real representations of finite real groups · real representations of finite real groups 3...

Real Representations of Finite Real Groups

LAM Chi Ming

A Thesis Submitted to the Graduate School of

The Chinese University of Hong Kong

(Division of Mathematics)

in Partial Fulfillment of the Requirements for the

Degree of Master of Philosophy

(M. Phil.)

©The Chinese University of Hong Kong

August 2000

The Chinese University of Hong Kong holds the copyright of this thesis. Any

person(s) intending to use a part or whole of the materials in the thesis in a

proposed publication must seek copyright release from the Dean of the Graduate

School. .

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Real Representations of Finite Real Groups i

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my supervisors Prof. Lam

Sill Por and Prof. Shum Kar Ping for their aid, comments, encouragement and

stimulation throughout the whole period of my postgraduate study and on the

preparation of this thesis. I also want to thank everyone who has concerned on

my thesis.

Real Representations of Finite Real Groups ii

Abstract

The main purpose of this thesis is to develop a version of Brauer induction

theorem on Real groups. The notion of a Real group was introduced by Sir

Michael Atiyah. This was introduced together with the notion of equivariant Real

X - t h e o r y which was intended as a technical tool in proving results about equiv-

ariant XO—theory. At that time, there were results about equi variant AT—theory

whose equivariant KO analogues were unable to obtain. However, by introduc-

ing the notion of equivariant Real i^T-theory and Real representation theory, Sir

Michael and some of his collaborators were able to prove the Real analogues of

those results using the same method used for equivariant A'-theory, and then by

specializing, the results about equivariant /CO-theory were obtained.

Chapter one is the introduction of Real groups and Real representations. We

take the viewpoint of Karoubi who introduced more general notions of Real groups

and Real representations. The purpose of chapter two is to prove our Brauer in-

duction theorem on Real representations. We introduce the hyperelementary

subgroups of a Real group at first and then we prove that every Real represen-

tation of a Real group is a Z-l inear combination of those induced from complex

and Real representations of the hyperelementary subgroups.

® _ M

本論文的主要目的是提出一個Real表示論上的Brauer誘導定理• Real 群和 equivariant Real K-理論最初是由 Sir Michael Atiyah 提出.當時，一些在equivariant K-理論的証明，無法直接套用在equivariant K0-理論上•但是’當Sir Michael與他的同事提出equivariant Real K-理論及

Real表示論後’他們能夠直接使用equivariant K-理論的証明而得出Real K-理論上類似的結果•特別地’ equivariant K0-理論是equivariant Real K-理論的特例.

第一章，我們先介紹Real群和Real表示•我們採用了 Karoubi 的觀點，因爲他的定義比較槪括•第二章，我們首先定義Real群的

hyperelementary子群•隨即，我們証明在Real表示論上的Brauer 誘導定理:每一個Real表示都是一個由有限個hyperelementary子群的Real 表示或複表示誘導生成的Real表示的整係數M.性組合• •

Contents

Acknowledgments [

Abstract jj

Introduction 3

1 Introduction to Real Groups and Real representa-

tions 5

1.1 Real Groups and Real representations 6

1.2 RR(G,e) 10

1.3 Examples of Real representations 15

1.3.1 Cyclic groups 17

1.3.2 Dihedral groups 18

1.3.3 Other examples I9

2 Brauer induction Theorem on Real representations 22

2.1 Real induction 22

2.2 p-hyperelementary subgroups 27

2.3 Brauer induction Theorem on Real Representations • 29

1

Real Representations of Finite Real Groups 2

2.4 Monomial Real Representations 42

Bibliography 47


Introduction

i^T-theory was defined by Atiyah when he studied vector bundles over a com-

pact space X. Complex vector bundles and real vector bundles over a compact

space X give the complex group, K{X), and real i^-group, KO(X), respec-

tively. The homotopy groups of infinite unitary group and orthogonal group were

given by Bott. It led someone to define the cohomology theories, K*{X) and

KO*{X). They are two important concepts in topology. For instance, Atiyah

and Singer solved the index problem by K*{X). Moreover, Adams has solved the

vector field problem on spheres by evaluating suitable cohomology operations on

KO*{X).

It was believed by Atiyah, Singer and Segal that one can generalize the com-

plex index theorem to other situations. In the case of a family of real elliptic

operators parameterized by a compact manifold X, the target group should be

KO{X). However, Atiyah and Singer [At-S] believed that in one of the intermedi-

ate group, they had to introduce an involution on the space. So, they considered

a new kind of vector bundles, which is called Real vector bundles. This was the

first example of Real theory. The originally purpose of Real i^- theory was as

a technical tool to solve the index problem for families of real elliptic operators.

On the other hand, Atiyah evaluated K(BG) where G was finite. Moreover,

Atiyah and Hirzebruch evaluated K{BG) where G was a compact and connected

groups. Atiyah and Segal wanted to do the same thing for compact Lie group and

KO-theory. In [At-Se], they evaluated K*(EG x X/G) and KO*{EG x X/G)

using the language of equivariant theories. The idea of the proof was the follow-

ing. They considered the circle group at first. Then they could evaluate the same

thing for torus group by mathematical induction. The next step was to deal


with the unitary group. In complex iiT-theory, they used holomorphic induc-

tion to complete it. However, holomorphic induction doesn't exist for orthogonal

AT-theory. So, they considered equivariant Real theory which the holomor-

phic induction exists if the group is the unitary group with complex conjugation

on entries as the involution.

These two examples show why we consider Real /^- theory and equivariant

Real A'-theory. The coefficient group of the latter is the Real representation

ring. Reader who are interested in Real K-theory, equivariant Real /^- theory

and Real representation rings may refer to [At 1], [At 2] and [At-Se .

When C. B. Thomas studied group actions on sphere, he had to consider the

image of the homomorphism R{G) — K(BG) and RO(G) — KO{BG) where G

was a finite group. The basic properties of first homomorphism may find in [Ad 2 .

For the second homomorphism, Thomas [Th] suggested a general study by Real

iT-theory and Real representations. In [Th], Thomas suggested a strategy and

verified this strategy was valid for orthogonal representations. He also suggested

a version of Brauer induction theorem for Real representations would furnish the

proof.

In the paper [L-L-S] under preparation, Lam and his collaborators will p r o

vide two other versions of the Real Brauer induction theorem, enabling them to

complete the program suggested by Thomas. Our version here differs from those

two given in [L-L-S] by using fewer hyperelementary groups.

I thank Prof. S. P. Lam for his guidance in preparing this thesis. He introducd

me to the theory of Real representations; it was Prof. Lam who has conceived

the statement of the present version of the Brauer induction theorem for Real

repreesentations proved in this thesis; he explained to me his idea of how to

adapt the proof of the Brauer induction theorem for complex representations

(such as given in say Serre's book) to the Real case; some of the key steps in the

proof are also due to him; he also suggested me to prove Proposition 2.4.3. The


historical background given in this introduction is a modificaction of a similar

passage in a private manuscript of Prof. Lam and I thank him for allowing me

to use it here.

Chapter 1

Introduction to Real Groups and

Real representations

1.1 Real Groups and Real representations

In this thesis, we assume all groups G are finite and all vector spaces V over

complex field C are finite dimensional.

Definition 1.1.1 A Real group is a pair (G, e) where G is a group andeiG-^

Z2 is a group homomorphism.

Remark 1.1.2 The original definition due to Karonbi [Ka] requires that e is

surjective. But in this thesis, we deal mainly with Brauer induction theorem for

脑I representations in which we have to consider subgroups ofG and it is possible

脑仇e subgroup lies inside kere and restricting e to this subgroup is no longer

surjective. Therefore, it is convenient to use the present definition to include

these subgroups.

Definition 1.1.3 Let (G，e) be a Real group. Define G^ ：= kere and G^

A Real representation of (G,e) is a pair {V,p), where V is a vector

印ace over C and p : G x V V is a group action on V such that

6


(1) \/g E G。，g acts linearly on V, that is, p{g, au-\-bv) = a p{g,u) + b p{g,v).

