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Dépôt Institutionnel de l’Université libre de Bruxelles / Université libre de Bruxelles Institutional Repository

Thèse de doctorat/ PhD Thesis Citation APA:

Sharif, A. H. (1970). Representations of finite groups (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/214952/1/5f5ff32b-4890-4722-ae04-90faa5dbd6cb.txt

(English version below)

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REPRESENTATIONS OF FINITE GROUPS

Director of the Thesis; Professor F.6IN6EN

This Thesis is presented to obtain Doctorale

in Science in the group of pure Mathematics

Anwarul Haque SHARIF

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ET DE PHYSIQUE

THE FREE UNIVERSITY OF BRUSSELS

Faculty ol Science

REPRESENTATIONS OF FINITE GROUPS

51^.‘JH

Director of the Thesis; Professor F.BINGEN

Tliis Thesis is presented to obtain Doctorale

in Science in the group of pure Mathejnatics

Anwarul Haque SHARIF

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I

AcJcnowledgemcnts .

At the outset I like to thank Professer Franz Bingen who has very kindly allowed me to work under him. It would be- come quite impossible to préparé this thesis without bis kind help^ He also helped me financially^

I shall like to thank Mrs, F. Bingen for her générons help in kind for my family at Brussels. Finally^ let me thank the Ministry of National Education and Culture of Belgian Govt. for granting me a fellowship during my stay in

Belgium ^

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INTRODUCTION.

In this Work , we shall consider the follovjing things .

Let G = Cp X Cp be a group where p is prime. Let V be a represen tation space, over a field F of characteristic p, for this group Assume that V is a finite dimensional vector space over a field F of characteristic p.

Let us take A • V ■ ^ V where A^ = I in F and find a matrix P

B such that AB = BA and also B^ = I in F . Let us dénoté the p

generatros of the group Cp x Cp by a and b^ Then the defining relations of the group hâve the form a^ = b^ = e and ab = ba. Since AB = BA and A^ = B^ = I in F , the correspondance a ---7 A

P

b --- B , may be extended to a homomorphism of the group G into the group of non-singular matrices over a field F of characteris tic p,

Consequently the matrices A and B détermine the représentation o the group G = Cp x Cp , that is, the représentation for the group Cp X Cp will be given by the vectors of the forms («<.0) corres- ponding to A and (0^|5) corresponding to B .

The following notations will be used throughout the whole work : = Ker(A-I),

W. = (A-I)V

A J

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III ^

In the first chapter, we hâve collected some preliminary pro- perties of représentation spaces and linear algebra.

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CONTENTS

Cha2ter_I.

Characteristic and Functorial subspaces p. 1

§ 1. Characteristic subspaces p, 1

§ 2. Functorial subspaces p. 4

Appendix to Chapter I p. 8

Çhapter_II.

Study of a class of modular représentations of the

group Cp X Cp P . 14

§ 1. Introduction p 14

§ 2, Définitions of some subspaces of the représentation

space p, 15

J 3^ Further analysis of the représentation space p. 22 Çhapter_III^

Standard form of the matrix (B-I) p, 29

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1.

CHAPTER I.

CHARACTERISTIC AND FUNCTORIAL SUBSPACES.

§ 1. Characteristic subspaces. Définition :

A subspace will be called characteristic if and only if it is in­ variant by ail automorphisms of the représentation space.

Proposition 1 .

Every characteristic subspace is invariant. Proof :

Let V be a représentation space and C3< be an automorphism of V. Let W be a characteristic subspace of V.

Take x € W then o((x) € W. Hence «<(W) C W C V.

Now , since G is commutative and A and B are automorphisms of the représentation space then A(W) C VJ and B (VJ) C VJ. Hence W is an invariant subspace.

Proposition 2 :

If W is a characteristic subspace in V then (A-I)W, (B-I)IJ^ ker^(A-I) and ker^^(B-I) are characteristic subspaces in V.

Proof :

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This implies o<(x) = o((A-I)y = (A-I)o(.(y) é. (A-I)VJ ^ [o<A = A «< and «K(A-D = (A-I)o(^ , so , X 6 (A-I)W implies «<(x) €. (A-I)W.

Again, o< is also an automorphism of V.

So, when o((x) (A-I)W then oC^(<3^(x)) = x (A-I)W.

Hence c< is an automorphism of (A-I)W and (A-I)W is a character.- istic subspace in V.

Now, the analogous proof gives that (B-I)W is also a characteristic subspace in V.

2) Take x é. ker^^(A-I) then (A-I)(x) = 0 and

(A-I)o<(x) =o^(A-I)(x) = 0 implies o<(x) €. ker(A-I). Since x £ W theno(,(x) £ VJ.

Hence o^(x) <=- ker,,(A-I) ,

VI

Again , o< is also an automorphism of V.

So, when o<(x) £ ker^^(A-I) then oC ^(c<(x)) = x €. ker^(A-I). Hence c< is an automorphism of ker,^(A-I) and ker^^^(A-I) is a characteristic subspace in V.

Now ^ the analogous proof of (2) gives that ker,.^(B-I) is a characteristic subspace in V.

Proposition 3

The sum and intersection of a family of characteristic subspaces are characteristic subspaces.

Proof :

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3.

Take x e 2L then x = + X2 + ... +

andoC(x) = oC(x^) + ciC(x2) + ... + ot^(Xj^) £, + V!^ ... This implies <^(x) C ^ W. , • X 1 so , X € ^ W. implies o(,(x) € VJ.. ^ • X «X X X

Again , o<^ is also an automorphism of V.

So , when <?((x) €. ^L then oC^(°^(x)) = x è ^ . Hence o( is an automorphism of ^ W. and 2_ VJ. is a

• X «X

X X

istic subspace in V,

+ VJ,

character

2) Take y €

n

VJ ^ then y €: j V ^ ^2 , . . * , Y £ VJj^ and

O

o< (y) € VJ^ , o<(y) é. VJ2 , .. . , =<(y) ^ VJj^ which implies o(.(y) ^ so, y £■ 1^) VJ. implies c<(y) £ O VJ..

Again ^ is also an automorphism of V.

So, when o<(y) £ then o(^(o((y)) = y £ O .

VJ

Hence is an automorphism of istic subspace in V.

O

W.^ and VJ^ is a

character-Proposition 4 .

If VJ’ and VJ" are two characteristic subspaces in V and

= -^x' £ VJ* I (B-I) x’ C W"^ then VJ^ is a characteristic sub­ space in V.

Proof :

Let «<. be an automorphism of V.

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_1

Again^ «< is also an automophism of V. _ i

So ^ when e<(y) <£- VI^ then o<. WCy)) = y .

Hence o( is an automophism of and is a characteristic sub- space in V.

§ 2. Functorial subspaces. Définition .

Let M be the category of représentation spaces of the A subspace functor F of M is a covariant functor of M such that for ail V H ; F(V) is a subsnace of V and

group G. into itself

F(V) where ; V --- » VJ

; F(V) ---> F(W) ,

F(V) is then called the functorial subspace of V corresponding to F. Définition •

A subspace v/ill be called fully invariant if and only if it is in­ variant by ail endomorphisms of the représentation space.

And every fully invariant subspace is characteristic.

Proposition 5 ;

Every functorial subspace is fully invariant. Proof :

Let F be a subspace functor and be an automorphism of V. Let Vf: V --- ? V then : F(V) ---> F(V) .

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5

Définition .

If S and T are subspace functors then define (S+THV) = S(V) + T(V) , (S+T)(f) = f|s(V)+T(V)’ and (S HtHV) = S(V) A T(V), (S n T)(f) = f S(V) n T(V) Proposition 6 •.

(S+T) and (S H T) are subspace functors^ Prgof .

We hâve that

S(V) C V and T(V), so, S(V) + T(V) C V.

Let us take an identity map V ——> V then S(e) = Id S(V) and (S+T)(e) = Id T(e) = Id , T(V) S(V)+T(V)•

Again, let f and g be two morphisms of M then, S(f) = f|s(v) ’ = f and (S+T)(f) = f T(V) ,S(V)+T(V) ^ S(f O g) = f O 5|s(v) > T(f O g) = f O c|t(v)» and (S+T)(f o g) = f o g = f S(V)+T(V) S(V)+T(V) ° - S(V)+T(V) Hence (S+T) is a subspace functor.

