takes only real values; then Cxlx> = 1, so d1 = 2. Considering the

coefficient of T2 in formula (1) of (1.3), we get Cx~j1) = = #={ildi = 2}, where x~ is the second symmetric character (in the notation of [15]). But <x~l1) = ~CCx!x) + vCx)) = ~C1+vCx)), so 1+vCx) = 2Cx~l1) E 2IN, and therefore v(x) = 1. Hence G is a real reflection group. This shows that (a) implies (b). The rest of the proof of (vi) is easy.

(1.5) G is an n-dimensional reflection group in V. Let P be a subset of V. Put Gp= {g E Glgp = p for all p E P}. Then Gp is a

reflection sub group of G (see [ 20] ). If m is the dimension of the vector space spanned by P, then Gp is at most (n-m)-dimensional. If G is reducible, then G is a direct product of reflection subgroups which are irreducible in dimension smaller than n. Therefore we can restrict ourselves to the determination

of irreducible reflection groups. It is clear that it is only the conjugacy class of G we are interested in.

( 1. 6) It is well known that there exists a unitary inner product (

I )

on

V

invariant under

G;

hence we may assume that G is a subgroup of U(V), the group of all unitary transformations with respect to a unitary inner product. From now on we will make this

assumption. One can prove that two finite subgroups of U(V) are conjugate in U(V) if and only if they are conjugate in Gl(V).

Furthermore, G is real if and only if there is an orthonormal

basis of V such that the matrix of any element in G with respect to this basis has real coefficients only.

DEFINITIONS. A (base) root of a reflection in V is an eigenvector (of le~gth 1) corresponding to the unique nontrivial eigenvalue of the reflection.

A (base) root of G is a (base) root of a reflection in G. Let s be a reflection in V of order d

>

1; there is a nonzero vector a E V and a primitive d-th root of unity ~ such that, putting

(1)

we have s

=

s r•

a'"'

(xEV),

(2) We will also write s instead of s if

s

=

exp(2Tiid-1 ) ,

a,d

a,s

and s instead of s ~·

a a,L

If t is any unitary transformation of V, we have the equality (3) ts t-1

- ~ •

a,s

ta,l;;

1 Define 0

8 : V ~IN by oG(v)

=

!Gwl

where W

=

v (v E V). Then oG(v)

>

1 if and only if v is a root of G. In this case o

8(v) is the order of the cyclic group generated by the

reflections in G with root v; if a is a root of G, the number oG(a) will be called the order of a (with respect to G).

In the rest of this chapter, G is a reflection group in the unitary space V (as always G

<

U( V )) with degrees d1 ,d2 , • • • , dn.

The following observations concerning linear characters of G are due to T.A. Springer.

(1.7) If a is a nonzero element of V, we denote by la the linear

LEMMA. Let a,b be roots of G, Zet

s

be a root of unity, and let c E [* be such that s r.l.

=

clb. Then either c

=

1, or

a.'~ D c

=

s-

1 and a E

Uh.

PROOF. Since 1-b = s r"lb = 1 b' we have that b is an

eigen-c a,~ sa,l;; .

vector of s . with eigenvalue~. Therefore c = 1,l;;-1 • If c =

s-

1 ,

a'

s

the dimension of the eigenspace of s r corresponding to t is 1,

a'~ so b is a multiple of a.

(1.8) For each reflection s of G we fix a base root a in such a way s

(1)

( 2 )

that if s and s' are reflections of G with Ua

=

Ua , , we have

s s

as

=

as 1 •

Put U

=

{asls is a reflection of G} and P

=

{Uasls is a reflection of G}. Note that G acts on P and that there is a natural map

T : P ~ U such that, for L E P, we have T(L)

=

a #a E L n U.

If O is an orbit of Gin P, define f

0 ES by f0

=

_LEO IT 1 ( ); T L moreover, define x₀ G ~ U by x₀Cg)

=

IT (<let si)-1

uas

.eo

l

if s1 , 8 2 , • • • , sr are reflections of G with g = 8 1 s2 • • • sr.

It follows from (i) of the proposition below that

x

0 is well defined.

PROPOSITION. Fix an orbit 01 of Gin

P.

(i) If 8 is a reflection in G with nontrivial eigenvalu~ l;;,

then sf ₀

=

l f i f Ua ~ 01

1

· t

~ ~

f i f ua s e

o

1

01 s

C ii)

x

01 is a linear character of G with pX ( T) =

TI

0

1

l;

01

(iii) any linear character of G is the product of some

x

0 (0 orbit in P).

PROOF. (i) Let. L₁ ,~ , • • • ,Lr be an orbit of s in P, so sLi =

=

Li+l ( i E r-1) and 8~

=

L1 • Put h

=

II 1 (L·)' and Lr+l

=

L1

iEr i: l

Since there are c. E ~· such that sT(L

1. )

=

c.T(L. 1) , we have l l l+_ s.h

=

( IT iEr

c.

)h 1- , and

If II

iEr

c.

*

1, then (2) and (1.7) imply that sr (and therefore

l

s, to;) has base root a

=

T(L1 j, whence h =la , and (using (1))

s s

s.h

=

r;;-1

h.

As to (iii), let$ be a nontrivial linear character of G, and let f E S be a nonzero homogeneous polynomial of minimal degree such that g.f = cp(g)f (g E G). It is clear from (1.3) that

such a polynomial exists. Suppose s E G is a reflection with

cf>(s)

*

1. For any v E V with (vjas)

=

O, we have f(v) = f(s-1v) = = (s.f)(v) = cp(s)f(v), so f(v)

=

O. Therefore f must be divisible by la , and also by lT(L) for any Lin the G~orbit 0 of Vas.

Hence 8 f is divisible by f

0• Write f 1 ·for the quotient of f by

f

₀

,and cp1 for the quotient of cp by

Xo·

If f1 is a non-constant,

then f 1 is a nonzero homogeneous polynomial of minimal degree

such that g.f1

=

cf>1 (g)f1 · (g E G), and of degree strictly lower

than the degree ~f f. Induction finishes the prbof. Finally, (ii) is a consequence of the preceding.