(2) \/g G G^, g acts anti-linearly on V, that is, p{g, au bv) = a pig^u) +

for all a, 6 G C and u,v eV. We say that the degree of (V, p) is the dimension

ofV.

Remark 1.1.4 If there is no confusion, we will use V instead of {V, p) and g . v

or pgV instead of p{g,v). If e is trivial, then the Real representation of {G, e)

coincides with the complex representation of G. So, we are mainly interested in

the case where e is surjecUve. Suppose e is surjective. Obviously, G^ is a normal

subgroup of index 2 in G and the order of G is even. However, not all group

with even order can be equipped a surjective e. For instance, An {n > 4) hasn't a

normal subgroup of index 2.

Given two Real representations of (G, e), we can produce a Real representation

by direct sum or tensor product.

Definition 1.1.5 Suppose and are two Real representations of

(G,e). Their direct sum, (V i 6 V2,/？丄 ®p2)，^^^ tensor product,

p^), are defined by

：= p\9,vi)®p'^{9,V2) and

for all p € G, f i G Vi and 1)2 G V2.

The direct sum and tensor product of arbitrary finite number of Real repre-

sentations are defined inductively, that is,

© …© ：= (pi © …e f f © and

. pi <S)... <S> f/" := (/9I (g)…(g)广1)�“"


Definition 1.1.6 Let (Vi, /？) and (V2, p^) be two Real representations of {G, e).

Vi is said to be isomorphic to V2 if there exists a linear isomorphism T : Vi —> V2

such that T o p^{g,v) = p^{g,T{v)) for all g e G and v e V. We denote it by

Vi = V2-

Definition 1.1.7 Let (V, p) be a Real representation of {G, e) and W be a sub-

space ofV. W is said a Real subrepresentation ofV if PgW G W for all g ^ G

and w eW. We denote it byW<V.

Remark 1.1.8 Obviously, V and {0} are Real subrepresentations of V. They

are called the trivial Real subrepresentations ofV.

Definition 1.1.9 {V^p) is irreducible if

(1) V is not {0} and

(2) V has trivial Real subrepresentations only;

If V has a non-trivial subrepresentation, we say that V is reducible.

A natural question is raised. If K is a Real representation of (G, e) and W <V,

does there exist a complement U oi W such that U <V. This question can be

answered by the following lemma.

Lemma 1.1.10 Let {V,p) be a Real representation of (G, e). If W < V, then

there exists a complement U of W such that U <V.

Proof: Let W be arbitrary complement of W in V. Then there exists a projec-

tion p from V into W along W'. Define

qgg

(1) Since g • p - is linear V^ G G, so is p^.


(2) pO maps V into W.

(3) P^X = X V.T G W.

Therefore, is a projection from V into W, and there exists a complement U of

W corresponding to this projection

Moreover, we have =

For any u e U and g e G, we have • u) = g • p^(u) = 0. Therefore, g - u £U

and U <V. •

In the theory of complex representations, Maschk's Theorem tells us that

every complex representation is isomorphic to a direct sum of its irreducible

subrepresentations. We have a similar result in Real representation.

Corollary 1.1.11 (Analogue of Maschke's Theorem) Every Real represen-

tation V is isomorphic to a direct sum of irreducible Real subrepresentations of

V. That is, F = © • • • © Wn, where each Wi is irreducible.

Proof: By mathematical induction on the dimension of V and lemma 1.1.10. •

Similar to the case of complex representations, this decomposition is not

unique.

Example 1.1.12 Consider a Real group, {Ds, e), where D^ =< {r, = r^ =

1, rrr = r^} > is the dihedral group of order 6 and e : D^ D^/ < r > is the

quotient homomorphism. Let C^ := {(a,6)|a,h G C}, which is a 2-dimensional

vector space over C. Define a Real representation, (C^, .)，of {Ds, e) by

(1) g . (a, b) = (a, b) if g e D^,

(2) g • {a, b) = {a, b) if g e D\


Then every one dimensional vector subspace of C^ is a Real suhrepresentation

of {Ds,e). Thus, this representation have a plethora of decompositions of (C^, •).

However, we shall prove that any two decompositions of V are isomorphic in

next section.

1.2 RR{G, e)

Let {Vi)i=i be a system of representatives of non-isomorphic irreducible complex

representation of a group G, where h is the number of conjugacy classes of G.

The Group of Virtual Representations of G, denoted by R{G), is the free

abelian group generated by (V i)jLi. An element in R{G) is a different of two

complex representations of G. Moreover, {R{G), ©，0) is a ring, where © and 0

are the direct sum and tensor product of complex representations respectively.

Similarly, we can define the group of virtual Real representations of a Real

group (G，e).

Definition 1.2.1 Let (Vi)iî be a system of representatives of non-isomorphic

irreducible Real representation of a Real group {G, e). The free abelian group,

RR{G,e), generated by (Vî)ie/，is called the Group of Virtual Real represen-

tations of{G,e), that is, RR(G, e) = ©ê/ZV；. An element in RR(G,e) is called

a virtual Real representation o/(G,e).

Remark 1.2.2 In fact, a virtual Real representation is a difference of two Real

representations. Moreover, e), ©, 0 ) is a ring, where ® and <S) are the

direct sum and tensor product of Real representations respectively.

So far, we don't know whether the number of non-isomorphic irreducible Real

representations of a Real group (G, e) is finite or not. However, we shall prove

that it is finite at the end of this section.


Suppose (G, e) is a Real group. Let p : G x V V he a Real representation

of (G, e). Consider the restricting group action on p\go \ G^ y. V ^ V.

Obviously, {V,p\go) is a complex representation of and we define the R e a l

cha rac t e r , or simply character, of a Real representation (K, p) to be the complex

character of {V, p\g^). Therefore, we can define a map Resô : RR{G, e) —>• R{G^)

by restricting every Real representation of (G, e) on G � . Moreover,

Lemma 1.2.3 Resô is a group homomorphism, which is called the Complex-

ication map of (G, e).

Suppose e is surjective. Take a complex representation Vq in R(G^). Since the

index of G^ in G is 2, we can induce it to a complex representation mdgo(Vo):=

VQ © TVO, where r is a fixed element in G^. Note that , VQ is isomorphic to TVQ as

a vector space. The index r is used to distinguish them only.

Definit ion 1.2.4 Let (V, +,.) be a vector space overC. The vector space {V, +，o)，

where cov '=c-v Vc € C and v eV, is called the conjugation of V.

We define a new vector space Indô{Vo) •= Vq where rVo is the conju-

gation of tVo. Moreover, we can define Indô{Vo) to be a Real representation of

(G, e) by the following group action:

Fix a T in G\ Write G = G^ U t G � . Define p : G x Indô{Vo) — Indô{Vo) by

(1) pg{v © Tw) = g • V ® T(r—ipT . w) and

(2) Prg{v ® rw) = {rgr) -werig- v);

where g G C®, v EVQ and rw G TVQ.

Lemma 1.2.5 {IndQo{Vo), p) defined above is a Real representation of (G, e).

Since the construction of /ndgo(Vo) depends on the chosen representative in

G^. Therefore, we may have different Real representations for same VQ G R{G^)

but different choice of r . However, all of them are isomorphic.


Lemma 1.2.6 The Real representation Indô{Vo) defined in lemma 1.2.5 does

not depend on the choice of representative up to isomorphic.

Proof: Suppose r and T' are two elements in G^. We have two Real representa-

tions (Vi，pi) and (V^2,p2)’ where Vi := M) ® ^ and V2 := VQ 0 T'VQ. Consider

the linear mapping F :Vi V2 defined by

Fiui e TU2) =Uie (Pr O

where ui G VQ, R«2 G TVQ.