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Définition •

If S and T are subspacc functors then S o T is defined by S O T(V) = S(T(V)) and S O T(f) = fI 3 Q T(V) S(T(V))• Proposition 7 : (S O T) is a subspace functor. Proof : We hâve that S(T(V)) C T(V) C V.

Let us take an identity map V ——> V then S O T(e) = J.

Again ^ let f and g be two 'Jnorphisms of M then, S O T(f) = f|‘s(T(V))

S O T(f O g) = f O cjg ^ = f O r,js(T(V))' Hence (S o T) is a subspace functor.

Définition ;

Let R, S and T are subspace functors. Assume T satisfies the con­ dition T(Z^X^) = ^T(X^) where U(V) is defined by largest subspace X of R(V) such that T(X) C S(V).

Proposition 8 :

U is a subspace functor. Proof :

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7

Now^ U(V) =2Lt(W) where VI is such that T(W) CZ R(V).

This sum is not empty because TCO) = 0, whence U(V) is well defined. Id

Now, let us take an identity map V- for ail V € M.

Let f and g be two inorphisms of M then

= ^lu(V)

UCf O g) = f O gju(v)- Hence U is a subspace functor. Définition <

For every représentation space, let us put oC(V) = (A-I)V and ' (f) =

Proposition 9 : o( is a subspace functor. V then U(e) = Id (A-I)V U(V) Proof : We hâve that^ o< (V) c V.

Let us take an identity map V---^ V then o<(e) = Again, let f and g be two morphisma of M then

= ^j(A-I)V O g) = f O gj(A_i)v

Hence is a subspace functor where cC will usually be denoted by (A-I).

Remark :

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APPENDIX TO CHAPTER I. 1) Let f : V --- >• W is injective and V = V* {T) V"

then f(V) = f(V*) ® f(V”). Proof :

One has that f(V) = f(V») + f(V").

Take x £ f(V*) H f(V") then there exista a unique y € V such that X = f(y) and y €. V ’ and also y €. V".

This implies y € V H V" =

|oj

and x = f(y) = 0. Hence f(V*) A f(V") =

|o)

and f(V) = f(V») (+) fCV’).

2) Let V = l J {+) S and e'. = e. + s. with e. U s. € S and ^ W

111 l'i 1 then V = W* @ S. Proof : Let W = ^ • • • . ®-(-j S = ^ * * * > ^n^ W H S = ^0^ and W + S = V. Hence V = V/' + S.

Again ^ let x £ W* A S v;ith x = x* + x'' where x’ £ W and x" £ S. then VJ' + S C V.

Take y t V with y = y* + y" where y' € VJ and y” £ S

Since x £ S and x" £ S then

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9.

3) Let f : V--- J-X and g : V --- »X where both f and g are injective and f(V) + g(V) = X. Take f(V) O g(V) = X^ with f^(X^) = and

g ^(X^) = Z^. Put n and assume that f(V^) = g(V^) then one has the décomposition V = 0 <+) ® such that the block représentations of f and g become

0 0 °\ ’ g = /J 0 0 ' 0 I 0 ( 0 0 I

M

0 0 I 0 0 0 0 0 0 0 0 I 0 0 1 0 0 0 I ! 0 0 0 0

\

0 0 0 0

/

0 0 0 1

for suitable choice of the bases of the subspaces (J is the Jordan form of f g).

Proof :

One has that C Y^.

So, one can define a supplément of in Y^ such that Y^ = 0 S^.

Now_ f(Y^) = f(V^) 0 f(S^) and g(Z^) = f(Y^) = X^ where f is injective.

Let = |x eZ^ g(x) e f(S^)| = g"^f(S^). Now, g(Z^) = f(V^) 0 f(S^)

= g(V^) 0 g(T^).

This implies dim g(Z^) = dim g(V^) + dim g(T^). Again , C Z^ and C Z^.

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Hence 0 .

Now . Vj = ^0} , n Tj = ^oj. Take x i H then x C which implies x 6 and x 6 A ^ Hence ^

Again, C V and define as a supplément of (V^ + Sj + T^) in V then V = 0 0 0 U^. But X = f(V) + g(V) = [f(V^) 0 f(Si> 0 f(Tj) 0 f(U^)] + jg(V^) 0 g(S^) 0 g(T^) = f(Vj) 0 f(S^) 0 f(Tj) © f(Uj) 0 g(S^) ® g(U^) where g(V^) = f(V^) and g(Tj) = f(S^).

Now^ let us suitably choose the bases for the subspaces in V such that the matrices for f and g become

f=/l 0 0 o\ 0 10 0 and g = 0 0 10 0 0 0 0 0 0 0 1 0 0 0 0, 0 0 0

\

0 0 10 0 0 0 0 0 10 0 0 0 0 0 0 0 0 1

J

-1

where J is the Jordan form of f g. ^eneral case.

If g(Vj)

A

f(Vj^) then let us say f(V^)

f)

g(V^) = with

f (X^) = Y2 and g ■*'(X2) = Z2. Put ^2 ~ ^2 ^ ^2 assume that

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f = / I 0 ' 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

\

0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ^ J 0 0 0 0 / 0 0 I 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 \ 0 0 0 0 0

lo

0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0

for suitable choice of the bases of the subspaces (J is the Jordan form of f g).

Proof :

The analogous proof will give that = V2 ® S2 ® T2 ® U2. Let f f V and g 1 V, = ^ then X X.

Now let us extend this to V then V = V2 (î) S2 e T2 ® U2 ®

f(V^) = fCVj) © f(S2) © f(T2> © f(U2>, g(V^) = g(V2> © g(S2> 0 g(T2) © g(U2> , where g iV= f(V2>

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Now, f(V^) + g(V^) = f(V2> 0 f(S2) 0 f(T2> 0 0 g(S2> 0 g(U2)CX^. Again, f(Y^) = f(V^) 0 f(S^) = f(V2> © f(S2) © f(T2> 0 f(U2> © f(S^) = and g(Z^) = g(V^) 0 g(T^) = g(V2) 0g(S2> © g(T2) © g(U2> © g(T^) = f(V2) 0 g(S2> 0 f(S2) © g(U2> 0 f(S^) = X^. Hence X = f(V) + g(V) = f(V2> 0 f(S2> 0 f(T2> © f(U2> ©g(S2) © g(U2) 0 f(S^) 0 f(T^) 0 g(S^) © f(U^) © g(U^) .

Now ^ let us choose the bases for the subspaces in V such that the matrices for f and g become

f = 0 0 0 0 0 0 1 0 0 0 0 0 and g = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Now, by induction one bas that

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13. and V = ® (S^ ® ... (±) 0 (T^ © ... (3) ® (U^ 0 . . . © U^+i> where g(V_^.j) = f(V^.j) and g(T^) = f(S^) (i = 1 , ... n+1). Then X = f(V^^^) 0 [f(S^) © ... +

[f(T^) 0 . .. ® f(Tn+i>^

1

® !

[fCUj)® ...©f(U„.j)] [g(S^) Q) ... ® g(s^^i)]

1

[g(u^)0 ... ® g(U^+i>l

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CHAPTER II.

STUDY OF A CLASS OF MODULAR REPRESENTATIONS

OF THE GROUP C x C . __________________ 2_____ 2

§ 1. Introduction.

Let V be a représentation space of G = x over a field F of characteristic P. A and B will be the représentative maps of the two standard generators of G. Since A^ = = I 1 is the only eigen value of A and B and (A-1)^ = (B-D^ = 0. If A-I = B-I = 0. the représentation of G is the constant représentation. In what follows we shall study the next easiest case namely that of those of représentations of G for which (A-I)

G = C2 X C2 , this covers ail cases.

(B-I)^ = 0. VJhen

Since (A-I) =0, there is base of V such that A = / In 0 0 \

m

m

m /

and D Denoting by V2 , one has V =

Decomposing B in blocks according to this subdivision of V and taking into account that AB = BA , one gets that

B = /b^ 0

"3^

B4 B2 B,

0

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15. We shall define some subspaces in V in order to put the matrix B in

some simpler standard form. We shall define the subspaces such that = L0M@N, = P0Q(î)RandV3 = X©Y0Z.

5 2 ^ Définitions of some subspaces of the représentation space .