(1.9) o

8(T(L)) being independent of the choice of LE 0 for a given G-orbit 0 of P, we will also write o₈(0) for tbis number.

COROLLARY. G/[G,G] is a direct produat of cyclic groups of orders o

₈

CO)~ 0 running through the G-orbits in P.

The proof follows from G/[G,G] ~ Hom(G,U) and the fact that if

01 , 02 , • • • , 0 are different

m ' /..

integers such that X~1

x

₀2

/.. 1 2

=

X m

=

1. Orn

orbits in P, and A.₁ ,1..2 , • • • ,A. are /.. ~ /.. . /.. m X m = 1, then X 1 =

x

2

=

Orn 01 02

(1.10) DEFINITIONS. A vector v E Vis called reguZar with respect to

G, if (via)* O, for any root a of G (in other words, if no

reflection of G fixes v, or equivalently, if G{v}

=

1, cf. (1.5)).

A transformation g E G is called reguZar

if

g has a regular

eigenvector. The regular degrees of G form the set of numbers,

minimal with respect to inclusion, such that the order of any regular element of G is a divisor of an element in this set and,vica versa, any divisor of an element in this set is the

order of a regular element of G. Of the many interesting properties of regular elements we will only mention a few. For their proofs and other properties we refer to [18].

THEOREM. Let (; be a primitive d-th root of unity. Let g E G be

regular with regular eigenvector v E

V

and related eigenvalue r;.

Denote by W the eigenspace {x E Vjgx

=

r;x} of g in V. Then

(i) d is the order of g; moreover, g has eigenvalues

,...1-d1 ,...1-d2 ,...1-dn.

':> ,.., , • • • ,.., '

(ii) dim W

=

ffe{ild is divisor of di};

(iii) the restriction to W of the centralizer of g ~n G defines

an isomorphism onto a reflection group in W, whose degrees

are the di· divisible by d and whose order is IT di·; (iv) the conjugacy class of g consists of all elements of dldi G

having dim

W

eigenvalues (;.

(1.11) EXAMPLES. (i) The polynomial J

=

IT lo(a)-l satisfies (1.4)(iv). aEU a

If G is real, then o(a)

=

·2 for all roots a E G. However, this condition is not sufficient for G to be real (as will turn out later on, see (5.2)).

(ii) Let V = ~n, and let (

I )

be the standard unitary inner product. Fix d1 ,d2 , • • • ,d _n E IN\{1}, and p u t s . = sE. d· _{l l ' l} ( i =

= 1,2, ... ,n). Now G =

<

s1 ,s2 , • • • ,s _n

>

is a reducible reflection

group. Choose

x.

ES homogeneous of degree 1 such that X.(E.)

=

l l J

=

o ..

(i,j = 1,2, . . . ,n). The f

0 are nonzero constant

l ]

multiples of the Xi.

Furthermore,

X~

1

,x~2

, ... ,Xdn form a set of algebraically n

independent homogeneous polynomials generating SG, so d₁ ,d_{2 , • • •} ,d are the degrees of G. This shows that (1.4)(vi) does not

necessarily hold if G is reducible. An element of G is regular if and only if all its eigenvalues are equal. Thus there is only one regular degree, namely gcd ( d1 ,d2 , • • • ,dn).

(iii) Let G ~ U₂ (~) be the reflection group generated by

_{(10 10)}

and

(~_

1

~)

, where(;= exp(2nim-1 ) . Then G is conjugate to

n

W(I₂ (m)) (in the notation of [ 3] ), the dihedral group of order 2m. Let

x.

be as in (ii). The algebraically independent homogeneous

l

polynomials X1X2 and

xr

+

xr

generate

s

8 , so the degrees of G

are 2,m. Since G is irreducible it follows from (1.4)(vi), once

m m . .

again, that G is real. Furthermore, J

=

X - Y is a polynomial satisfying (1.4)(iv).

If m is odd, then J is the only f

0; if m

=

2p for some p E IN, then there are two f

0, namely

xP - yP

and

xP

+

yP

(again, up to a constant multiple). Finally, the regular degrees of G are 2,m if m is odd, and m if m is even.

§2. THE IMPRIMITIVE CASE

V is the unitary space (n equipped with standard unitary inner product (

I ),

and S i s as in (1.2). In accordance with (1.6), a reflection (group) is always assµmed to be·unitary.

(2.1) DEFINITIONS. A group G of unitary automorphisms of V is called imprimitive i f Vis a direct sum V

=

V1 @ V2 @ . . . @ Vt of

nontrivial proper linear subspaces V. (1 ~ i ~ t) of V such that l

{V. [i Et} is invariant under G. In this situation the family

l

,.-(Vi) 1 :e;;;: i ~ t is called a system of imprimitivity for G. If such a direct splitting of V does not exist, G is called primitive

( cf. [ 7] , [ 9

J ) •

A

polynomial p E S is called semi-invariant with respect to

G

if i t affords a linear character e:: of G, i.e. if g.p

=

e::(g)p ,for any g E G.

(2.2) PROPOSITION. Let G be an irreducible imprimitive reflection group

in V (n ~ 2), and let (Vi)l :e;;;: i ~ t be a system of imprimitivity

for

G.

Then

( i ) dim V. = 1 for each i E _!, and t = n; there are distinct l

linear homogeneous polynomials 1₁ ,1_{2 , • • •} ,1 (not even equal n

up to a constant factor) such that 11 .12 • • • • • 1 _n is a

semi-invariant homogeneous polynomial of degree n in S;

(ii) for any reflection s E G we have either sVi

=

Vi for all

i E n, or there are i

*

j (1 ~ i , j :e;;;: n) such that any root of s i s contained in

v.

+ V., sV.

=

v.,

sVk

=

Vk for all

l J l J

k

·*

i , j , and s i s of order 2;

(iii) let ~ : G ~ Sn be the homomorphism that assigns to g E G the permutation a E Sn defined by gVi = Vcr(i) for any i En.