Since Pi = pI V" G G � , we have / 4 o ( 心 ) - i =片成("JJ-i(乂)- i =

By this equation, we have F o p^^ = p^ o F ^g e G. •

By lemmas 1.2.5 and 1.2.6, we see that Indô ： R(G^) — RR{G, e) defined by

Indô{VO) •= Vb ® tVo, where Vq e is a well-defined mapping. Moreover,

Lemma 1.2.7 Indô is a group homomorphism, which is called the Realifica-

tion map of {G, e).

Remark 1.2.8 The Realification map was given in [Fa] at first. This homomor-

phism is very important Firstly, it is a crucial step to prove Resô is injective

(see Theorem 1.2.11). Moreover, it is used to prove that every Real character of

{G, e) is a sum of at most 2 complex characters of G^ (see lemma 1.3.2).

Lemma 1.2.9 LetVQ G R{G^) andx be its character. Suppose V = Indô(Vo)=

Vq 0 TVq and X' is its character. Then

x'ig) = xig) + x(T-V) = xio) + xir-'g-'r)

for all g G G�.


Proof: Let (aij(g)) be a matrix representation of g in VQ, where g G G^. Note

that the matrix representation of ^ in Vq 0 tVq is {aij{g)) © {aij{r~^gr)). Let

{ / i , / „ , } and {ei, ....,6^} be bases for VQ and TVQ respectively. Thus,

n

j=i n

j=i

Then the matrix representation of g with respect to the basis {/i，....’ /n，ei，"..’ e„}

is (aij(g)) © (aijir-^gr)) and its character is x(g) + Since x is a com-

plex character, we know that = x(y) implies = + =

x ( f f ) + x(r-'g-'T) •

We are going to prove our main result in this chapter.

Lemma 1.2.10 For any V G RR{G,e), Ind% o Res%oiy) =

Proof: Take a Real representation V e RR{G,e). Let Vb = Resô{V) and

V = /ndgo(K)) = with the group action p'. Define F :V ^V ^V by

F(Vi e V2) ：= (vi + V2) e (p； O (/V)-l”l - �（/?t)-1”2)

for all V i e V =

By direct verification, F is a linear mapping and F o ( p g ^ pg) = p'ôF for all

g & G. F is surjective since Viui 0 tw2 G V,

+ PrW2) © - prW2)) = Wi ® TW2

Moreover, dim V = dim(V 0 V) implies F is injective. •

Theorem 1.2.11 Resô is injective.


Proof: Suppose Res%o{V) = ResôiW), where V,W e RR{G,e). We have

2V = Indjo o = I减o�Resô{W) = 2W

Since RR(G, e) is a free abelian group, so V = VK. •

Theorem 1.2.11 tells us that a Real representation is characterized by its char-

acter. Therefore, we can prove whether two Real representations are isomorphic

by verifying their characters.

P r o p o s i t i o n 1.2.12 Suppose K, G RR{G,e) and K), VVq G R{G^). Then

(1) Res%o{V Res%,{V) © Res%o{W).

(2) Res%o{V ^W)^ ResôiV) 0 Resô{W).

(3) IndôiVo e Wo) = IndôiVo) 0 /n略

(4) Ind%o{VS ® WQ) ^ IndôiVo 0 W^).

(5) IndôiVo) ® Ind%o{W^) = /ncig�(V�公 Wq) © /nrfgo(K)* � ^ô).

where VQ is the contragredient complex representation ofVo.

Proof: Showing that they have same characters can prove these. •

Theorem 1.2.11 is also useful to deduce many properties of RR[G, e).

Corollary 1.2.13 Two decompositions of a Real representation into a direct sum

of irreducible Real subrepresentations are isomorphic.

We can prove that RR{G, e) is a finitely generated free abelian group by

theorem 1.2.11. Let us quote a lemma in group theory at first.


Lemma 1.2.14 (The Nielsen-Schreier Theorem) If W is a subgroup of a

free group F, then W is a free group. Moreover, if W has finite index m in F,

the rank of W is precisely nm + 1 — m，where n is the rank of F.

Proof: [Ro, p.159] •

Corollary 1.2.15 Every subgroup of a finitely generated free abelian group is

finitely generated.

Theorem 1.2.16 RR{G,e) is a finitely generated free abelian group.

Proof: Since Res^o ： RR{G, e) is a monomorphism, so RR{G, e) is iso-

morphic to a subgroup of Since R{G^) is a finitely generated free abelian

group ([Se, Ch 2]), by corollary 1.2.15, RR{G, e) is also finitely generated. •

Corollary 1.2.17 A Real Group (G, e) has a finite number of non-isomorphic

irreducible Real representations.

Proof: Since {Vi)ia is a basis for RR{G, e) and two bases of a free abelian group

have same cardinality ([Hii, p72]). •

1.3 Examples of Real representations

In this section, we are going to look for all Real representations of certain Real

groups. Since RR[G, e) = R{G) if e is not surjective, we assume e is surjective in

this section.

Let V = Spanc{ei,....’ e^}. For any v eV.we write v = i^iei H h =

(?；1’ ."� ) .We define v := ( t ^ , . . . . , ^ ) =可 e ! + … +


For any Real group (G, e). Let V = Spanc{ei}. Fix a r G Define a group

action p : G xV V by

p{g, v) = V and

pirg, v) = V

where g e G^ and v e V.

By direct verification, p is a group action and its character 三 1. We call p the

Real- l -representation of (G’e). We may think that the Real-l-representation

is a "Real" extension of complex-1-representation of G � . It is natural to ask if x

is a complex representation in whether there is a Real representation such

that their characters are same. The following lemma is an useful tool to decide

which X hasn't such extension.

Lemma 1.3.1 Let (V，p) be a Real representation o/(G, e) and \ be its character.

If 9, he G�such that h = t-^t，then x{h) = xig).

Proof: Note that h = T'^gr is a linear mapping from V into V. Suppose

has a matrix representation with respect to a basis { e j , e n } ,

then the matrix representation of h with respect to a basis is

0^)ij.=i，.."，n. Thus , x{h) = xig)- •

Lemma 1.3.2 Let {V,p) be an irreducible Real representation of (G, e) and x be

its character. Then x is a sum of at most 2 irreducible complex characters ofG^.

Proof: Let Res%oV = where W is an irreducible complex representation.

Then

2V = Ind% o Res%oV

=Ind%o{W © U)

• = Ind^oW e Ind^oU


Since V is irreducible, we have /rzc/goVl = V or 2V. •

Therefore, if V is an irreducible Real representation, then V is induced from an

irreducible complex representation of G � o r ResôV is a complex representation

of GO.

1.3.1 Cyclic groups

In this subsection, we are going to look for all Real representations of (Z2n,e).

Since Z2n has only one normal subgroup, say 2Z2n, of index 2, it can be equipped

one surjective e only. ..

Suppose n is odd. By [Se, Ch 5], 2Z2n = Z„, has exactly n non-isomorphic

irreducible complex representations, say Xi (z = 0,1, - 1), given by

= u/k (1.1)

where u; is a primitive nth-roots of unity and Z„ = < a >. Obviously, xo is the

Real-l-representation. For each Xz, 2 0, we can induce it a Real representation,

啦,by complexication map. Obviously, i j j i = Xi + X n - i and 也 = ^ 斗 So, we

have

Z x o e + X n - i ) ) C R R { Z 2 n , e )

Suppose X = ^ oXo + ttiXi + • •. + fln-iXn-i G RR{Z2n, e). W.L.O.G., we may

assume ai > a^-u for i = 1， . • • •， t hen

X = FTOXO + fln-l(Xl + Xn-l) + • • • + ajn+l) (X(n-l) + 2 2 2

+(ai - a n - i ) X i + ••• + ( a i n ^ 一 Q(n+i))Y(n-i) € RR(Z2n,e).

This implies 6> := 6iXi + -• • + 6(n-i) Y(n-i) 6 RR{Z2n,e), where k = a^-a^-i >

0, for i = 1，….，学.Assume there is some j e { 1 , … . , s u c h that bj + 0.

