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On the vanishing of a pure product in a (G,6) space Sing, Kuldip 1967

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THE UNIVERSITY OP BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY o f KULDIP SINGH B.A. Pun jab U n i v e r s i t y , I n d i a , 1952. M.A. Pun jab U n i v e r s i t y , I n d i a , 1954. MONDAY, A p r i l 2 4 t h , 1967 a t Jt.JO p.m. I n Room 104,. M a t h e m a t i c s B u i l d i n g COMMITTEE I N CHARGE Cha i rman : I . MoT. Cowan J . B r e n n e r M. S i o n E. L u f t C. A . Swanson B. N. Moy ls R. Wes tw ick P. R a s t a l l E x t e r n a l Examine r : M a r v i n Marcus Depar tmen t o f M a t h e m a t i c s U n i v e r s i t y o f C a l i f o r n i a S a n t a B a r b a r a Research S u p e r v i s o r : R. Westwick. ON THE VANISHING OF A PURE PRODUCT I N A {G,<r) SPACE A b s t r a c t We b e g i n by c o n s t r u c t i n g a v e c t o r space o v e r a f i e l d F , w h i c h we c a l l a (G,<r) space o f t h e s e t W = V-^xVgX. . . x V n , a c a r t e s i a n p r o d u c t , where i s a f i n i t e - d i m e n s i o n a l v e c t o r space o v e r an a r b i t r a r y f i e l d F , G i s a subgroup o f t h e f u l l symmet r i c g r o u p S n and cr i s a l i n e a r c h a r a c t e r o f G . T h i s space g e n e r a l i z e s t h e spaces c a l l e d t h e symmetry c l a s s o f t e n s o r s d e f i n e d by Marcus and Newman [1]. We can o b t a i n t he c l a s s i c a l s p a c e s , namely t h e Tensor s p a c e , t h e Grassman space and t h e symmet r i c space , by p a r t i c u l a r i z -i n g t h e g r o u p G and t h e l i n e a r c h a r a c t e r cr i n o u r (G,<r) space . I f ( v - ^ V g , . . . , v ) € W , we s h a l l d e n o t e t h e "decomposab le^ ' e l e m e n t i n o u r space b y v l A v 2 A " * ° A v n a n d c a l 1 it the (G,o-) p r o d u c t o r t he Pure p r o d u c t i f t h e r e i s no c o n f u s i o n r e g a r d i n g G and <r , o f t h e v e c t o r s v l ' v 2 , , , , , v n " T h i s c o r r e s p o n d s t o t h e t e n s o r p r o d u c t , t h e skew symmet r i c p r o d u c t and t h e symmet r i c p r o d u c t i n t h e c l a s s i c a l spaces . The pu rpose o f t h i s t h e s i s i s t o d e t e r -mine a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e v a n i s h i n g o f t h e (G,cr) p r o d u c t o f t h e v e c t o r s v i ' v 2 * * * * ' v n i n t h e S e n e r a 3 - case . The r e s u l t s f o r t h e c l a s s i c a l spaces a r e w e l l - k n o w n and a r e deduced f r o m o u r m a i n t h e o r e m . We use t h e " u n i v e r s a l mapp ing p r o p e r t y " o f t h e (G,C") space t o p r o v e t h e n e c e s s i t y o f ou r c o n d i t i o n * These c o n d i t i o n s a r e s t a t e d i n t e rms o f d e t e r m i n a n t - l i k e f u n c t i o n s o f t h e m a t r i c e s a s s o c i a t e d w i t h t h e s e t o f v e c t o r s v , , v 0 , . . . , v . GRADUATE P o i n t S e t T o p o l o g y A l g e b r a i c T o p o l o g y Complex A n a l y s i s . T h e o r y o f Groups T h e o r y o f R i n g s STUDIES S. C