Then ~ is surjeetive and admits a section T : S ~ G, which

n is a homomorphism;

(iv)

v.

l

v.

_{for aZl} l , ] _(i

*

J ' 1 ~ l,] ~ n);

l _J

(v) _{i f} v is a base root of G of order _2' and w is a base root of G of order

>

2, then !Cvjw)j E _{0,2 -1 2_{} .}

PROOF. (i) Let i be such that dim Vi> 1; because G is irreducible,

there is a J

*

i and a reflections E G such that sV.

=

v ..

It l J

follows that dim(V.

n

V.)

=

dim(sV. n Vl.) > O, contradicting

v.

n

v.

=

{O}. To establish the last part of (i), fix unitary

l J

a. E

v.,

and take for l · the linear homogeneous polynomial with

l l l

l·(a.)

=

1 and l·(a-)

=

1 for j * i .

l l l J

(ii) Let s E G be a reflection with base root a and nontrivial eigenvalue

s·

such that

CalV1)

*

D and a

E

V1.

permutation of indices) sV1

=

V2. Take 0

*

x. l

Then (up to a E V. ( i = 1,2)

l

that sx1

=

x2 . There is a J E n with s 2 x1 E V. ;

. J thus s 2 x1 E

such

E ( <ra + (Cx_{1 )}

n

V.

=

(V ₁ + V_{2 )}

n

VJ· . This implies that J

=

1 or 2.

J . 2

Since a E V1

u

V2 ' we have s X1

=

X1 and

s

2

=

1; so

s

= -1 and s is of order 2. Furthermore, the root a is a scalar multiple of x₁ -~ , in particular a E V₁ + V_{2 . If} (a IV.)

*

0 for some i

>

2,

l

there is a j

>

2 such that sV.

=

V.; thus a E CV. + V.) n (V1 + V2 )

=

l J l J

=

{O}, which is impossible. So

Ca

IV-)

=

0 for any i

>

2. In l

particular sVk

=

Vk for k

>

2, and (ii) is proved.

(iii) The _irreducibility -of G implies that for each J

>

1 there exists a reflections. with s.V1

=

V. (necessarily of order 2). The

J J J

image of sj E Gunder~ is the element (1j) E Sn (according to (ii)). It is known that the set {(1j)lj

=

2,3, . . . ,n} generates Sn. Finally, note that the restriction of ~ to the reflection subgroup

<::

S2 , S3 , • • • , S

n

>

of G is an isomorphism. n .

(iv) Note that (x,y) i-+ ~ 1. ( x) 1. ( y) (where 1_{1 ,} 12 , ... , 1 are as in

i=1 l l n

(i)) defines a unitary inner product on V, fixed by G. Since such a unitary inner product must be a strictly positive scalar multiple of (

I ),

the required result is readily' deduced.

(v) is clear from (ii) and (iv).

(2.3) If we do not mention an explicit basis, we shall always identify a linear transformation of (Cn with its matrix with respect to the standard basis.

Let TI be the group of all n x n-permutation matrices; let n

ACm,p,n), where pjm (m,p E IN), be the group of all n x n-matrices

Ca .. )

₁ _... . . ;,., such that a. .

=

e. o .. ,

where

er:1

=

1 for each

lJ ~ i,J ~ n . lJ l lJ i

p-1m

i E n, and ( det (a . . ) ) = 1. Then Tin normalizes A(m ,p ,n). Define lJ

G(m,p,n)

=

A(rn,p,n).ITn; this is a semi-direct product. It is not hard to see that G(m,p,n) is an imprimitive reflection group in ~n, with system of imprimitivity (~E.). E

(2.4) THEOREM. Let n ~ 2, and let G be an irreducible imprimitive reflection group in

V.

Then G is conjugate to G(m,p,n) for some

m,p E IN with p!m. Furthermore, G(m,p,n) is irreducible i f and only i f m

>

1 and (m,p,n)

*

(2,2,2). (By conjugacy we mean conjugacy within U(V))

PROOF. Let G be as stated. There is an orthonormal basis e1 ,e2 , • • • ,en with the properties that the Vi=

form a system of imprimitivity for G, and that is a reflection s. E G such that se1 = e. (cf.

J J

~.e. (1 ~ i ~ n)

l l

for each j

>

1 there (2.2)). Without

changing the conjugacy class of G we may put e. = e:. (i.e.

l l

e1 ,e2 , • • • ,en is the standard basis). It follows from (2.2) that

rrn is a subgroup of G. Let q be the order of the cyclic group generated by the reflections that leave

v+

pointwise fixed

(so q = 0

8Ce1) with notations of (1.6)). Then A(q,1,n) is a sub-. group of G.

According to (2.2), the only reflections outside A(q,1,n).ITn are the s' E G with s'e. = ee. where

e

E U\{1} and i

*

j, and s'ek =

l J

f o r k * i , j . Up to conjugacy by an element of IT , we may take

n

l = 1 and j = 2. Let s = s2 E G be the reflection with se₁ = e_{2 •}

Then (ss')e1

=

8e1 and (ss')e2

=

e-1

e2 , so e is a root of unity.

Let m be the maximum of all orders of elements st E G where t is a reflection such that tV1 = V2 • It is not difficult to see that

A(m,m,n) is a subgroup of G, commuting with A(q,1,n), and that q!m. Putting p = q-1 .m, we have A(m,p,n) = A(q,1,n).A(m,m,n); so G(rn,p,n) = A(m,p,n).ITn is a subgroup of G. Since all reflections of G are contained in this subgroup, the subgroup must be equal to

G itself, in other words G

=

G(rn,p,n).

In order to prove the second statement of the theorem, suppose that G = G(m,p,n) leaves invariant a nontrivial proper linear subspace W of V. Since W is also a IT -invariant subspace of V,

n

we know from [15;p.29, 301, for instance, that W = ~(e:₁ +e:₂ + up to an interchange of Wand

w1.