2.1. Proposition 1 : (B-I)W^ = (A-I)Wg. Proof •

Take x €. VI^ then x = (A-I)y , for ail y € V. This implies (B-I)x = (B-D(A-I)y

= (A-D(B-I)y

€ (A-I)Wg.

Hence (B-I)W^ C. (A-I)Wg.

For symmetry reasons we hâve that (A-I)Wg C (B-I)W^.

Hence (B-I)W^ = (A-I)Wg.

Let us define Q as (B-I)W^ = (A-I)Wg.

2.2. One has (B-I)VJ^C Define P as a supplément of Q in

VIO Vg. Then ^ ~ Q ® P- This last space is a functorial subspace in V by Prop, 6 of Chap. I. Now, together vjith a supplé­ ment R that will be specified later on, one gets that

= P0 Q © R.

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Then (B-I)V^ ^ ^ 0 Q (±) H Ç: (B-I)V^.

Mow^ Wg is a functorial subspace in V by Prop. 6 of Chap^ I and ^ Vg + Wg 0 is also a functorial subspace in V by Prop _ 6 of Chap. I.

One gets (B-I)V^ + ^ Vg C Wg ^ O Vg ^

Define as a supplément of (B-I)V^ + W^ D Vg in Wg H + W^ ^ Vg. Then Wg n O Vg = [cB-I)V^ * Cl Vg] ®

= P @ Q © M <±) O. where (B-I)L = 0 = (A-I)L .

a a

Again . ^ is a functorial subspace in V by Prop. 6 of Chap. I. One has Wg ^ + W^ ^ Vg C fl Vg.

Define as a supplément of Wg ^ "*■ ^ '^A ^ • Then ^ Vg = [Wg ^ D Vg] ® Lg

= P®Q0M©L^©Lg where (B-I)L. = 0 = (A-I)L, .

b b

2 3. Let us construct ^ | ^B-I) x £ P + qJ.

One has W^ + V^ D Vg C . Define as a supplément of ‘■'^A AAB ^ ^'*1 where L = L 0 L, (±) L .1 a bc

Then = (W^ + ^ Vg) 0 = P @ 0 © R 0 M ® 0

= P ® Q © R © M© L where (A-I)M^ = 0.

Let us define P = (B-I)L vjhere P = P © P, and P, is a

a c a ^ b b

supplément of P^ in P.

Mow, (B-I)M. = (B-I)W. + (B-I)L = Q + P = Q©P .

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17

Proposition 2 :

Proof :

We hâve defined that P = (B-I)L .

3. C

This implies dim P = dim(B-I)L .

3 C

Since V„ H L = îol then B-I : L --- î

B c t. J c

Hence (B-I) gives an isomorphism of

P is injective . a

with P . a 2.4, We hâve that C V^.

Hence (B-I)W^C (B-I)V^ implies Q C(B-I)V^.

Again M C. (B-I)V, and P ^(B-I)V. by définitions.

* f\ 3. A

Hence P + Q + M C (B-I)V..

a A

Take y €. (B-I)V^c: P + Q + M then

y = 7]^ + y2 + 73 with y^ C. P^ y^ € Q and y^ M. Mow, y - y2 - 73 = 7i £ (B-I)V^ P = P^. Hence

y = y^ +

72

+

73

^

+ Q + î'î

(B-I)V. = P + Q + M = P ©Q0M. A3 3 2.5. Proposition 3 :

Let N' be any supplément of in then (B-I) gives an isomorphism of N* with M,

Proof :

One has N' C V^.

This implies (B-I)H' C (B-I)V. = P + Q + M. A 3

Now. (B-I)n*. = p. + q- + m. vjith p. Ê P , q. Q m. £

’ 1^1 1 '^1 a’ ^1^-1 M

N n ' > n

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So there existe 1. ^ L and f. ^ VJ. such that p. = (B-I)l. and

) 1 c 1 A ^1. 1

q. = (B-I)f..

Hence (B-I)(n'. - !• - f.) = m. ^ M. X 1 1 1

Let us put n. = n*. -1. - f. and take N = \ n. n \ which

1

1

1

1

L 1 } • * • J gj

implies that (B-I)N C M where 0 N by (2) of appendix to Chap. I and = L®M®îK+)P0Q0R.

Now, (A-I)n. = (A-I)n'. - (A-I)l. - (A-I)f. = 0.

Again ^ (B-I)V^ = (+) (B-I)N where (B-I) is injective. This implies dim(B-I)V^ = dim(B-I)M^ + dim(B-I)N and

dim(P^0QGM) - dim(P, 0 Q) = dim(B-I)W.

O. a

Hence dim(B-I)N = dim. M.

Since (B-I)N CM then (R-I).M = M.

Since N = -^0^ then B-I : M --- 7 M is injective. Hence (B-I) gives an isomorphism of N with M.

J 6. Proposition 4 ;

Let Y be any supplément of VJ^ 0 in VJg ‘then (A-I) gives ap iso- morphism of Y with Q.

Proof :

Vie hâve that Y C Wg.

Hence (A-I)Y C (A-I)Wg = Q. Again = Wg

Hence (A-I)VJg = (A-I)Y which implies Q = (A-I)Y and dim Q = dim (A-I)Y.

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19.

2.7. Proposition 5 ;

Let H = C Vg

j

(A-I) X €. P j- D ^ ^3* Define as a supplé­ ment of n Vg in H. Then 1) [(V^n Vb * Wb>] =<0 = H-2) Vb = n Vb * Wg. Proof : 1) X C V„ and (A-I)X C P. C D C

But (A-I)(V^ ^ = Q and P 0 Q = |^oj. Hence (A-I)(V^ ^ '^b ^ (''a ^''b " ''b» >'o‘^ ''a ^''b -But X^ is a supplément of ^ ^3 H. Hence X ^ (V. H + W„) = îo c A B B I 1 2) One bas X + V. H V„ + W„ C V„ c A B B B •

Take y €. Vg then (A-I)y €. (A-I)VgC H Vg = P ® Q .

Hence (A-I)y ~ with (£ P and ^ Q. But Z2 ^ Q=(A-I)Wg. So ,there exists y2 ^ ^'^3 such that (A-I)y2 = 22*

Hence (A-I)(y-y2) = z^ £ P.

Again . there exists y’^ €. h such that (A-I)y'^ = z^ P^ Now, (A-I) (y-y2~y ’ ^) = 0 v/hich implies y'^ = y - y 2 ~ y\^ and y = y2+y’3 + y’i with y 2 ^ Wg , y ' ^ ^ Vg and y'^ £ H. Let us take y*^ = y^ + y’’^ with y^ £ X^, y’’^ ^ ^ ^3

y3 = y*3 + y"3 ۥ H Vg.

Hence y = y^ + y2 + yg and ^3 = X^ + Vg + Wg.

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2,8, Proposition 6 :

Let é. V

j

(B-I)z C Y ^ Vg, Let Z be any supplément of Vg in , Then

1) B-I : Z = Y.

2) (A-I)Z is a supplément of (P + Q) in W^. Proof :

1) One has that

Z C H. and (B-I)Z C (B-I)H. C Y

1 1

Açain , take y Y then y = (B-I)x, for some x €. V . This implies x é. and (B-I)x ^ (B-I)H^.

So ^ Y C (B-I)H^. Hence Y = (B-I)H^. Again^ = V^(+) Z.

Hence = (B-I)Z which implies Y = (B-I)Z and dim Y = dim (B-I)Z.

Since Vg A Z = then B-I : Z ---^ Y is injective. Hence (B-I) gives an isomorphism of Z with Y.

2) One has that (A-I)Z C VJ^.

Take x € (A-I)Z H (P + Q) then x £ P + Q C Vg which implies X £ Vg and (B-I)x = 0.

Again, x € (A-I)Z then x = (A-I)y, for ail y £. Z. This implies (B-I)x = (B-I)(A~I)y

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On the other hand

9 ^Z J. •

(B-D(A-I)Z = (A-D(B-I)Z = (A-I)Y = Q = (B-I)W^,

So for every x C . there exists y in (A-I)Z with (3-1) (x) = (B-I) (y Then one has that x - y <£. = P + Q and x = y + v with

V € P + Q.

This implies that = P ♦ Q * (A-I)Z.

As conclusion one gets = P (*) Q (?) (A-I ) Z.