Since 9 6 RR{Z2n,t), we have Xj + Xn-j is a "component" of 2(9 = IndôoResôO,


tiiat IS，Xn—j IS a “ component" of Q. It contradicts our assumption, so all bi = 0.

Hence,

R 卿 2n, e) = Zxo e + Xn-i))

Suppose n is even. 2Z2n = has also exactly n non-isomorphic irreducible

complex representations, say Xi {i = 0，l，....’n - 1) given in the equation 1.1.

Obviously, xo is the character of the Real- 1-representation. Moreover, we can

extend Xf to another Real representation with same character of Xf • By similar

method, we can prove

e) = Zxo e Zxn © + Xn-i))

1.3.2 Dihedral groups

Let Dn = < { r , r | r " = r^ = e^r'^rr = r—i} > be the dihedral group of order 2n.

Let 6 : Dn ^ Z2 be a homomorphism with kere = < r � • We are going to look

for all Real representations of this Real group e).

Lemma 1.3.3 Suppose {G, e) is a Real group. Let (V, p) be a complex representa-

tion of G^ such that it has a matrix representation M(g) satisfying =

= M{rgT) for all g e G^. Then there exists a Real representation of

[G, e) with same character of p.

Proof : Suppose the matrix representation M[g) is with respect to a basis {ei,....’ e„}

for V. Fix a r G we define an action 0 : G x V V hy

= M{g) ‘ and

= M(").(可，….’旬

for all g e G ^ and ....，Vn) = Viei + ——h VnCn € V .

Since M ( T — = = M(T"T) for all g e G。，it is a group action. •


In Dn, we have t — W = = r r ^ r . So, for any complex representation

{W,p) in R{Dl), we have Mir-'^gr) = M{g-^) = Mirgr) for all g G By

lemma 1.3.3, for any complex representation p of D^, we have a Real representa-

tion of (D„, e) with same character of p. In particular, each irreducible complex

representation of D^ has a “ Real" extension. Therefore,

RR{Dr.,e) = R{Dl)

If n is odd, Dn has only one normal subgroup of index 2. However, if n

is even, Dn has another normal subgroup, say K' =< {r^,r}〉，of index 2.

Consider a Real group (D4, e) with D^ D2 = Z2 x Z2. By [Se, Ch 5], D^ has

four non-isomorphic irreducible complex representations given by

where k,l == 0,1.

By lemma 1.3.3, we see that X(o,o) and x(i，o) are in RR{D^,e). By complexi-

cation map, we have X(o,i) + X(i’i) is in RR{D^,t). Therefore,

Zx(o，o) ® Zx(i’o) e Z(X(0,1) + X(1,1)) c RR[D4, e)

Since X(o,i) and X(i,i) do not satisfy the relation in lemma 1.3.1，we have

e) = ZX(0,0) e Zx(i’o) e U x m + x(i，i))

1.3.3 Other examples

Let (G’e) be a Real group with G � = Qg, where Qg = {±1，士i, 士j, i / c j i ? =

f = k2 = -1 and ijk = 1} is the Quaternion group of order 8. Qs has five

non-isomorphic irreducible complex representations given by:


1 i i k

Xo 1 1 1 1 1

XI 1 1 1 - 1 - 1

X2 1 1 -1 1 -1

X3 1 1 - 1 - 1 1

( l o \ / - I 0 \ / z 0 \ / o - l \ / o - A X4

V^ 1 / V 0 - 1 / V1 � 乂 v ^ 0 ;

Suppose G = Q^^ < r >, where r^ = = i and t -丄六= k , and we

take e to be the quotient homomorphism from G onto G/Qg. It is easy to check

that

Zxo ® Zxi e Z(X2 + xs) ® 2Zx4 C RR{G, e)

Since X2,X3,X2 + X4,X3 + X4 do not satisfy the relation in lemma 1.3.1, they

are not in RR{G, e). Since T_HT = i, by direct verification, there isn't a Real

representation such that its character is equal to that of \4- Hence,

職G, e) = Zxo e Zxi e Z(X2 + X3) e 2Zx4

Let 54 and A4 be the permutation group and alternating group on {a, b, c, rf}.

Since has exactly one subgroup A4 of index 2，we have exactly one Real group

(54, e) with = A4. Let x = (ab){cd) and t = (abc). Then A^ has four non-

isomorphic irreducible complex representations given by:

e X t 亡2

Xo 1 1 1 1

X l 1 1 …

X2 1 1 w

/ 1 0 0 \ ( 0 1 0 \ / 0 0 1 \ / 0 1 0 \

X3 0 1 0 1 0 0 1 0 0 0 0 1

\ 0 0 1 / \ - 1 - 1 - 1 / V 0 1 0 y \ 1 0 0 ^


where u = e 3 .

By lemma 1.3.3’ we see that xo, Xi, X2 are in RR{S4, e). By direct verification,

there isn't a Real representation such that its character is equal to that of Xs-

Therefore, e) = Zxo e Zxi e 1X2 © 2ZX3

Chapter 2

Brauer induction Theorem on

Real representations

2.1 Real induction

Let (G,e) be a Real group and H he a subgroup of G. Obviously, e\H is a

homomorphism.

Definition 2.1.1 is said a Real subgroup o/(G,e). For simplicity, we

denote it by {H, e) < {G,e).

Definition 2.1.2 Suppose {H, e) < {G,e) and {V, p) is a Real representation of

(G, e). (V,P\H) IS called the restricted Rea l representation of (V, p) on H.

We denote it by res%V.

Suppose (G, e) is a Real group with surjective e and {H, t) < (G, e). Let {W, 6)

be a Real representation of {H, e).

Suppose t\H is trivial. For any V E RR{H, e), since V is a complex represen-

tation of H, we can induce it to m r f g V G RR{G^, e) by complex induction. By

complexication map, we have a Real representation Ind^o o i n d g V G RR{G, e).

22


That is, we have a group homomorphism:

IND^:RR{H,e) /?i?(GO，e)'“二含。/?/ (G，e) (2.1)

Proposit ion 2.1.3 Suppose e|i/ is trivial. Let V G RR{H, e) with character xv

and X be character of IND^V. Then, for all g G G^,

= { i n d ' S x v m + {indfSxv){r-'g-'r)

On the other hand, suppose is surjective. Let (V, p) be a Real representa-

tion of (G, e) and VK be a Real subrepresentation of res%V. Let sH be a left coset

of H in G. Define Ws ：= PsW. Wg is independent of the choice of representative

since pshW = PsPhW = psW. Moreover, for any g eG, pg permutes {M^JseG///-

Thus, V' = ^seG/H is also a Real subrepresentation of (G, e).

Definit ion 2.1.4 A Real representation {V,p) of (G, e) is said to be induced by

a Real subrepresentation {W,9) of {H, e) if V = ®seG/HWs. We denote it by

V = ind^W. (We also use the notation, ind, to denote the induction of complex

representations since, in principle, they are same.)

Conversely, suppose (H, e) < (G, e) and {W, 6) is a Real representation of

{H,e). Take a left transversal {>�}二i to H in G, where n = |G : and all

Si £ G^. Construct a vector space V := ^'l^^SiW. For any t/ g G and Si e S, we

have unique elements hg(i) £ H and G S such that gsi = � . W e define

a group action of (7 on V by: n n

9 • (Zl^t'^O := XI 〜⑷切i)， i=l i=l

where g eG, SiWi € SiW for all i.

Lemma 2.1.5 The vector space V equipped with the above group action is a Real

representation of {G,e). Moreover, {V,p) is induced by the Real representation

{W,9) of{H,e).


Lemma 2.1.6 Suppose {V, p) is induced by {W,e) and {V\p') is a Real repre-

sentation of (G, e). Let f : W V' be a linear map such that f{0{h,w))=

p'{hj{w)) for all h e H and w e W. Then there exists a unique linear map

F \V ^V such that F\w = / and F o p = p' o F.