As A(m,p,n) stabilizes

~(e:₁ +e:₂ + ••. +e: ), _n all diagonal coefficients of an element in A(m,p,n) must be equal. It is not hard to deduce from this that

(m,p,n) E {(1,1,n),(2,2~2)}.

On the other hand, it is obvious that G(1,1,n) and G(2,2,2) are reducible in V.

+e: ) n

(2.5) REMARKS. (i) G(m,m,2) is conjugate to W(I2 (m)), the Coxeter group

'

corresponding to type I2(m) (notation of [ 3] ). This group is

reducible only if m .;:;;;: 2. For the other Coxeter groups the notation will be similar.

G(1,1,n)

=

Tin,operating on the hyperplane (£1+E2+ +En)l of

~n,

represents W(An_ 1 ) (n ~ 2).

G(2,1,n) represents W(Bn)

=

W(Cn) Cn ~ 2). G(2,2,n) represents W(Dn) (n ~ 3).

W(A) is primitive if n

>

_{2, and W(A2 )_is conjugate to W(I2 (3)).} n

The above groups form the set of all real reflection groups appearing in (2.4).

(ii) Let X₁ ,X_{2 , • • •} ,X E S be as in (1.11)(ii). The first n-1

n

elementary symmetric polynomials in (X~). E (i.e. X~ + X~ + •.. + Xm

n i i n n'

2: X~Xl'.1-, ... , L TI Xj) and (X1 X2 • • • X )q, where q = p-1 .m, form

i <j i J i

=

1 j =Fi · n ·

a set of G(m,p,n)-invariant homogeneous algebraically independent polynomials; the product of their degrees equals m.2m ... (n-1)m.qn

=

= q.mn- 1 .n! = p-1

• mn.n! = !GCm,p,n)

I.

By (1.2) the degrees of

G(m,p,n) are m,2m, ... ,(n-1)m,qn. One of the consequences is that !ZCGCm,p,n))I

=

q.gcd(p,n). Finally, X1X2 • • • Xn is the

semi-invariant associated with the canonical system of irnprimitivity.

(iii) G(m,m,n) and G(p-1_{m,1,n) are reflection subgroups of}

G(m,p ,n).

Civ) G(4,4,2) is conjugate to G(2,1,2). These two groups form the only pair of conjugates in the set of all irreducible G(m)p,n), as can be seen with the help of the invariant polynomials of (ii).

(v) If p

=

1 or rn, it is possible to choose n generating

reflections for G(m,p,n): take the reflections of order 2 with roots £1 -£2 , £2 -£3 , • • • , £n_

1-£n, and

)the reflection of order 2 with root (£1 - exp(2~im-1 _)£

2 ) if p

=

rn, la reflection of order m with root £1 if p

=

1.

If p

*

1,m, take n generating reflections for G(m,m,n) and an additional reflection of order p-1m with root £1 to obtain n+1

generating reflections for G(m,p,n).

Cvi) Put G

=

G(m,p,n) and, as always, q

=

p-1

Suppose n

=

2; P consists of gcd(2,m)(gcd(2,q))-1 G-orbits of

lengtr~ m.gcd(2,q).(gcd(2,m))-1

and, if p =f=. m, of one more G-orbit,

in fact {Ue:_{1 ,} 11Je:_{2 }} of length 2. If n

>

2, then G admits one orbit in P of length ~mn(n-1); if, moreover, p

=

m, this is the single orbit in

P;

if p =f=. m,

P

contains one more orbit, of length n.

(vii) If q is even or m is odd, then the f ₀ of G(m,p,2) are

Xml - xm ₂ an , un ess p d 1

=

m, x x 1 2 . Ir r q is o . dd an d m

=

2k

,

t en t e h h f₀ of G(m,p,2) are

X~

-

X~, X~

+

X~

and, unless p

=

m, X1X2 • Finally,

if n

>

2, then the f

0 of G(m,p,n) are _i<j IT (X~ _l - X~) and, unless _J

x .

_n

(2.6) LEMMA. Let G be an irreducible reflection group in V.

If

G has a reflection subgroup which is primitive in dimension r

>

1 and not conjugate to W(Ar), then G itself is primit;ive.

PROOF. Let H be a reflection subgroup of Gas described in the assumptions, and denote by W the orthogonal complement of VH. We may assume that r

<

n.

Suppose that G is imprimitive with system of imprimitivity

L1 ,L2 , • • • ,Ln. Since dim W

=

r, we have that His primitive, and

therefore irreducible, in W. If Li ~ W for some i E ~' then HLi spans W, so the Lj with Lj ~ W form a system of imprimitivity for H in W, unless W = Li; but W = Li would imply that r = 1 (because of (2.2)(i)), which is assumed to be false. Thus (1) Lj ~ W for all j E ~·

Let s EH be a reflection with root a E V. Note that a E W. Tf

sLi

=

Li for every i E ~' then a E Lj for some j E ~' and Lj ~ W, which is impossible because of (1); so sL₁

=

L2 (up to a

permutation of indices), and s is of order 2 (see (2.2)(ii)). If

a' E V is a root of another reflection s' of H such that s'L1

=

~,

then (by (2.2)(ii)) ([:a+ <ta' = 1₁ + L_{2 ;} since a,a' E W, this yields L₁ ,L₂ CW, contradicting (1).

We conclude that there are no reflections s E H such that sLi

=

Li for any i E ~' and that for i,j E ~ (i =f=. j) there is at most one

reflections EH with sL. =Loa By now, it is obvious from (2.2)

l J

(iii) that there exists t En such that His conjugate to G(1,1,t). Because H is r-dimensional, we have that t

=

r+1, and that H is conjugate to W(Ar).

(2.7)' (EMMA. Suppose G

=

G(m,p,n} is irreducible (p!m and n ~ 2). Then

G .has

a

unique system of imprimitivity provided that

(m,p,n) fi.. {(2,1,2),(4,4,2),(3,3,3),(2,2,4)}.

PROOF. The L.