2.9. We hâve that B-I : Z = Y and (A-I)Y = Q.

This implies dim Z = dim Y = dim Q and dim(A-I)Z ^ dim Q and dim Z dim(A-I)Z. Hence dim Z = dim(A-I)Z and Z H = ^o|.

Proposition 7 :

Let us define R = (A-I)Z then 1) A-I : Z = R.

2) B-I : R = Q. Proof :

1) We hâve defined that R = (A-I)Z which implies dim R = dim(A-I)Z. But Z - |o|. Indeed (B-I) (Z n V^) C Y O and (B-I) restricted to Z is injective.

So, A-I ; Z --- » R is injective.

Hence (A-I) gives an isomorphism of Z with R. 2) We hâve from 1 that R = (A-I)Z. This implies

(B-I)R = (B-D(A-I)Z = (A-D(B-I)Z = (A-I)Y = Q and dim(B-I)R= dim P Since Vg D R = j o|then B-I : R --- ^ Q is injective.

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3. Further analysis of the représentation space.

3,1^ Let us form + (B-I)V^ = P 0 Q @ R <+) M vjhich is a functorial subspace in V by Prop. 6 of Chap. I.

ment of (V. + V^) in U vjhere X, = X 0 X’ .

AB b c d

Then U = (V^ + V^) 0 X’^. Nov7 (A-I)X' CW-.

3 d A

This implies (A-I)z = y with z X’^ and y é, and (B-D(A-I)z = (A-D(B-I)z = 0 where (B-I)z Ê V^. Hence (A-I)X' C .

d B

Now, (A-I)X' C W. n = P + Q and (A-I)e'. = v. + w. with

» dAD 111

e P, £ Q and X*^ = , e'^].

But we bave defined that (A-I)Wg = Q. So, thera exists f. € W„ such that (A-I)f. = w. £ Q.

Hence (A-I)(e'^ - f^) = € P.

Let us put e". = e’. - f. and take X", = -(g",, ... e" 1 v;hich implies (A-I)X”^ C P.

Now, (B-I)e"^ = (B-I)e'^ - (B-I)f^. = (B-I)e'^£

Hence e”^ £ U and U = (V^ + Vg) © X''^ by (2) of appendix to Chap. I.

Again, (B-I)X"^ C and (B-I)X"^ ^ ^''^b • Hence (B-I)X"^ C C\ Wg = P + Q + Il

But V76 hâve found that P = (B-I)L , M = (3-i)M and Q = (3-I)U.

a c ’ ^ A

Let U |x € V I (B-I)x £ O + Vg. Define X'^ as a

supple-= P + P, + Q + M. a b

Now (B-I)e". = X. + x„ + x_ + X,, vyith x^ £ P , x„ £- P, , x- C Q

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23.

So . there exists and f^ 6. M such that (D-I)f^ = ^ (B-Dfg = Xg and (B-I)f^ = x^^.

Hence (B-I)(e"^ - - f^^) = ^2 ^

Let us put = e'^ - and take ^ • • • ^ which implies (B-I)X^<1 P^.

Now, (A-I)e. = (A-I)e". - (A-Df. - (A-I)f, - (A-I)f,^ = (A-I)e"^ Ç. P which implies (A-I)X^ C P.

Again^ (B-I)e^ = (B-I)e"^ - (B-I)f^ - (B-Dfg - ( B-I ) f • This implies €; U and U = © X^ by (2) of appendix to Chap,

Remark ? (A-DU and (B-I)U will be found in the Chap. III.

3.2. Let ~ ^ ^ I ^ ^ Befine X’^ as a supplément of U in U., where X = X’ 0 ^1 a ^ bk*

Then U. = U 0 X’ . J. 3. Now (A-I)X* O W..

y 3 A

This implies (A-I)z = y with z X' and y €- W. and (B-D(A-I)z = (A-D(B-I)z = 0 i^here (B-I)z e V^. Hence (A-I)X' C v„.

3 D

Mow (A-I)X' ^ VJ. = P + Q and

> a B A

(A-I)e'. = V. + w. vjith v. ^ P w. è Q and X* = \e*' ... e '.

X X X X *x ^ a(^lj »t

But we hâve defined that (A-I)VJg = Q.

So there exists f. G W„ i X B such that (A-I)f. = w.,XX Hence (A-I)(e'^ - f^) = ^P.

Let us put e". = e*. - f. and take X" = ie''> ... e ". implies (A-DX” C P.

3

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Now, (B-I)e"^ = (B-De'^ - (B-I)f^ = (B-I)e’^é . This implies e" . € U., and = U 0 X" by (2) of appendix to Chap. I

X ^ JL â.

Apain (B-I)X" C M. <r V. and (B-I)X" c:: W„

» a 1 A a B •

Hence (B-DX" C M. H d L + M + P + Q = L + M + P + P^ + Q.

a 1 D a b

Now J (B-De"^ = + X2 + Xg + + Xg with x^ <=- L _ X2 ^ M, X3 Ê P^, X, £P^, XjÊ Q and X"^ = ....

But we hâve already found that (B-I)L = P , (B-I)N = M and C â.

(B-I)VJ. = Q. So there exists f» N, f~(=- L and è W» such

A - Z’3c 5A

that (B-I)f2 = X2 , = X3 and (B-Df^ = x^. Hence (B-I)(e". - f„ - f_ - f,) = x., + x,, L + P, I2dbl4 b'

Let us put e. = e". - - f., - fr and take X = W- , e.^V

' 1 1 2 3 5 a(^l.;***’t)

which implies (B-I)X C L + P, .a b

Now (A-I)e. = (A-I)e". = (A-I)f„ - (A-I)f„ - (A-I)f.

>1 1 2 3 b

= (A-I)e"^.

Hence (A-I)X C P and (B-I)X C L + P^

a a b *

Again ^ (B-I)e^ = (B-I)e"^ - (B-I)f2 - CB-Df^ - (B-Df^ e which implies e. é. U., and U. = U © X by (2) of appendix to

X X J. a Chan I, * • • 3 3. Proposition 8 ; 1)

(B-i)x^

a

p^ = (o).

2) (B-I)X^ n fcB-DV^ + “a ^ ''b] = i'’}-Proof : 1) Take y é. (B-I)X 0 p, So a b • On the other hand U = \z ^ V

y = (B-I)x with x X . a

|(B-I)z é M2land X^ is a supplément 1*

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25.

But (B-I)x £. ^ ^ ^2 ' implies that x ^ U and X é U n X^ = ^oj-. Hence y = 0 and (B-I)X^ “ 1®}'

2) (B-I)X^ O jj:3-I)V^^ + VJ^ n Vg] C (L + P^) r\ J^B-DV^ + OV C P, O P, n (B-I)X^

b b a

3 4 . VJe know that L C_ VJ„ A V. .

• a B A

Hence for given <c Wg H V^, v?e hâve that y^ = (B-Dz^ £■ and (A-I)y^ = 0.

Now Z. £ U. where U. = U 0 X' = (V. + V„)(±)X' ®X» .

jol X aAB aci

Proposition 9 ; V = 0 Z.

Proof .

Take x ê H Z then (B-I)x £ Y and (B-I)x «£ ^a*

This implies (B-I)x £ Y O = ^0^

Again x £ Z and x £ Vg H z = ^0^ (Z = supplément of Vg in H^) . Hence x = 0 and H Z = ^0^*

Again. take z <£ V then (B-I)z £ W„ C L + P + Q + M + Y.

7 D a

This implies (B-I)z = y^ + Y2 + ^3 * ^4 * ^5 * = (B-I)z.1 ; Z. £ R C U1 5 II C M > ) (B-I)z2, Z. e L e U 2 c ; C O II (B-DZg , z~ £ N ^ U >7 » II (B-I)z^, II L O > ^ (B-Dzr 5 ? Z5 eu^, C T ) 11 (B-I)z-6 J Zg £ Z.

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Hence = Now 2 = Z Hence V = 0 Z Proposition 10 : A-I : X = P Proof : We hâve that

(A-I)X^ C P^ (A-I)X^ C P and (A-I)X^CP which impliee (A-I)X CP where X = X ® X <ï) X ,

a c d*

We also know that = U (+) X^ which implies (A-I)U, = (A-DU + (A-I)X C

P

+

Q.

X â.