Proof: Uniqueness of F: Note that V = For all x e SiW, we have

pj^^x G W and

This determines F on SiW, and so on V. Thus, F is unique.

Existence of F: Since V is a direct sum of SiW, define F : V V' hy

n n

i=l i=l By direct checking, F is a linear map and F o p = p' o f . •

Proposition 2.1.7 Let {W,9) be a Real representation of a Real subgroup {H,e)

of (G, e). Then there exists a Real representation {V, p) of (G, e) induced by {H, 0).

Moreover, it is unique up to isomorphism.

Proof: The existence can be proved by lemma 2.1.5. So, it remains to prove

uniqueness. Suppose {V, p) and {V',p') are two Real representations induced by

{W,e). Consider the inclusion map i ： W ^ V'. Since i o e{h,w) = p'{h,i{w))

for ail h e H, w e W, by lemma 2.1.6, there exists a linear map F \ V V'

such that F\w is an identity map on W and F o p = p' o p. Since F o p s \ W ) =

P>F{W) = and = F is surjective. Moreover, F is injective

since dim(l^) = dim(K')- •

Proposition 2.1.8 indffj : RR{H, e) RR{G, e) is a group homomorphism.


Therefore, we have another group homomorphism

IND^^ : RR{H,E) RR{G,E) (2.2)

Remark 2.1.9 Equations 2.1 and 2.2 give a group homomorphism

IND^ : RR(H, e) RR{G,E)

for any Real subgroup {H, e) of (G,e). This homomorphism is called the Rea l

i n d u c t i o n from (i/ , e) to {G, e).

Proposi t ion 2.1.10 Suppose {V,p) is an induced Real representation of {W, 6).

Let xv and Xvy be characters of {V, p) and {W, 9) respectively. Let R he a left

transversal to H^ in G^. Then for any g G G^, we have

Xvig) = Xwit-'gt) = ~ Xwit-'gt). S.t. t-igtGH teGO s.t. t-igteH

Moreover,

H Xw{t-^gt) = Xvig) + XV {g).

teG s.t. t-igteH

Proof: Let R 二 {si,. . . . ,s„}, where Si G G^ for all i. Then V = ©^^^s-P . Por

any g G we have gsi = � 沒 � ’ where S g � e R and 她）G H, for all i.

To determine Xvig), we can use a basis of V which is a union of bases of QiW.

冲 ) — i gives zero diagonal terms, so xvig) = 喊分 S f ) .

For any t in t can be written in a unique way Sih, where Si e R and h e H^.

Therefore,

teGO s.t. t-^gt€H

= Xw{{sih)-'^g{sih)) SiER and hGff�s.t. (sih)-^gi3ih)eH

gSi) {Since xw is a class function of H^) SiER and h€H° s.t. s'^gsiEH

= Y . Xwis-'gsi) Si&R s.t. s广gsiGH

= I 丑


On the other hand, if h' € H^, then Xw{{h')-^g{h')) = Xw{9), for all g in G � .

Thus,

[ Xw{t-'gt) teG s.t. t-^gteH

二 xwir^gt) + Xwit'^gi)

teG^s.t. t-igtGH t砂s.t. t-^gteH

= Y1 Xwif'^gt) + Y1 Xw{{h')-H-'^gth') {where h' G G^) teG^s.t. t-'^gteH teG^s.t. t~^gteH

= E xwit-'gt) + MFW teG^s.t. t-^gteH teG^s.t. t-igteH

•

Corollary 2.1.11 Suppose W e RR(H, e) with surjective e and V = ind%W.

Define WQ ：= Res^o{W) and Vb := Res^o{V). Then VQ ^ ind%l{Wo).

Proof: Let xv and Xv be the characters of V and W respectively. For all

g, y G G。，g-�g e His g-�g G H^. So, the complex character of ind%{Wo) is

{ind%xw){y) = Y^ Xw{t~^yt)

t&G^ s.t. t-^yteH^

s.t. t-^yteH =xv{y),

for all y G G^. Since each complex representation is characterized by its charac-

ter, so Vo = ind^l{Wo). 口


2.2 p-hyperelementary subgroups

Let {G, e) be a Real group. Take a p'-element x in G � . Define N{x) := {a e

= x}U{b G G^lb'^xb = By direct verification, N(x) is a subgroup

of G with a normal subgroup, say < a: >. Consider the quotient homomorphism

TT : N{x) — N[x)l <x>. Take a Sylow-p-subgroup P工 in N{x)/ < x � . Define

H{x) := TT一i(/y and call it a p-hyperelementary subgroup associated to x.

Lemma 2.2.1 \H{x)\ = • | < x > |. In fact, H(x) is a semi-direct product of

N{x)p by a normal subgroup < x >, where N{x)p is a Sylow-p-subgroup of N{x).

Proof: Since TT is surjective, H{x)/ < x >= 7r{H{x)) = P^ and = \H{x):<

•T > I = By Sylow 1-st Theorem, there exists a Sylow-p-subgroup N{x)p

of H{x). Since TT is surjective and that | < .T > | and p are relatively prime,

N{x)p is also a Sylow-p-subgroup of N(x). Moreover, (| < rr > |’p) = 1 implies

<x> nN{x)p = {1}, that is, H{x) =< x > xN{x)p. •

We are going to show some properties of a p-hyperelementary subgroup H{x).

Proposit ion 2.2.2 H{x) nGO is an elementary subgroup ([Se, Ch 10]) of G^

associated to x.

Proof: H{x) n G � = (< .T > KN{X\) n G � = < x > x(N(x)p n G � ) . •

Suppose H(x) n 0. Consider the group action

defined by

h o x^ = h-^x^h

Obviously, H{xf is the isotopy subgroup of x under this group action. Since

H{x) n — 0，this action is transitive and

.\H{x) : H{x)'\ = \{hox\h e H{x)}\ = \{x,x-'}\ = 2


Hence, \H{xf\ = i ^ .

Suppose Let H{x) is a p-hyperelementary subgroup. Suppose | i iXT)=

\ where c = | < RR > | and = \H{x)l < .T > Obviously, < X > < H{xf^ so

= c-pk for some k € { 0 , 1 ， T h i s implies \H{x) ： //"(x)。！ = 矢—2.

Therefore, H{x) n = 0 and H(x) is a subgroup of G^.

Suppose p = 2. Let x be a 2'-element in G^ and H{x) be a 2-hyperelementary

subgroup. Write N{x) = N{xf U N{x)\

(1) Suppose N{xy = 0. Obviously, N{x) = N(xf and H(x) is a subgroup of

GO. •

(2) Suppose N[xy~ — 0. Consider the quotient homomorphism TT : N{x)—

N{x)/ < - T � . If lN(x)/ < . T > I = 2 时 im with (m，2) = 1, then |iV(:r)| =

2",+icm’ where (cm, 2) = 1. Since N(xy ^ 0，so ： = 2 and

iV(:r)0| = 2^cm. Moreover, 7r(ff(x)) is a Sylow-2-subgroup of N(x)/ < x >,

so = 2"+ic.

Assume H(x) is a subgroup of N(.Tf, then \H{x)\ divides \N{xf\. This

implies f is an integer. It is impossible since (m, 2) = 1. Therefore

NixY n Hix) + 0 and \H{x) : / f ( : r )� | = 2.

Let's summarize what we have discussed in a proposition.

Proposition 2.2.3 Let H{x) be a p-hyperelementary subgroup of{G, e). I f p = 2

•d N{xy + 0，then \H{x) : H{xf\ = 2. Otherwise, H{x) is a subgroup o / G � .

Recall that every x G G^ can be written in a unique way x = x^Xr, where

^n is a p-element, is a p-e lement in G^ ([Se, Ch 10]). Let C^ be a cyclic

subgroup generated by x. Consider the quotient homomorphism TT ： N(xr)

• I < ^ r > . Obviously, x^ e < :r > < N{xr). Since rr, is a p-element, we see

that order of 7r(.T„) is a power of p and 7r{x^) is in certain Sylow-p-subgroup,


say P . . , of N{xr)/ < . t , � . Then we take H{xr) := 7 r - i (P , J , which is also a

p-hyperelementary subgroup associated to Since TT is surjective, x^ e H{xr).