=

~E. (i En) constitute a system of imprimitivity

l l ..,....

for G(m,p,n). Let P be as in (1.8). First of all, we will pay attention to the case that an orbit of p gives rise to another system of imprimitivity. By (2.S)(vi), we then have either ( 1) n

₌

2 = m gcd( 2 ,q). (gcd( 2 ,m) )-1

, or

(2) n

>

2 and mn(n-1)

=

2n.

( 3)

If (1) occurs, we have (m,p,n) E {(2,1,2),(4,4,2)}; (2) leads to a contradiction with m

>

1 (cf. (2.4)). The conclusion is that none of the groups G in question has a system of imprimitivity afforded by roots different from the canonical system.

Let us assume that

V

1

,V

2 , • • • ,Vn is a system of imprimitivity

different from 11 ,L2 , • • • ,Ln and not corresponding to an orbit of

P. Let 1₁ ,12 , • • • ,ln be defined with respect to V1 ,V2 , • • • ,Vn as

in the proof of (2.2)(i), and put f

=

1112 ln. Suppose that f is a semi-invariant but not an invariant. It follows, by an argument similar to the one in the proof of (1.8)(iii), that f is the product of an invariant and some £

0 (0 orbit in P). Since deg(f)

=

n, the irreducibility of G implies that there is an orbit 0 in P of length n such that f

=

£₀; this is contradictory to our assumption. Therefore f is an invariant homogeneous polynomial. Because f ~ ~.X1 X2 • • • Xn, there must be an a E ~such that

f-aX1X2 Xn is a nonzero homogeneous G-invariant polynomial

in x~,xr,

...

,xm (cf. (2.S)(ii)). Hence m divides n.

n

Put 11

=

a1X1+ a2

X

2 + .•• +a X, r.

=

#{i E

n!a-n n!a-n J - i

and ro

=

#{i E ~lai

*

O}. Let j E {0,1,2, ... ,n}.

=- c..} for j E

J

Since the

stabilizer in IT of ~1₁ is of order~ r.!(n-r.)!, and since the

n ·. J J

Iln-orbit of ~1₁ has ~t most n elements, we have n ~ n! . (r].!)-l . ((n-r.)!)- 1

=

en).

J r j

This implies that r.

=

1,

J n-1, n for any j E {0,1, ... ,n}. Note that r0

*

1. Suppose r0

=

n-1. Using a combinatorial

we get that the stabilizer of ~1_{1 in G(m,p,n) is of order}

argument,

~ mq ( n-1) ! ,

wh _{ence n -}2 _m n-1 _{.q.n . . m.q. n-}' ( ( ·1)')-1 _.

₌

_m n-2 _{.n; son~} __.. _{2, and} 11 E ~X1, contrary to the assumption that V₁ ,V_{2 , • • •} ,V is

n

of the <Cl1-stabilizer in G(m,p,n) is smaller than or equal to

' ~ n-1 ' ( ')-1 n-2 T h . h

I

m.n., son~ m .q.n . . m.n.

=

m .q. oget er wit m n, this implies that (m,p,n) E {(2,1~2),(2,2,4),(3,3,3)}.

(2.8) REMARK. To G(2,1,2) correspond, apart from the canonical one, two systems of imprimitivity, namely

<C(E_{1+E 2 ),Q;(£1-E 2 ) with semi-invariant} X~ - X~, and

<C(E1.f.iE2),1:(E_{1-iE 2 ) with} invariant

x:

+

Xi

(compare (1.11)(iii)). To G(3,3,3) corresponds (C(E_{1+E 2 +E3} ),(C(£_{1+WE 2 +W}2E₃ ),<C(E₁+w2

E2 +wE3 )

with invariant X~ + X~ +

x: -

3X1

X

2 X3 •

To G(2 ,2 ,4) correspond <t(E₁ +E_{2 +£ 3 +£ 4 )} ,(C(q +£_{2 ....,£ 3 -£4 )} ,~(e:_{1 -e:2} +E₃·-e:_{4 ) ,} (C(E1~E:2-E3+e:4) with invariant

X1

+ X~ +

x:

+

x:

-- 2

C

xi

Xi

+

xi

x~ +

xi x!

+

xi

x~ +

xi x!

+ x~

x!

>

+ 8X1 X2 X3 X4 , and the

system that can be obtained from the preceding one by substitution of _{-E 1}for _{£ 1} with invariant that can be obtained from the

preceding one by substitution of -X₁ for X1 •

Moreover one can prove that these are all non-canonical systems

of imprimitivity in the respective cases.

(2.9) PROPOSITION. Let 1

<

m

<

n, let G be a primitive refleation group in

V,

and let

H

be an imprimitive irreduaibZe m-dimensional (i.e. imprimitive irreduaible in dimension mas defined in (1.1))

refleation subgroup of

G.

Suppose, moreover, that

H

has a unique system of imprimitivity 11 ,L2 , ..• ,1m in (VH)l. Then G contains a reflection so suah that

<

H,s0

>

is a primitive

(m+1)-dimensional refleation subgroup of

G.

PROOF. Put W = (VH)l. Now dim W : rn; the proof goes by induction with respect to n-m.

Suppose m

=

n-1. Put L

=

VH. Note that 11

,~

, • • • ,L form a system

n n

of imprimitivity for H in V. The required result is a direct consequence of the observation that this is the only system of imprimitivity for H in V consisting of 1-dimensional linear sub-spaces. In order to prove this observation, let V1 ,V2 , • • • ,Vn be

another such system. Reasoning as in the proof of (2.6), we obtain that either H is conjugate to G(1,1,n) or there is a j En with Vj CW. As G(1,1,n) is primitive in dimension n-1, we must have that Vi CW for all but one i E ~;the uniqueness of 11 ,12 , • • • ,Ln

(2.7) L~MHA. Suppose G

=

G(m,p,n) is irreducible (plm and n

>

2). Then

G

has a ~nique system of imprimitivity provided that

(1) ( 2 )

( 3)

( m, p , n) (:i. { ( 2 , 1 , 2 ) , (4 , 4 , 2 ) , ( 3 , 3 , 3 ) , ( 2 , 2 , 4 ) } .