l e v e l a n d J . V. W h i t t a k e r M. S i o n R. Ree D„ C. Murdoch ON THE VANISHING OF A PURE PRODUCT IN A (G,o) SPACE by KULDIP SINGH M.A., Punjab University, India, 1954 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA February, 19^7 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t l i a L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and sl .udy I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tmen t o r by h i s r e p r e s e n t a t i v e s I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d v / i t h o u t my w r i t te.n perm i ss ion , D e p a r t m e n t o f 1^] <xik tYV^O.'tt C6 The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada - i i -T h e s i s S u p e r v i s o r ; Dr . R. Wes tw ick . ABSTRACT We b e g i n by c o n s t r u c t i n g a v e c t o r space o v e r a f i e l d ' F w h i c h we c a l l a ( G , a ) space o f t h e s e t W = V-^xVgX' • • X v " n > a c a r t e s i a n p r o d u c t , where V. i s a f i n i t e - d i m e n s i o n a l v e c t o r space o v e r an a r b i t r a r y f i e l d F , G i s a subgroup ~Of the f u l l s y m m e t r i c group S n and o i s a l i n e a r c h a r a c t e r o f G . T h i s space g e n e r a l i z e s t h e spaces c a l l e d t h e symmetry c l a s s o f t e n s o r s d e f i n e d by Marcus and Newman [1]. We can o b t a i n t h e c l a s s i c a l spaces, namely t h e Tensor space , t h e Grassman space and t h e s y m m e t r i c s p a c e , b y p a r t i c u l a r i z i n g t h e group G and t h e l i n e a r c h a r a c t e r a i n ou r ( G s a ) space. I f ( v - j ^Vg* . . . , v ) € W , we s h a l l deno te t h e "decomposab le " e lement i n our space by v-jAVgA. . . A v n and c a l l i t t h e ( G , a ) p r o d u c t o r t h e Pure p r o d u c t i f t h e r e i s no con -f u s i o n r e g a r d i n g G and a , o f t h e v e c t o r s v-, , v 0 , . . . , v . A. d n T h i s c o r r e s p o n d s t o t h e t e n s o r p r o d u c t , t h e skew symmet r i c p r o d u c t and t h e symmet r i c p r o d u c t i n t h e c l a s s i c a l spaces . The p u r p o s e o f t h i s t h e s i s i s t o - d e t e r m i n e a ' ' necessa ry and s u f f i c i e n t . c o n d i t i o n f o r t h e v a n i s h i n g o f t h e ( G , a ) p r o d u c t o f t h e v e c t o r s v - p V g , . . . 3v i n t h e g e n e r a l case . The r e s u l t s f o r t h e c l a s s i c a l spaces a r e w e l l - k n o w n and a r e d e d u c e d . f r o m our ma in t h e o r e m . - i i i -We use the ' ' u n i v e r s a l mapp ing p r o p e r t y " o f t h e ( G , a ) space t o p r o v e t h e n e c e s s i t y o f our c o n d i t i o n . These c o n d i t i o n s a r e s t a t e d i n te rms o f d e t e r m i n a n t l i k e f u n c t i o n s o f t h e m a t r i c e s associated w i t h t h e s e t o f v e c t o r s v l * v 2 ' " " v n . - i v -TABLE OP CONTENTS page INTRODUCTION 1 CHAPTER I 1 . Vector spaces spanned by sets 5 2. (G,a) spaces 7 3. Special cases of the (G,a) spaces 14 CHAPTER I I 1. Notations 16 2. Statement of the Problem 17 3. Embedding Theorem 17 4 . Some Definitions 21 5. Representation