Hence, xr^x^ € H{xr). Moreover, .T = x^xr G H{xr) and C^ < H{xr). That is,

Proposi t ion 2.2.4 Every cyclic subgroup of G^ is in certain p-hyperelementary

subgroup of {G, e).

2.3 Brauer induction Theorem on Real Repre-

sentations

The main purpose of this thesis is to generalize Brauer induction theorem on

complex representations to that on Real representations. The statement of Brauer

induction theorem on complex representations is:

Theorem 2.3.1 (Brauer induction theorem on complex representations) Every

complex representation of a group G is a Z-linear combination of complex rep-

resentations induced from complex representations of elementary subgroups ofG.

Proof: [C-R, Ch 40] •

The proof of our Brauer induction theorem on Real representations is based

on that on complex representations by Roquette and Brauer-Tate ([Se，Ch 10])

Firstly, we state the statement of our Brauer induction theorem on Real repre-

sentations.

Theorem 2.3.2 (Brauer induction Theorem on Real representations) Every Real

representation of a Real group (G, e) is a linear comhination with integer coeffi-

dents of Real representations induced from Real representations of hyperelemen-

tary subgroups by the Real Induction.

Real Representations of Finite Real Groups ^q

Firstly, we shall prove the following theorem. In fact, Brauer induction The-

orem on Real representations is a consequence of this theorem.

Theorem 2.3.3 Suppose (G，e) is a Real group. For any prime p, let Vp is the

subgroup of RR{G, e) generated by those Real representations induced from Real

and complex representations of p-hyperelementary subgroups of{G, e) by the Real

induction. Then the index of Vp in RR{G, e) is finite and prime to p.

Proof: (Theorem 2.3.3 4 Theorem 2.3.2)

Let V := E p Vp, where Y.^ runs through all prime p. It suffices to show that

y = RR{G, e). Note that, Vp is a subgroup of K and V i s a subgroup of RR[G, e).

So, \RRiG,e) ： V l • ： = \RR{G, e) : by assumption, which is prime to

P- So, \RRiG, e) : is also prime to p. Since p is an arbitrary prime number, so

\RRiG,e) : Kl = 1. 口

Let X{p) be the collection of all p-hyperelementary subgroups of (G, e). Ob-

viously, the group Vp is the image of the homomorphism

• : ®Hexip)RR{H,e) — RR{G,e)

Moreover,

Lemma 2.3.4 Vp is an ideal in RR{G, e).

Proof: K G I； iff K = EhIND^M), where H is a p-hyperelementary

subgroup of (G，e) and Wh G RR{H,e). So, it is enough to prove that, for

any U G RR{G, E) and p-hyperelementary subgroup H, U 0 IND^{Wh)=

IND%{{res%U)®WH).

Let xu, XWH be characters of U and WH respectively. Note that U ®U ^

Jnc/g�o Res%oU, so we have {xu + Xu){y) = Xu{y) + Xuir-'yr). This implies

Xu{y) = Xu(j-�T) = XU[T 一、〜、.


Suppose E\H is trivial That is, U 0 I N D ^ H ) = U <S) ( / n d g � �

and its character is

Xu(y) • {ind^H XwAy) +

=Xu{y) . ind^'xwAy) + Xu{T-'^y~^T) •

=ind%\xu{y) . XWniy)) + •

= � ( X u . XWH ) (y)

Obviously, it is the character of IND%{resÛ<S)WH). Since a Real representation

is characterized by its character, so U (g) IND§(WH) = IND§(resÛ 0 WH).

Suppose t\H is surjective. Then IND^Wh = ind^Wn and its charater is

indfâXwH- Therefore, the character of U <S) IND^{Wh) is

Xu{y) . {ind%'oXwH{y)) = ind^lixu . XivJ(y)

for all y G Obviously, the right hand side of the equation is the character

of IND^{res%U ® Wh)- Since their characters are same, U <S> IND^Wh =

•

Since Vp is an ideal in RR{G, e), to prove theorem 2.3.3’ it is enough to show

that there exists a Real representation Vi in V with character = I, prime to p. It

is because, for any V in RR{G, e),

T,小 I times T ’ vê … e V p

In fact, we shall prove:

Theorem 2.3.5 Let g = fH be the order ofG, with {p, I) = l . Then there exists

an element in Vp with character 三 I.

Let A be the subring of C generated by " - t h roots of unity, where g is the

order of G. This ring is free and finitely generated as a Z-module . Since el-

ements in A are all algebraic integers, we have Q fl = Z ([Se p 51]) The


quotient group A/Z is finitely generated and torsion-free, hence free. Thus, A / Z

has a basis, and by lifting this basis of A j Z to A, A has a basis, say {1, ei , . •. ’

Lemma 2.3.6 A® Res-.A® RR(G, e ) A R{G^) defined by

(A ® Res){Y^ a<S>V) Res^o{V)

is injective. That is, an element in A(S>RR{G, e) is characterized by its character.

Consider the homomorphism IND defined by tensoring with A, an yl-linear

map: -

A (8) IND : A 0 {®Hex{p)RR{H, e)) — A 0 RR{G, e)

defined by

(A ® I N D ) { Y ^ an (8) XH) := X ] a " (g) IND%{XH)

If ^ G ^ (8) then = (kXi, where Xi are characters of G^. For all

y £ we have:

= a i M y ) + X ] ""iXiir-'y-^r)

= + ^{r-^y-^r)

Since A has a basis {1, ei’ • •. ’ e„} containing the element 1，this implies

Lemma 2.3.7 The image of A <S) IND is A (S)Vp. Moreover,

Furthermore, since Vp is an ideal in RR(G, e) and A has a basis containing 1,

this implies:

Lemma 2.3.8 A®Vp is an ideal in A ^ RR{G, e).


Thus, to prove there exists a Real representation V e Vp with character

功三 /，it is enough to prove such V belonging to A(S)Vp (Since RR{G, e) has

the Real-1 -representation which character 三 1). That is, V can be written by

®脇、P� IND运 (CLHWH ) , where AN E A and WH G RR{H, e).

Definition 2.3.9 Let x,y be elements in G^. We say that x is conjugate to y

in G^, denoted by x^ y, if there is an element a e G^ such that a'^xa = y. x

is said to be Real conjugate to y in G。，denoted by x ~ y, if there is an element,

a ^ G^ such that a~^xa =

Remark 2.3.10 For each x G G�we define the Real conjugacy class, R工，of

in {G, e) by {y e C^ly ^ x} U {y G G'^ly - x}. Obviously, R工 is a union of at

most two conjugacy classes in G^.

Let .T be a p'-element in G^ and H{x) be a p-hyperelementary subgroup as-

sociated to .T. By lemma 2.2.1，we see that H[x) ¥ C工 tx N{x)p, where C^： is a

cyclic subgroup generated by x and N{x)j, is a Sylow-p-subgroup of N{x). We

are going to construct an element in ^ 0 such that its character 三 I, prime

to p. By proposition 2.2.3, we know that either H{x) < G � o r \H{x) ： H{x)^\ = 2

Case 1: Suppose H < G^.

Define by

c … = 工

I 0 otherwise

Obviously, e, = where 乂 runs through the set of irreducible

characters of CX. It follows that E A (G) RR(CX, e).

Since H(x) = Co： x N{x)p, we have the quotient homomorphism T ： H{x)

Cx defined by T{ab) = a for any a € C^.b e N{x)p.


Thus, we have an element e^oT = J：^ x{x-')x�:r e (g) RR{H{x), e).

By direct verification,

_ = 卜 / M 调外

0 otherwise

\

By complex induction, we have an element《工 :=(indg�� f ) ^ e).