PROOF. The L·

=

~E- (i En) constitute a system of imprimitivity

l l

-for G(m,p,n). Let P be as in (1.8). First of alr; we will pay attention to the case that an orbit of p _{gives rise to another}

system of imprimitivity. By (2.5)(vi), we then have either n = 2 = m gcd( 2 ,q). (gcd( 2 ,m) )-1

, or

n

>

2 and mn(n-1) = 2n.

If (1) occurs, we have (m,p,n) E {(2,1,2),(4,4,2)}; (2) leads to a contradiction with m

>

1 (cf. (2.4)). The conclusion is that none of the groups G in question has a system of imprimitivity afforded by roots different from the canonical system.

Let us assume that V1

,V

2 , • • • ~Vn is a system of imp~imitivity

different from L1 ,12 , • • • ,Ln and not corresponding to an orbit of

P. Let 11 ,12 , • • • ,ln be defined with respect to V1 ,V2 , • • • ,Vn as

in the proof of (2.2)(i), and put f

=

1112 1 . _n Suppose that

-f is a semi-invariant but not an invariant. It -follows, by an argument similar to the one in the proof of (1.B)(iii), that f

is the product of an invariant and some f

0 (0 orbit in P). Since deg(f)

=

n, the irreducibility of G implies that there is an orbit 0 in P of length n such that f

=

f

0 ; this is contradictory to our assumption. Therefore f is an invariant homogeneous polynomial.

Because f q ~.X₁ X₂ • • • Xn, there must be an a E [such that

f-aX1X₂ • • • Xn is a nonzero homogeneous G-invariant polynomial

in x~,xT,

...

,xm (cf. (2.5)(ii)). Hence m divides n.

n

Put 11 = a1X1+ a.2X2+ .•• +a. X, r- = #={i E n!a· =a..} for j E ~'

n n J - l J

and ro

=

#:{i E ~lai

*

O}. Let j E {0,1,2, . . . ,n}. Since the

stabilizer in IT of ~1₁ is of order< r.!(n-r.)!, and since the

n J J

IT _n -orbit of ~1₁ has at most n elements, we have n ~ n! . (r].!)-i • ((n-r.)!)-1

=

_en).

J rj

This implies that r.

=

1,

J . n-1, n for any j E {0,1, . . . ,n}. Note that r0

*

1. Suppose r0

=

n-1. Using a combinatorial

we get that the stabilizer of IC1₁ in G(rn,p,n) is of order

argument, ~ mq(n-1)!,

n-1 n-2

whence n

>

m .q.n! . (m.q.(n-1)!)-1

=

_{m .n; s o n < 2, and}

11 E <rX1 , contrary to the assumption that V_{1 ,} V_{2 , • • • ,} V is n

of the ~1₁ -stabilizer in G(rn,p,n) is·smaller than or equal to

1 -... n-1 r ( 1 ) - 1 - n- 2 T th · "th

I

m.n., s o n ? rn .q.n . . m.n. - m .q. oge er wi m n, this implies that (m,p,n) E {(2,1,2),(2,2,4)~(3,3,3)}.

(2.8) REMARK. To G(2,1,2) correspond, apart from the canonical one, two systems of imprimitivity, namely

<r(e:_{1+e: 2 ),Q;(e:1-e: 2 ) with semi-invariant} X~

- Xi,

and

<r(e:_{1+ie: 2 ),Q;(e:1-ie: 2 ) with} invariant

Xi

·+ X~ (compare (1.11)(iii)). To G(3,3,3) corresponds 1:(e:₁+e:₂ +t:₃ ),Q::(e:₁ +we:₂ +w2 e:₃ ),Q;(e:1+w

2

e:₂ +we:_{3 )} with invariant X~ +

x:

+

x: -

3X1X2X3 .

To G(2,2,4) correspond <t(t:₁ +e:_{2 +e: 3 +e:4 )} ,1:(e:₁ +e:2-E3-e:4) ,IC(e:₁ -e:2+e:3·-e:4), 1:(e:

₁

~t:

₂

-e:

₃

+e:

₄

) with invariant

Xi

+

Xi

+

x:

+

X!

-- 2

c

xi

x;

+

xi x;

+

xi x!

+

x; x;

+

x;

x~ +

x; x! )

+ 8X1 X2 X3 X4 , and the

system that can be obtained from the preceding one by substitution of -e:1for t:1 with invariant that can be obtairied from the

preceding one by substitution of

-X

1 for X1 •

Moreover one can prove that these are all non-canonical systems

of imprimitivity in the respective cases.

(2.9) PROPOSITION. Let 1

<

m

<

n, let G be a primitive refleation group in V, and let H he an imprimitive irreducible m-dimensional (i.e. imprimitive irreducible in dimension mas defined in (1.1))

reflection subgroup of

G.

Suppose, moreover, that

H

has a

. "' . . . t . . L L L . ( VH )l Th G

un~que system or ~mpr~m~ ~v~ty 1 , 2 , ••• , rn ~n • en

contains a reflection s0 suah that

<

H,s0

>

is a primitive

(m+1)-dimensional refleation subgroup of G.

PROOF. Put W

=

(VH)l. Now dim W

=

m; the proof goes by induction with respect to n-m.

Suppose m

=

n-1. Put L

_n

=

VH. Note that 11 ,L2 , • • • ,L _n form a system

of imprimitivity for Hin V. The required result is a direct consequence of the observation that this is the only system of imprimitivity for H in V consisting of 1-dimensional linear sub-spaces. In order to prove this observation, let V 1 , V2 , • • • , V n be

another such system. Reasoning as in the proof of (2.6), we obtain that either H is conjugate to G(1,1,n) or there is a j E n with Vj CW. As G(1,1,n) is primitive in dimension n-1, we must have that Vi CW for all but one i E ~; the uniqueness of 11 ,L2 , • • • ,Ln

Suppose m < n-1. Assume that there is no s0 as required. Let

s be a reflection in G with base root a such that a

e

WU

w1.