Theorem Form I 22 6. Computation of the Co-efficients 27 7. Representation Theorem Form I I 32 CHAPTER I I I 1. Construction of a mult i l i n e a r and (Gso) function 34 2„ Solution to the Problem 36 3. Basis of P(W5G3a) 43 4 . Decomposition of P(W«,G3a) 46 - V -page CHAPTER TV 1. P a r t i c u l a r i z i n g a 50 2. S p e c i a l i z a t i o n t o t h e c l a s s i c a l spaces 53 3. D imens ion of t h e c l a s s i c a l spaces 51*-4. A s u f f i c i e n t c o n d i t i o n f o r v^VgA'. . . A v n t o be z e r o 56 5. P a r t i c u l a r i z i n g V and G 58 BIBLIOGRAPHY 60 - v i -ACKNOWLEDGEMENTS I am g r e a t l y i n d e b t e d t o P r o f e s s o r R. Westw ick f o r s u g g e s t i n g t h e t h e s i s t o p i c and f o r h i s v a l u a b l e h e l p and encouragement d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . I a l s o w i s h t o t h a n k P r o f e s s o r s B, N. Moy ls and J . L. B r e n n e r f o r t h e i r c o n s t r u c t i v e c r i t i c i s m o f t h e d r a f t f o r m o f t h i s wo rk . The generous f i n a n c i a l s u p p o r t o f t h e N a t i o n a l Research C o u n c i l o f Canada and t h e U n i v e r s i t y o f B r i t i s h Co lumb ia , w h i c h made i t p o s s i b l e f o r me t o c a r r y o u t t h e n e c e s s a r y r e s e a r c h , i s g r a t e f u l l y acknowledged. - 1 -INTRODUCTION Let V-pVg*° o • 'jV be any f i n i t e dimensional vector spaces over an a r b i t r a r y f i e l d F and consider the tensor n product (g> . I t i s wel l known that the tensor product i = l v-pvg$. . .®vn i s zero i f and only i f v^ i s zero for some i , X< i < n . S i m i l a r l y i f AV , where V i s any f i n i t e dimensional vector space over an ar b i t r a r y f i e l d F , i s the Crassman space, then the skew-symmetric product v-^AVgA. • • Av n i s zero i f and only i f v-^,Vg,...,vn are l i n e a r l y dependent [3] . Again i f 7 i s a f i n i t e dimensional unitary space and. 7^n^ i s the symmetric product space, then the symmetric product v l ' v 2 ' " ' ' v n i s z e r o i f a n y o n l y i f v i is'zero for some I , 1 1 i <. n [1]. The aim of the present thesis i s to define a suitable generalization of these three kinds of products and give a necessary and s u f f i c i e n t condition that a "pure product" vanishes The general theory contains each of the three foregoing spaces, which we s h a l l c a l l the c l a s s i c a l space's, as special cases. The s t a r t i n g point i s the construction of the (G,a) space of the set ¥ , which we denote by P(W,G,a) where G i s a subgroup of the f u l l symmetric group •S n of degree n , W = VnXVoX.-.xV i s the cartesian product of any f i n i t e - 2 -d i m e n s i o n a l v e c t o r spaces V\ ove r an a r b i t r a r y f i e l d P , such t h a t = v g ( i ) f o r 1 = l j 2 , . . . , n and f o r a l l g i n G. T h i s space g e n e r a l i z e s t h e c l a s s i c a l s p a c e s , as w e l l as t h e symmetry c l a s s o f t e n s o r s d e f i n e d by Marcus and Newman [ l ] . I f ( v - ^ V g , . . . , v n ) i s i n W , we deno te i t s image under % , where X