For any y G

. 9eG0s.t.g--^ygeHix)

(1) Since 0 x � g - � g � ^ 0 iff g-�yg G xN{x)p, we see that for all y e G。，

_ = 淑丨到

where B = {g e G^lg-'yg e xN{x)p}.

If 力 e B, so tN{x)p C B. Therefore, B is a disjoint union of left coset

of N{x)p in GO and order of B is a multiple of order of N{x)p. Since

= c . we have 认y) e Z for all y e

(2) If 2/ is a ；/-element of G � a n d g G g-^yg is also a p'-element of G � . If

9~^y9 e H{x), then it belongs to C工，and we have T j ^ g S g ) = 0 whenever

g S g + It follows that 油、=Oiiy^ X.

(3)

渊 = ^ 极、 ) = . 丨 { " € G0|广.Tg = �’g^G^s.t.g-^xg=x 万(…

Since N(x)p is a Sylow-p-subgroup of N{x),

� ^ w U a , 0 — p )

Induce to ^^ by complexication map. Obviously, V'x G A 0 e) with

character

for all y e G^. ipx has the following properties:


(1) M y ) e Z for all y e GO since G Z.

(2) Suppose 2/is a p'-element in G � s u c h that y % x and y Z x. Since &(y) + 0 GO

iff .T � y , so 认y) = 0 and ^^(广工？/-〜)=0. Therefore 么(2/) = 0.

(3) U r - ' x - ^ r ) = = M的, t h a t is •工 is constant

on R工.It remains to evaluate 么(.T:). Since H{x) < G � ’ by proposition

2.2.3，we know that either p — 2 or p = 2 with N{xy = 0, that is, .T 兴 x.

(a) Suppose p 2.

M ^ ) = ^XCT) + =似工、or

Since, p and 2 are relatively prime, we see that

, 0 [mod p)

(b) Suppose p = 2 with x ^ x.

i. Suppose X % and x ~ rr—i.

姻 = 側 + U t - ' x - ' t ) = U ^ )本 0 (mod 2)

since x - rr-i and rr-i % x imply “(广丄：?:〜)=0.

ii. Suppose x and x ^

，働 = + Ur-'x-'r) = CrOr)丰 0 (mod 2)

since x ^ x implies 计 - i j r V ) = 0. qO

iii. Suppose x �

秘、 = U r ) + & V ) = & ⑷ 丰 0 [mod 2)

• since rr—i 年 x implies &(7•-ix—V) = 0.


Case 2: \H{x) : H{x)^\ = 2’ that is, x - x.

Define 0工 by

I 0 otherwise

Obviously, e^ = where J ]^ runs through the set of irreducible

characters of Cx. It follows that 6x ^ A 0 RR(Cx, e).

Let N�x�S is a Sylow-2-subgroup of N{xf such that H{xf ：= C：, x 7V(.T)§.

Thus, we have the quotient homomorphism T : H{xf — C工 defined by T(ab) = a

for any a e e N{x)l. Therefore, we have an element T^ := o T =

Y.^ oTeA0 RR{H{xf, e). .

For each x � 了 € RR{H{x)^,€), it is a complex representation of H{xf of

degree 1. Let W^ be a representation space of x. Since the dimension of W^ is

one, choose a basis {ei} for W^, we define a map

c x : 聊） X ly^ ly^

by

= Px{h)w and

cTxirh^w) = p^{h)w

where h G H{x)^ and w = w • ci e W^.

Since T is a projection from //(.T)0 onto C工 and r- '^xr = x - \ by direct

verification, this map is a group action. Obviously, its character is x. Therefore,

we have an element 如5：父乂(厂丄)�G A®RR{H,t). Moreover, its character

is equal to that of =疋.


Induce if 工 to i!；^ e RR{G, e) by the Real induction. Then, for all y e

we have

= 陣 ) o geGOs.t.g-lygeH(x)

= _ £ _ . B 聊 ) � l

where B := {g e e

(1) UteB, then tN{x)^2 C B. So is amutiple of |iV(:r)引.Thus, “ ( y ) g Z.

(2) If 2/ is a 2'-element of and g e G。，g-^yg is also a 2'-element of G � . If

e H�,then it belongs to C；，and we have (Mg—iyg) = 0 whenever

g - � g — X. It follows that = 0 if rr 笑 y.

(3)

= N { x f =

幸 0 (jnod 2)

Let's summarize what we have done in a proposition.

Proposition 2.3.11 Let x heap，-element in G � a n d H{x) he a p-hyperelementary

subgroup associated to x. Then there exists tp^ e A (^Vp such that

(1) My) e Z for all y G G�.

� My) = 0 for all p'-element y in G�such that y is not in R工.

(3) M工)=Mt-1 厂、),0 (jnod p).


In the theory of complex representations, we have the following result.

Lemma 2.3.12 Let x be an element of A® with integer values, let x e G^

and Xr be the p'—component of x. Then

三 X{xr) (mod p)

Proof: See [Se, Ch 10] •

We have a similar result in Real representations.

Lemma 2.3.13 Let xp he an element in A (g> RR(G, e) with integer values. Let

X e G^ and Xr be the p '-component of x. Then

功(:r) = xjj{xr) {mod p)

Proof: Since 妙 = E ^ i X i ^ A 0 with integer values, where a, G A and

Xi are distinct characters of G^. Since Xi{x)三 —d p) for all i, we have

ip{x)三 (m,od p). 口

Proposit ion 2.3.14 There exists an element ippe A (S)Vp such that

'^p{y) , 0 (mod p)

for all y G G�.

Proof: Let {xi)ia be a system of representatives of Real p ' -conjugacy classes

(those classes consisting of p'-elements) in By proposition 2.3.11，for any Xi,

there exists an element ipi e A (^Vp such that

论(巧）二 I odp) i f j = i

I =0 if j


Define • := E i e /钱 G A (g) V；. By proposition 2.3.11 again, we see that

values of 妙p are integer. For rr G G � , the p ' -component of x is conjugate to a

unique Xi, by lemma 2.3.13,

树:r)三 ip{xi) = ijji{xi)丰 0 (mod p).

•

We quote two propositions from [Se, Ch 9, 10]. Let x e G^ and C be a cyclic

group generated by x with order c. Define Q�on C by

(

a , � c if y generates C

y 0 otherwise

Proposit ion 2.3.15 If order of G^ is g^, then

c

where C runs through all the cyclic subgroups of G^.

Proof: Put e'c = in 趕(fie). For xeG，we have

物 = I E y^GO s.t. y-^xyeC

E c J/GGO s.t. y-^xy gen. C

= { y ^ generates C} *

Since, for each y G y-^xy generates a unique cyclic subgroup of G^. So, we

have:

I ] 物 = =夕0.

C y€GO

•


Proposition 2.3.16 Each class function on G^ with integer values divisible by

90, where go is the order ofG^, is an A- linear combination of characters induced

from characters of cyclic subgroups of G^.

Proof: Write f = gx, where x is a class function with integer values. Keep the

notation of proposition 2.3.15,

c

whence

f = goX = Y A . = E 饥 • res^\) c c •

It remains to show that BE . RES^\ ^ A® R{C) for each C. But the values of

Xc •= Oc . r e s g � x are divisible by the order of C, so if ^ is a character of C,

we have < x c A > € A, which shows that xc is an A—linear combination of

characters of C, whence x c ^ ^ 0 . •

Lemma 2.3.17 Let f be a complex-valued function on G � w h i c h is constant on

脑 I conjugacy class of (G’e). If all values o f f are integer divisible by g,

—ere g is the order ofG, then there exists an dement K e ^ (g) V；, such that its

characters is f .

Proof: Since g = 2 • go, where go is the order of G � ’ so | is a cla^s function on

G^ with integer values divisible by QQ. By lemma 2.3.16,

^cind^'xc c

where runs through all cyclic subgroups of G。，ac e A and xc G R{C).

Since f is constant on each Real conjugacy class of G � , so is f For each cyclic

subgroup C of suppose C is generated by x. Write x = x以.where

are p- and ；/—dements of :r in G�respectively. By lemma 2.2.4, there exists a


p-hyperelementary subgroup HE associated to XR such that C < HE. We have

2 different cases.