Again ^ ^ ^1 ^ ~ ~ ^^A

implies dim + dim Z - dim = dim P + dim Q + dim R and dim - dim = dim P + dim Q.

Hence (A-I)U^ = P + Q.

Now^ + X + Y and (A-I)U^ = (A-I)X + (A-I)Y v;here (A-I)X C P and (A-I)Y = Q and P A Q = ^o|.

This implies (A-I)U^ = (A-I)X 0 (A-I)Y and dim (A-I)U^ = dim (A-I)X + dim (A-I)Y.

But dim (A-I)Y = dim Q and so dim (A-I)X = dim P. Since (A-I)XC P then (A-I)X = P.

Now. (A-I)V = and (A-I)Z = R when and dim Z=dim R.

then A-I : X ---^ P is injective Hence (A-I) gives an isomorphism of X with P

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27.

Proposition 11 ;

Let us define L’ = (B-I)X Then

a â. 1) L* is a supplément of (B-I)V. + W. O V„ in 17. H V„ + ^ V. a O Pi D li f\ 2) B-I : X = L’ . a a Proof :

1) We know that (B-I)X^ n ^ V = © 'J. Take t € ^ then t = (B-I)s = (B-I)u + (B-I)z where

s = U + Z with s è V U 6. U- and z Z. ' 1

But (B-I)utM^ € and (B-I)z <Z- Y. Again, t é. and t-(B-I)u which implies (B-I)z € where (B-I)z ^

Hence (B-I)z é. ^ Y = -[ol and Z = 0.

Nov7 t = (B-I)u with U (£. U. = V. + V„ + X + X, and

’ 1 A 15 a d

t G. (B-I)X^ + (B-I)V^ + (B-I)X^ = (B-I)X^ + (B-I)V^ + 17^ A , So for every t € VJ„ ^ V. + W. H v„ , there exists s. €. (B-I)X

da/id’ 1 a*

s2 G (B-I)V^ and Sg d VJ^ ^ Vg such that t = s^ + S2 + Sg. Hence + Wg = (3-I)X^ + j^(B-I)V^ + 17^ fl Vg]

= (B-I)X^0 jjB-I)V^ + V7^ = L'^e {(B-I)V^ * '"a

2) We hâve defined that L* = (B-I)X vjhich implies

â â.

dim L' = dim (B-I)X . 3. Si

Since V„ 0 X = j of then B-I : X —^ L’ is injective.

U a (. J a a

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5. Now the block représentations for A and B become ( A=/ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 o' I 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 0 0 'o 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 LO 0 0 0 0 0 0 I t B = I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I I 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 * 0 0 0 0 0 0 0 0 0 I I 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I I 0 0 0 0 0 0 0 0 0 0 0 0 I

;

Here will be shovm explicitly in the matrix later on Again we know that if two vector spaces V’ and V" hâve the

respective finite dimensions n' and n", thon the rank r of any linear map t : V* --- ) V" is at most the smaller of n’ and n". For each such map t there is a basis b' of V' and a basis b''

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29

CHAPTER III.

STANDARD FORM OF THE MATRIX B-I

1. Introduction . One has that

Q C (A-I)Vg n (B-I)V^ (A-I)Vg O Wg C (A-I)Vg and Q C (A-I)Vg r> (B-I)V^ C (B-I)V^ n C (B-I)V^. 2. Proposition 1 ; Let us de fine ^ 1) P,CP^. 2) (A-I)Vg n P^ = P^. 3) (A-I)Vj^ n (B-I)V^ = Q0p^. Proof :

1) This is clear from the définition.

2) We hâve defined that P = (B-I)L which inplies P = (B-I)L O P But M Q PC and M = (B-I)N Q = (B-I)R v/here M O P =

J ’ B i

Again ^ (A-I)Vg fUCB-DV^r» P) = ((A-DV^O (B-I)V^) H P = ((A-DVgfï (B-i)v^) n P^ = P^.

Hence (A-I)V„ fl P = P- .D a J.

3) One has C (A-DV^ O (B-I)V^ and Q C (A-DV^ H (B-I)V^ then Q + P^ C (A-I)Vg P> (B-I)V^C + Q.

and Q n P =

[o].

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Take x G (A-I)Vg n(D-I)V^ then x = y + z with y This implies x - y = z G: (A-I)V H (R-I)V. and Z £ ri[(A-I)Vg ri(B-I)V^l = P^.

Hence x=y+z€Q+P^ and (A-I)Vg H (B-I)V^ = This sum is a direct sum since Q + P = Q © P and Hence (A-DV^^ O (B-I)V^ = Q © P^ where (B-I)P^ =

3 _ Proposition 2 . Let us define P,. ^4 -) (A-i)Vj n Wg = n [(A-DVg Q® P^® P4-then €. Q and z £. P . ^ a

Q + Pi.

p^ c

P. 0 = (A-I)P^_ Proof :

1) This is clear from the définition.

2) Take x € (A-I)Vg ^ C P + Q then x = y + z with y C Q, z G P and z = z' + z” with z' G P and z" £ P,

a b •

This implies x - y = z G (A-I)V_ O W and D D z' £ ((A-I)V„ n W„) n P = P..

a li al

Now, X - y - z’ = z" G ((A-DV^H H P^ = P,^ which implies X = y + z’ + z’’ € Q + P^ + P^ and (A-DV^ H V7g = Q + P^ P^^ ^ This is a direct sum because Q + P +P, = 0©P (±)P,

^ a b ^a b*

Hence (A-I)Vj^ 0 =

Q 0

® P4 where (B-I)P,^ = 0 = (A-I)Pj^.

4 ^ Proposition 3 .

h-t ua define PV = Pb bk (A-KV., then - > '■'bC Pb-

> P4C. P'bC Pfe. .’) =

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Proof ;

31.

1) This is clear from the définition

2) C (A-I)Vg n Wg C (A-I)Vg and P'j^ = ^ (A-DV^ which imp3ic: P CP’

4 ^ b •

3) This is obvions.

4) Take x C (A-I)Vg C. P + Q then. x = y + z with y £ Q and z €. P This implies x - y = z £ (A-DV^^ Take z = z’ + z" with z' C P, and z" € P, . Now, z' £ (A-I)V„ 0 P = P, .b Bal

Hence x-y-z* = z"£ (A-I)Vg ^ = P’jj which implies

X = y + z' + z" € Q + P^ + P^ This sum is a direct sum since Q * . Pb = Q®P^©Pb.

Hence (A-DVj, = Q®Pj®P'b where (B-I)P'b = 0 = (A-I)P' . Now, P^ C P’j^ and define P^ as a supplément of P^^ in P'^ then P'b = ^4 P5 where (B-I)Pg = 0 = (A-I)P^.

Hence (A-I)Vg =

Q 0

P^ 0 P4 <±) P5 •

.5. Now. (A-I)Vg = Q0P^0P40P5.

Also (A-I)V„ = (A-I)X + (A-I)W„ = (A-I)X + Q where (A-I)X C p

B c B c c

and P n Q = Joj. This implies (A-DV^^ = (A-I)X^ 0 Q and ‘ (A-I)X^ = (A-DVg n P = P^ ® P^ 0 Pg.

Mow ^ (A-I)H = (A-IHV^HVg) + (A-I)X^ = (A-I)X^ = P^0Pj^©Pj.. Again^ (A-I)H^ = (A-I)Vg + (A-I)Z = (0 ® P^ 0 0 Pg)+ R

= Pi® P40P5 ® R0Q.

We hâve that (A-I)X^ = P;j^ ® P4 © P5 v;hich implies dim(A-I)X^ = dim(P^ 0 P4 © Pg).

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V/e hâve constructed that U = (V. + V„)(+)X,. This implies (B-I)U = (B-I)V^ + (B-I)X^.