i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n o f W i n t o P(W,G,a) , by v-jAVgA. . . Av and c a l l i t t h e ( G , o ) p r o d u c t o r t h e p u r e p r o d u c t o f t h e v e c t o r s V p V 2 , . . . , v n . T h i s p r o d u c t c o r r e s p o n d s t o t h e t e n s o r p r o d u c t , t h e skew-symmet r i c p r o d u c t and t h e symmet r i c p r o d u c t r e s p e c t i v e l y i n t h e case o f t h e t e n s o r s p a c e , t h e Grassman space and t h e symmet r i c p r o d u c t space. Then we c o n s i d e r t h e p r o b l e m o f d e t e r m i n i n g a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t t h e ( G , a ) p r o d u c t v - L A v 2 A . . . A v n i s z e r o . The method o f app roach i s as f o l l o w s : F i r s t t h e Embedding Theorem ~5> Chapter I I e n a b l e s us t o assume V-L = V 2 = . . . = V n = V ( s a y ) . We t a k e any b a s i s . y - ^ V g , . . . , y o f V and a s s o c i a t e w i t h any e lement ( v ^ , v 2 s • • • > v ) i n W , a s e t S o f n t u p l e s s = ( s ^ , s 2 , . .'. »&n) , 1 <_ s^ _< m ; 1 _< i _< n . We d e f i n e an e q u i v a l e n c e r e l a t i o n on s and deno te by E , a s e t o f r e p r e s e n t a t i v e s o f t he e q u i v a l e n c e c l a s s e s . We c o n s i d e r ( v - ^ , v 2 , . . . , v n ) ( } ) , where <J) i s a mapping o f W i n t o t h e f r e e space F(W) g e n e r a t e d by !" W' . By t h e R e p r e s e n t a t i o n Theorem Form I , 5 Chapter I I , we w r i t e ( v , , v ? , . . . . , v )<|) i n t h e - 3 -f o r m ( v - . , V g , . . . , v )(J) = u) + E a ( y , y , . . . , y )(J) : — ( * ) 1 d n S€E s s l s 2 s n where uu i s i n fi , a subspace o f P ( ¥ ) and a g i s i n F f o r a l l s i n E . I n 6, Chapter I I , we e v a l u a t e t h e c o - e f f i c i e n t s a_ i n te rms o f a d e t e r m i n a n t l i k e f u n c t i o n D s d e f i n e d on a s e t o f m a t r i c e s M g o b t a i n e d f r o m a m a t r i x M , where M i s a s s o c i a t e d w i t h ( v - ^ V g , . . . , v ) i n W , j w i t h r e s p e c t t o t h e b a s i s y-, * y 2 > • • • * y m o f v • W e r e w r i t e ( * ) by means o f t h e R e p r e s e n t a t i o n Theorem Form I I , 7, Chapter I I , as ( v , , v p , . . . ,v_)(j) = u) + Z D(M ) ( y '• , y_ , . . . >y ' )4> . 1 d n scE s s l s 2 s n 1 The main r e s u l t s a r e c o n t a i n e d i n 2, Chapter I I I . Lemma 2.1, Chapter I I I , g i v e s t h e s o l u t i o n t o t h e p r o b l e m i f ( v - ^ j V g , . . . , v n ) ( j ) s a t i s f i e s t h e p r o p e r t y P and Theorem . 2 . 3 , C h a p t e r : I I I , g i v e s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n i n t h e g e n e r a l case. I n t h i s c h a p t e r , we a l s o d e t e r m i n e a b a s i s o f our ( G , a ) space , w h i c h we use t o e x p r e s s . P(W,G,a) as t h e t e n s o r p r o d u c t o f ( G ^ a ^ ) spaces P ( ¥ i , G i , a ^ ) where G i i s a " d i s j o i n t " f a c t o r o f G (4, Chapter I I I ) . F i n a l l y i n Chapte r I V , we deduce some known r e s u l t s about t h e c l a s s i c a l spaces f r o m ou r ma in r e s u l t s