Note that < H,s > i s (m+1)-dimensional. Now< H,s > i s irreducible and imprimitive in W + sW. Furthermore, < H,s >has a unique

system of imprimitivity in W + sW, namely L1 ,12, . . . ,L ,L _{m m+ 1} = =

w

1

n

$W. Choose unitary vectors a. E L. ( i E m+1), and

l l

permute the indices to obtain sai

=

ai (i E m-1) and sam

=

am+i· Clearly, s is of order 2. Application of the induction hypothesis to< H,s >provides a reflection sr E G with base root, say, b E V such that< H,s,s' > i s primitive in W'

=

W

+ sW + s'W + s'sW. Let am+₂ be a unitary vector in

W'

n

(W+sW)

1 .

There is an

i E m+1 with s'a. ~ W+s~~ because the L· (j E m+1) form a single

~- l J

<

H,s >-orbit, there are g E

<

H,s >and a EU with gai

=

aam+i'

Replacing s' by gs 1_g-1 _{, we see that the assumption s'a fi. W+sW}

m

does not harm the generality. Because-< H,s' > is imprimitive in dimension m+1, we may assume that there are

A.,µ

E ~ with

- l

b

=

2 2 _{(a - A.a}

1 - µa +2). The imprimitivity of< H,ss's >

m m+ m

in dimension m+1 implies that 1 -

IA.!

2

=

l<ss'sa

la

)IE {0,1}.

1 . m m

Therefore IA.I

=

0,1, and b = 2-2_{(a - A.a}

1) or

, m m+ ·

2-2_{(a - µa +}

2), contradicting the primitivity of< H,s,s' >in

m m

§3. THE TWO-DIMENSIONAL CASE

V

= [

2 with standard unitary inner product.

In this chapter we shall identify µm and µm.1₂ form E IN.

(3.1) We present a description of all finite subgroups in Gl2 (~) (cf.

[10], [13]).

Let H,K be finite subgroups of Sl2 (C) such that K ~Hand H/K is

a cyclic group of order w, and assume an isomorphism ~ : µwd/µd -+ H/K is given.

Some definitions:

iJ; : Zl2 (<r) x Sl2 (([;) -+Gl2 ((£) is the usual product map,

µwdx$H

=

{(m,s) E µwdxHl$Cmµd)

=

sK}, and

(µwdlµd;HjK)~

=

$(µwdx~H).

The latter group is a finite subgroup of Gl2 ((£).Every finite

sub-group G of Gl2 (C) can be gotten in this way: Put µwd

=

G.Sl2 (C)

n

Zl2 (~), µd

=

G

n

Zl2 (~),

H = Sl2 (a::)

n

_G.Zl2 (~) , K = Sl2 (~)

n

G, and let

~ µwd/µd-+ H/K be the composition of the natural isomorphisms: µ d/µ,-+ (Zl2 (~)

n

G.Sl2(~))G/G

=

(G.Zl2 (~)

n

Sl2(C))G/G-+ H/K.

w a

Conjugation of G does not alter µwd and µd' and changes H and K into conjugates by the same element.

For each conjugacy class of finite subgroups of Sl₂ (~) we fix a representing element (cf. loc. cit.):

(

e27fim-1 the cyclic group of order m:

C

=

<

m 0

the binary dihedral group of order 4rn:

D

m

=

<

0 )

• -1

e-27fi.m

>,

(

~ ~)

,C2rn

>,

the binary tetrahedral group of order 24:

I=<...J:.(s s

3)

D

>,

12

E: €7 . ' 2 the binary octahedral group of order 48:

0

=

<

the binary icosahedral group of order 120:

I _

<

1 (

n

4

-n

-

rs

n2 -n3

where s

=

exp(ni4-1 ) , n

=

exp(27fi5-1 ) .

n2 -n3)

n -n4 (s3 0 _{) ,}

_{I > ,}

0 E:s 1

(n2 -n4

IS

1 -n

The choice of the representing elements is such that each group is in SU2 (~) and such that

C

2 ~

D

~

D

2 and

D2

~

T

~

Q.

Apart

, m m m

fr:om

cm

~

cmr'

these are the only normal inclusions with cyclic quotient. Thanks to this observation, i t is readily checked that if H is not cyclic, the conjugacy class of G

=

(µwdlµd;HIKl$ is independent of the choice of~- In that case we shall drop the index~ and write (µwdlµd; HIK) for G.

Note that G is irreducible if and only if H is non-cyclic. By now i t is not hard to prove the following

THEOREM. Any irreduaible finite subgroup of Gl2 (~) is conjugate to one of the following subgroups of U₂ (~):

(µ4qlµ2q; DmlC2m) µ2m ·

T

(µ4qlµ2q; D2mlDm) ' (µ6mlµ2m;TID2),

(µ2qlµ2q; Dm1Dml

=

µ2q · Dm µ2m ·

0

( µ 4 _q

I

µ ; _q

D I

_{m m}

c )

f

0 ::t' ( m' 2 )

=

1 ' ( µ 4, m

I

µ 2m ; _{_}

0

IT) '

where m,q E IN.

(3.2) Let H,K be subgroups of U₂ (G::) occurring in the list of (3.1) such that X ~ H; suppose that H is non-cyclic. The following statements concerning G

=

(µwdlµd;HIK) are easily verified.

( i ) G is imprimitive i f and only

i f

H

=

Dm for some m E IN;

(ii) Z(G)

=

G

n

Zl2 (~)

=

µd and G/Z(G) ::= H/Z(H)

=

H/µ2 ;

( i i i ) let m = pq

>

1; ther (

_Cµ

t

D IC

4q µ2q; ·~ . m)i! p even, q odd

,_

G(m,p,2) is conjugate to (µ

2

qlµq;Dm1D~)

i f

p odd, q even 2

µ2q ·

Dm

i f

p,q even (µ₄qlµq; Dml(m) i f m odd

(3.3) Let G

=

(µwdjµd;HjK) be as in (3.2). Suppose moreover that G is an irreducible reflection group with degrees d₁ ,d_{2 •} Denote by IT the projective group G/Z(G) operating on the projective complex line, and by n1 ,n2 ,n3 the orders of three conjugate

( 3 . 4 )

PROPOSITION.