By définition (B-I)U = (W^ + (B-I)V^) O where (E-I)V^C which implies (B-I)U = W. O W_ + (B-I)V. = (B-I)V. + W„ O, p

Ad a a b b •

But (B-I)V^ ■ 1°} implies (B-I)U = (B-I)V^0l'Jg H and dim (B-I)U = dim (B-I)V. + dim (W„ H p, )

A ü b •

Again (B-I)X, C P, VJ„ and (B-I)U = (B-I)V. + (B-I)X, which

I a D D A U

implies dim (B-I)U - dim (B-I)V^ = dim (Wg 0 p^) = dim (B-I)X^. Hence (B-I)X, = P. ^

d b B

Now^ Pg n Wg ^ (V/g n (A-I)Vg = Pj^. This implies (B-I)X^ O and (B-I)X^ ^'b ‘ ^4 •

.6. One has that Q C (A-I)U D(B-I)V^ C(B-I)V^ O C Wg O Proposition 4 : Let us define P* a P na DU 0 (B-I)V

a

then DP' CP. a a 2) PjC P'^. 3) (A-DU

O

(B-I)V^ = Q ® P' . Proof .

1) This is clear from the définition.

2) P^ C (A-DVg 0(B-DV^ O 0 (B-I)V^ and (A-D Vg C ( A-Dü. This, implies C (A-DU where CI (B-I)V^,

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33 3) Take x C (A-I)U ^(B-I)V. C Q + P then x = y + z with y Q

and Z € This implies x - y = z € (A-I)U H (B-I)V^ and Z <£ (A-DU n (B-I)V

Aj vO P = P’ . a a

Hence x=y+z6Q+P» and (A-I)U H = Q + P' .

a A a

This sum is a direct sum since Q + P = Q (+) P .

a a

Hence (A-I)U A (B-I)V. = Q0P' where (B-I)P' = 0 = (A-I)P' .

A a a a

Mow P-i C P* and define as a supplément of P. in P' then

-1- a £. la

P’a =

''a®P2-Hence (A-I)U H (b-I)V^ = Q 0 P^ 0 ?2 where (B-DP2 = 0 = (A-DP2

.7. Vie hâve that P^ P^ and P^^ (+) P^ C P^^. Let us say that

= Pi®P\ and = P^C±)P5©P\.

Mow^ (A-I)X^C P = P^0Pj5 = P"3 0P4(±) P5 ®P"j3. This implies (A-I)e^ = x^ + x”^ + X4 + Xg + x"j^ with e P^. x"^ 6 P"^, x^ é X. €. P, € P^ and X

■ ’ • • ■ • ®sl • '^5'“ ^5' b ''d

But we hâve already found that (A-I)X = P- (+) P, 0 Pc • c 1 H b

So, there exists e'^£ such that (A-I)e*^ = x^ + Xj^ + Xg Hence (A-I)(e. - e’.) = x" + x", <£ P" + P’’

1 1 a b a b •

Let us put e. = e. - e'. and take X 1 i i

(A-I)X, C P" + P'V.

d a b

= r'i. • • • • ®s] which implies

Mow (B-I)e. = (B-I)e. - (3-I)e'. = (B-I)e. and U = (V. + V„) 0X J ' '1 1 ' ' ~ 1

by (2) of appendix to Chap. I. Again one has that

A-I : X c B-I : X P P vjhere (B-I)X, C. P d b Let us define P^ = (A-I)X, H (B-I)X. where P C. P, and

c d d c b

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Now. (A-I)U = (A-I)X^ + (A-I)Wg + (A-I)X^ = (A-I)0 (A-I)Wg 0 (A-I)X^ because P = (A-I)X^(±) (A-I)X^(±) (A-I)X^ and P H Q =

Again P^^, P^ C (A-I)X^ which implies P4 ^ " [*^} ^5 ‘ Define Pg as a supplément o£ (P^ + P^) in (B-I)X^ then

(B-I)X^ = P4 0Pç,<3>P6 ^ ° ■ (A-I)Pg.

Now. (B-I)U = (B-I)V^ © (B-I)X^ = Q 0 P^0 P4 0 P^ © Pg © M and (B-I)U. = (B-I)U® (B-I)X^ = Q0P,0P., ®P 0P.(0M0L’ . Again (A-I)Vg C (A-I)U^ (A-I)U ^T(B-I)V^C (A-I)U and

(A-I)X, f) (B-I)X , C (A-DU.

d d

Hence (A-I)V„ + (A-I)U H (B-I)V. ♦ (A-I)X^ 0 (B-I)X^ C (A-I)U

D A d d

which implies Q 0 P^ 0 P2 © P4 ® Pg © P^ C (A-DU, Let us define P^ as a supplément of

(Pi 0 P2 0 P4 0 Pg 0 P^0 Q> (A-DU then

(A-DU = Pi ® P2 © P4 0 Pg 0 Pj, 0 P7 0 Q where (B-DP^ = 0 = (A-I)P^. Now, P'^C Wg and P^ C Wg where P^ H VJg = vjhich implies

P7O P'^ = |o] and P, ri = ^0}. Now. (A-DU = (A-I)X^ 0 (A-I)X^ © Q.

This implies dim (A-DU = dim (A-I)X^ + dim (A-I)X^ + dim Q and dim (A-DU - dim (A-DX - dim Q = dim (A-DX, = dim (P„ + P + P„)

c d 2 c 7 •

Now. (A-I)X^ C P and (A-DX^C (A-DU this implies (A-DX , C P n (A-DU =

d Pj ♦ p^ ♦ p^ * Pj ♦ Pc P7 and

(A-I)e^ = + X2 + + Xr + X + X- with 5 c 7 xi e. p^ , ^2 ^ Pj X, £ P 4 4 •> Xr £ Pc 5 5 » cX •£ P , X- Ê P„ and X, =

c ’ / 7 d (êi . . But we hâve found that (A-I)X^ = Pi + Pn +

c 1 4 Pg. So, there exists e*. <£. X such that (A-I)e’. = + x,, + x^.

1 c 1 1 4 5

Hence (A-D(e. - e'-) = x„ + x + x„ Ç P„ + P + P„.

(42)

Let us put - e'^ and take which implies (A-I)X*, C + P + P^.

a l c 1

Now, (B-I)e*^ = (B-I)ê^ - (B-De’^ = (B-I)ë^ and U = (V^ + Vg) ® by (2) of appendix to Chap. I and

dim (A-I)X^^, = dim (A-I)X, = dim (P_ + P + P^)

d d Ici

Hence (A-I)X* = P^ + P + P^ = P. 0P ©P„.

d / c / ^ c /

.8 . We know that P, © P-C P and P„ © Pr 0 Pc © P^ ® P C P, 1 I a 4 5 b / c D*

Define P^ as a supplément of (P^^P^) in P^ and Pg as a supplément of (P(^ 0 Pg © Pg © P^ © P^) in Pj^ then

P^ = Pj © P2 © P3 and

Pfc = P4® P5®P6®P7©Pe® Pg

where (B-DP^ = 0 = (A-I)P, and (B-I)P = 0 = <A-I)P„.

Now, (A-I)X^C P = p^0Pj^ = Pl©P20P3®P4©P5(^PeG)P70Pg © P^ 35 . and (A-I)e^ ~ x^ + X 2 + X3 " ^4 . Xj . Xg + X7 . Xg t X with c ^ Pi , ^2 e Pa2 5 X3 o g Po3 J ^4 1 '‘s ^ '’s . '‘e ^ '’e , *7 ^. X C O ^ 8, ^c e P and X = ^ c a [!e^ ,

But we hâve already found that (A-I)X^ = P^ 0 Pi^ © Pg and

(A-I)X*, = P«©P ©P„. So there exists e ’ . X and e*. €. X^,

d2c7j xc xd

such that (A-I)e'. = x. + x„ + x^ and (A-I)e^. = x„ + x + x„.

XX*+o X/'C/

Hence (A-I)(e. - e'. - e*.) = x- + Xc + x„ 6 P» + Pc + Pq.