by a s u i t a b l e s p e c i a l i z a t i o n . We a l s o s t a t e and p r o v e a c o n d i t i o n (4, Chapter I V ) w h i c h i s s u f f i c i e n t f o r t h e v a n i s h i n g o f a ( G , a ) - 4 -p r o d u c t ; v ^ A V g A . . . A v n i n t he g e n e r a l c a s e , and i s a l s o n e c e s s a r y , i f V i s u n i t a r y and G b e l o n g s t o a c e r t a i n c l a s s o f a b e l i a n groups (5> Chapter I V ) . - 5 -CHAPTER The a im o f t h i s c h a p t e r i s t o f i x t h e n o t a t i o n s and t h e t e r m i n o l o g y o f some w e l l - k n o w n c o n c e p t s . 1. V e c t o r spaces spanned by s e t s . L e t S be an a r b i t r a r y non-empty s e t and F an a r b i t r a r y f i e l d . L e t F ( S ) denote t h e s e t o f a l l mappings $ f r o m S i n t o F , such t h a t (s)i j i 4 0 f o r o n l y a f i n i t e number o f s e S . I f ^ and ty2 a re i n F ( S ) and a € F i we can d e f i n e t h e mappings ^ + $ 2 a n c * ^ n t h e u s u a l way by s e t t i n g ( s ) + t 2 ) '= ( s ) # 1 + ( s H 2 f o r a l l s e S . and ( s ) ( a ^ ) = a ( s ) ^ f o r a l l s e S . C l e a r l y ^ + $ 2 and a g a i n b e l o n g t o F ( S ) . One can e a s i l y v e r i f y t h a t t h e s e o p e r a t i o n s o f a d d i t i o n and s c a l a r m u l t i p l i c a t i o n g i v e F ( S ) the s t r u c t u r e o f a v e c t o r space ove r F „ T h i s v e c t o r space i s c a l l e d t h e f r e e space g e n e r a t e d by S over F . - 6 -1.1 Basis o f F ( S \ . For each s e S , c o n s i d e r the mapp ing • e s • : S > F ' , where ( t ) e _ = 6 , f o r a l l t € S and 6 , i s t h e K ronecke r d e l t a . Then e g e P(S) and t h e s e t B = t e g | s € S } i s a b a s i s o f F ( S ) . 1.2 D e f i n i t i o n : L e t (j) : S > F ( S ) , where ( s ) (j) = e s f o r a l l s e S . (j) i s a one-one mapping o f S o n t o B , and hence d im F ( S ) = c a r d i n a l i t y S . 1.3 P r o p o s i t i o n : Le t F ( S ) be t h e f r e e space g e n e r a t e d by S over F , U an a r b i t r a r y v e c t o r space over F and <p t h e mapping d e f i n e d i n 1„2„ I f f i s any mapping f r o m S i n t o U , t h e n t h e r e e x i s t s a un ique l i n e a r t r a n s f o r m a t i o n T o f F ( S ) i n t o U , w n i c h maK.es t h e f o l l o w i n g d i a g r a m U - 7 -c o m m u t a t i v e ; i . e . $)T = f P r o o f ; B = { e | s e S } i s a b a s i s o f F ( S ) by 1.1. D e f i n e 7 : B >U , by ( e s)T = ( s ) f , and e x t e n d i t l i n e a r l y t o a mapp ing , t o be deno ted a g a i n by T , on F ( S ) i n t o U . More e x p l i c i t l y (. E a e ) T = Z a„ ( e „ ) 7 , where A i s a f i n i t e v - . s s » s v s seA seA non -empty s u b s e t o f S and a_ e F f o r a l l s € A . Then s T i s t h e d e s i r e d mapping. 2. ( G , q ) spaces . L e t G be any p e r m u t a t i o n group on the s e t I = { l , 2 , . . . , n } ; i . e . , a subgroup o f t h e symmet r i c group S o f degree n. L e t F be any a r b i t r a r y f i e l d and a any l i n e a r c h a r a c t e r o f G ; i . e . , a i G- ^ F * i s a ( g r o u p ) homomorphism where F* i s t h e m u l t i p l i c a t i v e group o f F . For each i e I , l e t v\ be any f i n i t e d i m e n s i o n a l v e c t o r space over F ; and l e t ¥ = V"1 x V 2 x • • • x Vn , t h e c a r t e s i a n p r o d u c t o f t h e V i . 