(i) 2 .d. (d-1 d_{1 )} (d-1 d_{2 )}

=

I HI

and d

=

I

Z(G)

I

=

gcd(d₁ ,d2 ) ;

(ii) µ

1.G ~s a reflection group i f 1

=

~lcrn(2,wd);

(iii) µwd . His a reflection group i f and only

i f

wd2 ld₁d2 ;

(iv)

(v)

in this situation~ the degrees of µwd . Hare d.lcm(w,d-1_di)

(i

=

1,2);

the order of any reflection in G is a divisor of some n.

l ( i

=

1,2,3);

wdj 2 lcm(n₁ ,n₂ ,n_{3 ) .}

PROOF. (i) is an immediate consequence of (3.2)(ii) and (1.4)(ii). (ii) µ

1.G

=

< det g, gig E G >

=

< det·s, sis reflection in G >

=

=

<(det s-1 )s, sis reflection in G > i s indeed a reflection group. (iii) If µwd . H is a reflection group with degrees c_{1 ,} c_{2 ,} ·then

wd

=

gcd(c₁ ,c_{2 )} by (1G4)(ii), so w2 _d2 _{is a divisor of c}

1 c2 =

= jµwd .

HI

= wd₁d_{2 ,} and wd2 ld₁ d_{2 •} On the other hand, suppose that wd2 ld₁d_{2 •} Then w

=

gcd(w,d-1d1 ).gcd(w,d-1d2 ) and µwdH

=

µwd . G

has algebraically independent homogeneous invariant polynomials of degrees d.lcm(w,d-1_{di) (i}

=

_{1,2) (namely suitable powers of}

similar invariant polynomials of G). It follows that lcrn(w,d-1_d 1 )

. lcm(w,d-1d2 ) • d2

=

wd1d2

=

lµwd .

HI.

Now (1.2) finishes the

proof.

(iv) The image in IT of a reflection in G has the same order as the reflection itself and leaves fixed a point on the projective line.

(v) If g E G, then g can be written g

=

s₁s2 • • • sr where

s1 ,s2 , • • • ,sr are reflections in G. Therefore the order of det g

is a divisor of lcm(order(s1 ),order(s2 ) , • • • ,order(s )), which is _r

a divisor of lcm(n₁ ,n₂ ,n_{3 ) .} The conclusion is that wd

=

= order(µwd) j2 lcm(n1 ~n2 ,n3 ) .

THEOREM. Up to aonjugaay the primitive 2-dimensional refleation

groups a!'e _µ6m

.

_T

TID, l}

m

=

1,2, ( µ 6m

I

J.12m; µ4rn

.

0

OITl]

m

=

1,2,3,6, (µ4mlµ2m; µ2m

I

m

=

2,3,5,6,10,15,30.

PROOF. Let G

=

(µwdlµd; HIK) be as in (3.3); suppose, in addition, that G is primitive. Assume that d is even (this is allowed since K =f=. Dm, Cm, compare Theorem ( 3 .1) and ( 3. 2) ( i) ) . Let X1 , X_{2 ,} X₃ be indeterminates over (t. We fix homogeneous polynomials u1 ,u2 ,u3

(of degrees 6, 8, 12), v1 ,v2 ,v3 (of degrees ,8, 12, 18), w1 ,w2 ,w3

(of degrees 12, 20, 30) in S

=

S(V), and isomorphisms

r:

: :

~T

: such that t£[ U1 ; U2 , U3 ] ({;( V1 ,V2 ,V3] ({;(Wt ,Wz ,W3] 11:( U1 , U2 , U3 ] _,.. _,.. _,.. te[ X1 'X2 'X3 ] I (

xi

+ x~ +

xi )

a;[ X1 , X2 , X3 ] I ( Xi - X2 ( X~ - Xi ) ) a;[ X1 'X2 , X3 ] I ( x~ + x~ +

x; )

( 1 )

rL

sO

= _Q;[ V1 , V2 ,

v:d

s I

= i.C[ w1 ,w2 ,w3] and ( 2) p(u.)

=

o(v.)

=

-r(w.) ..

=

x.

( i

=

1,2,3). l i· l l

We refer to [13] for the existence of the polynomials and the maps. Now we will discuss the different possibilities for H.

(a) H

=

r.

From (3.3)(i) i t follows that 2ldl12. If G

=

µd .

T

and d

=

2,4, then the algebra of invariant polynomials of G is i.C[ u1 , u2 , u3 ] , i.C[ ui , u2 , u3

l ;

this, however, is contrary to ( 1. 2) .

Hence d

=

6,12. Suppose G

=

(µ

3dlµd;

TID

2 ). According to (3.3)(ii), µ 3d . G

=

=

µ₃d .

T

is a reflection group. As we have just seen, this implies

d

=

2,4.

(b) H

=

Q.

In view of (3.3)(i) we have 2!dl24. Suppose G

=

µd

.Q.

If d

=

2,6 the algebra of invariant polynomials of G is ~[v₁ ,v2 ,v3 ]

<I:[ vi , v2 , v3

l,

which is, again, in contradiction with ( 1. 2). So

d

=

4,8,12,24.

If G

=

(µ ₂dlµd;

O!T),

we have 2dl24 (according to (3.3)(v), since {n1 ,n2 ,n3 } = {2,3,4} in this case, cf. [13] ); so 2ldl12.

(c) H

=

I;

G

=

µd .

I,

and 2ldl60 by (3.3)(i). It remains to show that d =f=. 2; but this is clear since

sl

is not as required by (1.2) for

I

to be a reflection group.

Finally, a case-by-case argument, involving either invariants or generating reflections, yields that the groups in the theorem are