1 X 1 ooo^bo

Let us put 3nd take X^ = which

implies (A-I)X C P^ + Pc + Po. a O b O

Mow, (B-I)ë^ = (B-I)e^ - (B-I)e^ - (B-I)e*^ = (B-I)e^ - (B-I)e*^ 6 L + Pb

(43)

Again, P = (A-I)X = (A-I)X^ 0 (A-I)X^0 (A-I)X^ = (A-I)X 0(A-I)X 0 (A-I)X*,

a c a

which implies dim P = dim (A-I)X + dim (A-I)X + dim (A-I)X*, and

a c d

dim P - dim (A-I)X* - dim (A-I)X = dim (A-I)X

d c a = dim (P3 + Pg + Pg) where (A-I)X a O 0 O

C

P^

+

Pc + Pq. Hence (A-I)X^ = ?3 " ^6 " ^8 = ^3® ^6® ^8 * Remark ; Mow ^ Wg = Q 0 P^ (+) P^ © P^ 0 Pg 0 0 II © Y and O Wg = Q 0 P^ 0 P^ 0 P^ © Pg. 2.1. Introduction • VJe hâve that

(A-I)X^ = Pj0Pi^0Pg and (B-I)X^ = 0. (A-I)X^ = Po © Pc © Pq and (B-I)X = L’ .

a d D O a a

(A-I)X*^ = P2®Pj,©P, and (B-I)X*^ = P4 © P^ ® Pg . (A-I)L = 0 and (B-I)L = P = P- 0 P<, 0 Po

C c 3. X il O •

2.2

1) Let X^ X^

I

(A-I)x <£P^j. Now, (A-DX. CP n(A-I)X = P..

la cl

Take x^ P^ then there exists £ X^ such that x^ = (A-I)y^ € P^ and P^ C (A-I)X^.

Hence P^ = (A-I)X^ where (B-I)X^ = 0. 2) Let X^ = Jx e X^ I (A-I)x €. P^^ O Wg •.

Now' (A-I)X(, C (P n w„) n (A-I)X = P,..

t *+ D D c 4

(44)

37. Take x„ € P„ then there exists y„ é_ X such that

4 4 -^4 0

x^ = (A-I)y^ € P^n Wp and P^^ C (A-I)X^. Hence P^ = (A-I)Xj^ where (B-I)Xj^ = 0.

3) Now, X^ + X„ C X and this is a direct sum. Define X'r as a

14c b

supplément of (X^ + Xj^) in X^ then X^ = X^0X^C±)X*^. Now, (A-I)X’g C P^ + P^^ + Pg and

(A-I)e*^ = x^ + x^ + Xg with x^ €-P-^ ^ x,^ P^ ^ x^ ^5

X'g = ••• î ^'hj" * hâve found that P^ = (A-I)X^ and P^ = (A-I)Xj^. So, there exists y^ £ X^ and y^ € X^ such that

(A-I)y^ s x^ and (A-Dy^^ = x^^.

Hence (A-I)(e'^ " ~ ^4^ ” ^5 ^ ^5 ’

= e’^ - y^ - y^ and take X^ = ■ '•* ®hl implies (A-DXg C P^.

Now (B-I)e. = (B-I)e'. - (B-I)y. - (B-I)y,, = 0 and ^c ~ 0 ^4 © Xg by (2) of appendix to Chap. I_

Again , (A-I)X^ = (A-I)X^ 0 (A-I)X^^ 0 (A-I)X^ which implies dim (A-I)X^ = dim (A-I)X. + dim (A-I)X„ + dim (A-I)X.

c 1 4 5

and dim (A-I)X - dim (A-I)X. - dim (A-I)X,. = dim (A-I)X.

c 1 4 5

= dim P_. O Since (A-DX^C P^ then P^ = (A-I)X^ where (B-I)X^ = 0

2.3.

1) Let Xg = |x 7^ j (A-I)x ^ P^|. Now (A-I)X, ^ P n(A-I)X = P„.

J 3 a a 3

(45)

Hence = (A-DXg.

Movv

(B-I)Xo

^ de fine L,,

= (B-I)X_

where L,, dL L’ and

O a H O 4 a

(A-I)L^ = 0.

Now, (A-I)X^ C (Pk<^ ri (A-I)X = P_

Take Xg £ Pg then there exists yg ^ X^ such that Xg - (A-I)yg (A-I)Xg and Pg C(A-I)Xg.

Hence P_ = (A-DX^.

b b

Now ^ (B-I)Xg C L’^ = (+) where is a supplément of ^ in L'^.

This implies (B-I)e. = 1, + 1, with 1,, €. L,, 1, ^ L, and

i4k 44;kk

Xg = ^ ^ ‘ hâve defined that = (B-I)Xg. So^ there exists Xg € Xg such that (B-I)Xg = Ij^,

Hence (B-I)(e^ - Xg) = Ij, ^ . Let us put e^^^ = - Xg and take Xg = y ••• T which implies (B-I)Xg Cl

and Xg (+) Xg = Xg <+) Xg by (2) of appendix to Chap. I.

3) Nowj Xg (J) Xg = Xg ® Xg C X^ and define X'g as a supplément of

But vje hâve found that (A-I)X_ = P_ and (A-I)X- = P_.

do 6 6

Again (A-I)e- = (A-I)e.

5 X 1

N

ov

7 define Lr = (B-I)X

)

(B-DXf- where ^ L, .

6

5 k

(A-I)e. - (A-I)xg Pg - Pg C P3 0Pg®Pg

(Xg (±) Xg) in X^ then X^ = Xg ® Xg ® X'g = Xg © Xg (±) X'g.

Again, © Lg C L'^ and define Lg as a supplément of

(L^0Lg) in L'^ then L'^ =

0 Lg ® Lg .

Now, (A-I)X'gC, Pg + Pg + Pg and this implies

(46)

39. So, there exista ^ and Vg G. Xg such that (A-Dy^ = Xg and (A-I)yg = Xg.

Hence (A-I)(e'- - Vn ~ y= Xq <S. P Let us nut e. = e'. - y, “ Yr

1-^3-^b 8 8 ‘ X i 3 E

and take Xg = ^*2^, ..., e^| V7hich implies (A-I)Xg C Pg. Now ^ = Xg (J) Xg (+) Xg by (2) of appendix to Chap. I.

Again^ " (A-I)Xg0(A-I)Xg0(A-I)Xg which implies dim (A-I)X = dim (A-I)X« + dim (A-DX^ + dim (A-DX^ and

a 3 b 8

dim (A-I)X^ - dim (A-I)Xg - dim (A-I)Xg = dim (A-I)Xg = dim Pg. Since (A-I)Xg C Pg and dim (A-I)Xg = dim Pg then Pg = (A-I)Xg. Now (B-I)X’ CL’ = L|, 0 Lr 0 Le which implics

(B-I)e’. = 1„ + le + le with 1, € 1 4 5 b 4 l„ , 1. € L> 1. L. and 4’5 Dib 6 ^*8 bave defined that L,^ = (B-I)Xg and Le = (B-I)X_. So there exists x_ C X« and x_ ^ X^ such

b 6 > 3 3 6 6

that (B-I)x„ = 1,. and (B-Dx^ = Ir

3 4 6 5 •

Hence (B-I)(e'. - x, - x^) = le ^ Le Let us put ê. = e'. - - x,

X 3 O 66* ^1 i3(

and take Xg = ^e^ , . . . , which implies (B-I)Xg c Lg. Noi-J^ ^a “ ^3 ® ^6 ® ^8 " ^3® ^6® ^*8 appendix to Chap. I. Again (AI)i. = (AI)e’. (AI)x, (AI)Xe ^ X'

-J 1 1 O boob

Now^ (B-I)X^ = (B-I)Xg 0 (B-I)Xg0 (B-I)Xg which implies dim (B-I)X = dim (B-I)X„ + dim (B-I)X- + dim (B-I)X- and

a O b O

dim (B-I)X - dim (B-I)X- - dim (B-I)Xe = dim (B-I)X„ = dim Le.

a O b 8 6

Since (B-I)Xg C Lg and dim. (B-I)Xg = dim Lg then Lg = (B-I)Xg.

1) Let L^ = ^x é L^l (B-I)x € (A-I)X^|. Now, (B-DLe C P ri(A-I)X = P..

’ la cl

(47)

Take then there exists £. such that x^ = (B-I)y^e (A-I)X^ and C

Hence (B-I)L^ = P^ where (A-I)L^ = 0. 2) Let L3 = ^x C ) (B-I)x 6 (A-I)X^} ^

Now. (B-I)L. C P ^ (A-I)X = P„.

O a ad

Take x^ € Pg then there exists y^ <= such that Xg = (B-I)yg £ (A-I)X^ and Pg C(B-I)Lg.

Hence Pg = (B-I)Lg where (A-I)Lg = 0.

3) Novj^ + Lg C and this is a direct sum.

Define L’^ ss a supplément of (L^ + Lg) in then Le = L^ (+) L2 (£' Lg where (A-DL'^ = 0.