(1) If e is trivial on He, then we can induce xc to Oh�:= ind艺e RR{Hc, e)

by complex induction. By the Real induction, we have an element tpc €

RR{G, e) with character

M y ) = ind'^'jHciy) + ind'f^jHcir-^y-^T)

=in 趕 xc(y�+

for all y e G�.

� If e is surjective on He, then we can induce x c to � � ：二 Ind^^ o i n d g \ c G

RR(Hc,e) by complex induction and complexication map. By the Real

induction, we have an element 如 e RR{G, e) with character

M y ) = ind^Jniy)

= — � � C ( t - V I T )

for all y G G � .

Hence, we have ^ ：= T^c^cMv) ^ A^ RR{G,e). Since | is constant on

each Real conjugacy class of for all y G we have

， _ = Y^ac 吻 c(y) c

= a ^cind^\c{y) + Y acind!^\c{T-^y-^T) c c

={(") + { ( T � - V )

= { � Y � + { � Y � = M


•

Proof: (Theorem 2.3.5)

Let g = p^l be the order of G, with (p, I) = 1. Let ip be an element m A® V^

satisfies the conditions in lemma 2.3.7. Let N = be the order of the

group (Z/V^Z)*’ so that A"三 1�m,od p") for each integer A prime to p. Hence,

妙(.T)" = 1 {mod pn) for all x € Consider the function 一 1). By direct

verification, /(V^^ 一 1) is a complex-valued function, which is constant on each

Real conjugacy class of with integer values divisible by g. By lemma 2.3.17,

Ki^N - I ) e A® Vp. But A (g) Kp is an ideal of A 0 RR{G, e) and G ^ 0 Kp,

whence liP^ e A® Vp. Therefore, I e A® Vp. Since {A 0 Vp) n RRifi, e) = Vp

whence I G Vp. •

2.4 Monomial Real Representations

A complex representation K of G is said to be monomial if V is induced from a

complex representation with degree one of a subgroup of G.

Proposition 2.4.1 Every irreducible complex representation of a p-group is mono-

mial.

Proof: [Se, Ch 8] •

In the theory of orthogonal representations, we have a similar result.

Proposition 2.4.2 Let V be an irreducible orthogonal representation of a 2-

Oroup G. Then V is induced from a complex representation of subgroup ofG with

degree one or two.


Proof; [Lee, Lemma 4.5] •

The purpose of this section is to prove the following proposition.

Proposi t ion 2.4.3 Let G be a 2-group. IfV is an irreducible Real representation

of {G, e), then there exists {H, e) < {G, e) and WH G RR[H, e) with degree one or

two such that V = IND^Wh-

The proof of this proposition is based on the proof of the lemma 4.5 in [Lee .

In order to prove it, we have to quote some results in the theory of complex

representations. -

For any class functions / , p o f a group G, we define a complex-valued function

< • ’ . � b y

P€G

Lemma 2.4.4 (Frobenius) Let G be a group and H <G. Let x and fi he complex

characters of G and H respectively. Then

< X > G = < M, r e s g x >H

Proof: [Se, Ch 7] ^

Let V be an irreducible complex representations of G and res^V = ©？二。；*

be the canonical decomposition of res%V. Let l y be an irreducible complex

‘ representation of VQ with multiplicity m. Define

Gw ：= {g G G\g . Vo = Vo}

Obviously, H <Gw and V is a complex representation of Gw

Lemma 2.4.5 (Clifford) Keep the above notation. Suppose H is a normal sub-

group of G. 'Then


(1) resgy = \/o = rnW if Gw = G;

(2) V = ind%^Vo ifGw^G.

Proof: [Se, Ch 8] 口

Corollary 2.4.6 Keep the notation in lemma 2.4.5. Suppose H is a maximal

normal subgroup of a 2-group G. Then

(1) Tes^V = W if Gw = G;

(2) V = ind%W if Gw = H. •

Proof: Since 丑 is a maximal normal subgroup of a 2-group G, \G : H = 2.

Moreover, H < Gw < G, we have Gw = G or Gw = H.

If Gw — G, then Gw = H. Since V is irreducible, we have Vq = W and (2)

follows immediately.

Suppose Gw = G. By lemma 2.4.5, we have res%V = Vo = mW, where W is

an irreducible complex representation of H. Let xv and xw be characters of V

and W respectively. Obviously, r e s g x v = mxw- By lemma 2.4.4，we have

< Xv.ind^xw >G=< res^xv,Xw >H= m.

Therefore, Xv = m - ind^xw + for some 6 e R{G). However, V is irreducible,

so we have m = 1 and (9 三 0. Therefore, res^V = W. • *

Lemma 2.4.7 For any 2-group G, G has a maximial subgroup H which is cyclic

or G has an abelian normal subgroup A which is not cyclic.

Proof: [Go, p. 199] 口


Corollary 2.4.8 For any 2-group G, G has a maximal subgroup H which is

cyclic or G has a subgroup H of index < 2 with non-cyclic center.

Proof: Suppose G has an abelian normal subgroup A which is not cyclic.

Let A2 = (g e Alg"^ = e}. Obviously, A2 is a normal subgroup of G. Since

order of A > 4, we see that A2 is a vector space over Z2 of dimension > 1 . For

any two non-identity elements a, b in A2, we can produce a 2-dimensional vector

subspace V(a,b) :=< a > x < b > of 成.Thus, A2 has | ’ _ 1 � 2 , which is odd,

2-dimensional subspaces, where 2" is the order of A2. Let X be the collection of

all 2-dimensional subspaces of A2. Define a group action o ： G x X X by

9�V"(«，b) := g-^V(a, b)g =< g-�g > x < g � >

This action is defined since A2 is a normal subgroup of G. Note that

where Xg = {V eX\goV = V), GV, is the orbit of K under G and J： runs

through all orbits of this action with order > 2. Since the number of elements in

^ is odd, we see that there exists a normal subgroup, say < a > x < 6 >, of G.

Let H be the centralizer of < a > x < 6 > in G, that is,

H =�g & G\g~âg = a and g'^hg = h)

Then H has index < 2 in G since Aut{< a > x < 6 >) has Z2 as 2-Sylow

， subgroup. Obviously, < a > x < 6 > is in the center, Z[H), of H. 口

Proof:(Proposition 2.4.3) Let G be a 2-group of the smallest order for which

the proposition is false and V be an irreducible Real representation such that

^ — 腦 GHWH for ^\\H<G and G 卿 , E ) with degree one or two. Then

is a faithful Real representation since G is minimal. Let U be an irreducible

complex representation of resôV = ResôV.


Suppose V = Ind^oU. Since U is an irreducible complex representation of a 2-

group G®, by proposition 2.4.1, there exists a subgroup K and Wk, G RR{K, e),

with degree one such that U = ind^Wx. Thus, V = /nrfgo o ind^'WK =

IND%Wk. It contradicts the assumption.

Suppose V + That is, Res^oV is an irreducible complex representa-

tion and dim(i?esgo^) = dimV.

Let ^ be a maximal subgroup of G^. By cor 2.4.6, we have

(1) r e s g V = W or

(2) V = ind^'W.

where W is an irreducible complex representation of res^V.

(1) implies dim V = dim W. (2) implies dim 1/ = : . dim = 2. dim VK.

In any cases, dim V < 2 - dim W.

Suppose GO has a maximal subgroup H which is cyclic. Since every irreducible

complex representation of a cyclic group H is of degree one, we have 1

and dim V < 2 • dim l y = 2. So, we may regard V = IND^V.

Suppose GO has a subgroup H of index < 2 with non-cyclic center. Since cen-

ter of H is non-cyclic, so H hasn't a faithful irreducible complex representation.

However，K is a faithful Real representation of (C, e), so r e sgK is also a faithful

irreducible complex representation of H. 口

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