Again , (B-DL'2 ® ^2 ® ^3 implies

(B-I)e*^ = x^ + x^ + Xg with x^ 6- P^ ^ x^ ^ P2 ^ X3 C P3 and L*2 = '••j ^*d}‘ hâve found that (B-I)L^ = P^ and

(B-I)Lg = Pg. So, there exists 1^ £. and lg £- Lg such that (B-I)l^ = x^ and (B-I)lg = Xg.

Hence (B-I)(e*^ - 1^ - lg) = X2 ^ p2* Let us put

= e’^ - 1^ - lg and take ^ ‘ > ^dl '■^^Lch implies (B-DL2 C P^ where (A-DL2 = 0.

Mow, = L^®L'2©Lg = 0 L2 (±) Lg by (2) of appendix to Chap.I. Again ^ (B-I)L^ = (B-I)L^0 (B-DL2© (B-I)Lg which implies

(48)

41.

5. We hâve that

(A-I)X*^ = P2 0P^0P^ and (B-I)X*^ = P^ 0 P^ © Pg . Let D = (A-I)X*. + (B-I)X^, = P. 0 P„ © ® P

d a Z 4 O / c ’

î'îow A-I : X* , ^^ D J

J d

B-I . X* , ^--- ) D. d

VJe hâve already defined that P = (A-I)X*, A (3-I)X*,.

c d d

Let us say that (A-I)"^(P ) = X' and (B-I)~^(P ) = X” and define

c c

= X' n X" which implies (A-I)X C P ^ (B-I)X .

' V oc O

Now, X^ C X’ and define a supplément of X^ in X’ then X’ = X^ © X..

O Z

Let Pq = ^ Pç, I (A-I)“^(z) ^ xJ.

Now P C (A-I)X .

^ O O

Take y £ (A-I)X then y = (A-I)z' with z ' X V7hich implies z' C X'. This implies z' = (A-I) ^(y') with y' € P^ and y ' é. P^ and

(A-I)X C P O O •

lîence P = (A-I)X .

O O

Now, (A-I)X = P e P and (B-I)X CP.

' O O c oc

In case of (B-I)X = (A-I)X one has th' spécial case of (3) of

O O

appendix to Chap ^ I.

For a general case one has further to subdivide X But the form O •

of the matrix B will be analogous and there will be a finite num- ber of subspaces in X .

O

Now^ let us proceed v;ith the supposition of (B-I)X^ = (A-I)X^. Let X^ = ^x é: X" I (B-I)x Ê (A-DX^ j = ( B-I ) " ^ (A-I) X^ .

Again (B-I)X" = P = (A-I)X’ and (B-I)X" = (A-I)X © (A-I)X„

J c O 2

= (B-I)X 0 (B-I)X„,

O /

(49)

This implies dim (B-I)X" = dim (B-I)X^ + dim (B-I)X^.

Again^ X^ C X" and X^ C X’’ which implies X^ + X^ C X’’ and •:B-I)X^ + (B-I)X^ C (B-I)X".

Hence X" = X + X„ = X 0X„.

Nov;^ choose X2 in such a way that (B-DX^ C. + ?0.

Then put Pg ^ (A-DX2 = (B-DX^ where X^ = (B-I)'^(A-I) X2, Let us finally choose Xg as a supplément of (X* + X”) in X“^ such that (B-I)Xg C + Pg then (B-I)(X2 + Xg) = P^^ + Pg.

Mow, the block représentations for the matrices B and A are respecti­ ve ly I 0 0 0 0 0

"5T 0 0 T

0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c

6

I

0

0

0

0

0

0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

I 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

I

0

0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

0 0

0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0

0

0 I 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 I I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 I

0

0 0 0 0 0 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 U 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 A. 0 0 B.O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0^0 0 0 0 I 0 0 0 n 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0

c.o

0 D. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0'^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 J 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 n 0 I T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 “r 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I

0 0

0

0 0

0 0 0

0 0

0 0 0 0 0 0 0 0 _0.

JL

0 0 0 0 0 0 0 0 0 0 0 0 I

(50)
(51)

BIBLIOGRAPHY. 1) Curtis, C.W. and Reiner, I.

Représentation theory of finite groups and associative Algebras , Wiley-Inter-Science (New York, 1962)^

2) Hall, M. Jr.

The theory of groups , Macmillan (New York ^ 1C59) . 3) Burrow, M.

Représentation theory of finite groups Academie press (Nev; York 1965)

4) Maclane^ S. and Birkhoff , G.

Algebra^ Macmillan Company, (new édition, New York, 1968)^ 5) Gorenstein D

)

Finite groups, Harper and Rov/, publishers , (New York , 1968). 6) Feit W

Characters of finite groups, H^A.Benjamin, Inc. (New York, 1967). 7) Lang, S.

Algebra, Addison-VJesley publishing Company^ Inc. 1965. 8) Halmos P R

3 *

Finite dimensional vector spaces (second édition) D_ Van Nostrand Company, Inc. (New York. 1958) 9) Higman , D.G.

Indécomposable représentations at characteristic p, Duke Math. J. 21 (1954) p. 377-381.

0) Bashev V.A.

Représentations of the x Z^ group in the fiold of characteris­ tic 2

Dokl. Akad, Nauk 141 (1961) p. 1015-1018. 1) Krugljak. S.A.

(52)

45. 12 ) Lang ^ S ^

Linear Algebra, Addison-Wesley Publishing Company 195G. y

13) Freyd^

Abelian Categories

An Introduction to the Theory of Functors . îlarper & Rov7 Publishers New York 1964.*

14) Gantmacher , F.R.

The Theory of Matrices, Chelsea Publishing Company, New York _ M.Y^ (1959)^

15) Godement R. ?

Cours d'algèbre. Paris . Hermann, 1963. 16) Herstein I N.

;

Topics in Algebra. New York, Toronto, London . Blaisdell, 1964. 17) Kurosh A G

The Theory of Croups. Vol. I and vol. II ^ translated by K A. Hirsch (2nd English ed ) Nevj York . Chelsea, 1960 18) Zariski 0. and Samuel. P.

Commutative Algebra. Princeton, N.J. . Van Hostrand, vol. I, 1958 . vol. II. 1960.

19) Mitchell, B.

Theory of Categories. Nev; York . Academie Press, 1965.

20) Gabriel. P.. Des Catégories abéliennes, Bull. Soc. Math. France, 90 (1962) . 323-448.

21) Heller, A.

Homological Algebra in Abelian Categories Ann. of Math. 68 (1958) 484-525

22) Bourbaki, N.

Algèbre Commutative, Hermann, Paris . 1962. 23) Van Der Waerden.

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24) Buchsbaiim D.A.

Exact Categories and Duality , Trans. Am. Math. Soc. 80 (1955), 1-34. 25) Buchsbaum, D,A.

Satellites and universal functors Ann. of Math. 71 (1960), 199-209 26) Eilenberg S.

Abstract description of some basic functors. J. Indian Math. Soc^ 24 (1960^ 221-234. > 27) Roiter, A.V. On Categories of représentations. Ukrain, Math. J. 15 (1963)448-452. 28) Osima M. >

On the représentations of groups of finite order. Math. J. Okayama Univ. 1 (1952) 33-61.

29) Murnaghan^ F.

The theory of group représentations, Johns Hopkins ^Baltimore, 1938. 30) Relier, A. & Reiner, I.

Indécomposable représentations, III. J. Math. 5(1961) 314-323. 31) Heller, A. & Reiner^ I.

Indécomposable représentations of cyclic groups^ Bull. Am. Math. Soc. 68 (1962) 210-212.

32) Green J A

} • *

On the indécomposable représentations of a finite group. Math. Zeit. 70 (1959) 430-445.

33) Berman S D• • •

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Dans le cas où une version électronique native de la thèse existe, l’Université ne peut garantir que la présente version numérisée soit identique à la version électronique

Dans le cas où une version électronique native de la thèse existe, l’Université ne peut garantir que la présente version numérisée soit identique à la version électronique

Comme certains logiciels de la Toile peuvent requérir une activation par l'utilisateur, il est RECOMMANDÉ qu'il y ait au moins un champ Ecom visible par

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Si la lutte contre le terrorisme en Afrique de l’ouest est un enjeu pour le Maroc du fait du confl it saharien, il m’en demeure pas moins que l’intensifi cation des