2.1 D e f i n i t i o n ; ¥ i s c a l l e d a G-se t i f " and o n l y i f V i = vg ( i ) f o r a l l i e I and f o r a l l g e G ". We s h a l l assume i n t he s e q u e l , t h a t W i s a G - s e t , and deno te i t s g e n e r a l e l e m e n t s , w by ( w p W g j • • . > w n ) where w. e V / f o r a l l i e I . - 8 -2.2 D e f i n i t i o n ; L e t f be any mapp ing o f W i n t o U , where U i s any v e c t o r space ove r F . Then ( i ) if i s c a l l e d a m u l t i l i n e a r • f u n c t i o n i f f i s l i n e a r i n each o f t h e n - c o o r d i n a t e s , t h a t i s , i f ( w 1 , w 2 , . , a w ^ + p w ^ , . . . , w n ) f = a ( w 1 , . , w ± , . , w n ) f + P ( w 1 , . • , w ^ , . , w n ) f , f o r any w^w '^ i n , any a,P i n F and any , i e I . ( i i ) f i s c a l l e d a ( G , q ) f u n c t i o n i f ( w r w 2 , . . . , w n ) f = o ( g ) ( w g ( l ) , w g ( 2 ) , . , w g ( n ) ) f , • . f o r any w. e - arid g € G, . 2.3 D e f i n i t i o n : A v e c t o r space T over F i s c a l l e d a ( G , a ) space o f ¥ i f and o n l y i f t h e r e e x i s t s a mapping "TT o f W i n t o T w i t h t h e f o l l o w i n g p r o p e r t i e s ( i ) T i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n . ( i i ) Fc = { w t | w e W } i s a s p a n n i n g s e t o f T . ( i i i ) The space T has a u n i v e r s a l mapp ing p r o p e r t y , t h a t i s i f U i s any v e c t o r space ove r F and f i s an a r b i t r a r y m u l t i l i n e a r and ( G , a ) f u n c t i o n f r o m W i n t o U , t h e n t h e r e e x i s t s a un ique l i n e a r t r a n s f o r m a t i o n T : T- — , such t h a t t h e f o l l o w i n g d i a g r a m - 9 -W i s c o m m u t a t i v e j i . e . , = f 2.4 Theorem; G iven G,a and a G-se t W, t h e r e e x i s t s a ( G , a ) space o f ¥ . Any two ( G , o ) spaces o f ¥ a r e i s o m o r p h i c as v e c t o r spaces. P r o o f ; ¥e s h a l l p r o v e t h e l a t t e r s t a t e m e n t f i r s t . L e t T-j^  and T g be two (G,a) spaces o f > ¥ . Then t h e r e e x i s t m u l t i l i n e a r and ( G , a ) f u n c t i o n s : ¥ - T , and 1^ 2 ; ¥ ^ T 2 and un ique l i n e a r t r a n s f o r m a t i o n s T l ; Tg > ? ± and T 2 f o l l o w i n g d iag rams ?j f Tg ) w h i c h make t h e T l W T 2 -5 c o m m u t a t i v e ; i . ' e . r^2 =^2 a n d ^2T = T X . - 10 -Cons ide r t h e l i n e a r t r a n s f o r m a t i o n ^2ri °- T l > T 1 We s h a l l show, l^^i - ^\ > t h e i d e n t i t y map o f T-^. S ince a n d a re l i n e a r t r a n s f o r m a t i o n s o f • , i t i s s u f f i c i e n t t o show, t h a t ^2^1 = ^1 o n t t i e s p a n n i n g " s e t WT^ o f T 1 . Fo r w l ^ e Wfc^ , ( w ^ ) ^ ^ ) = ((^)Z{t2)\ = ( ( w ) ^ ) ^ = . ( w ) ^ = w t x . T h e r e f o r e X^X>\ = ^ l ' S i m i l a r l y , = ^2 t h e i d e n t i t y map o f T 2 . Hence T-^  and T 2 a re i s o m o r p h i c . * We n e x t show t h e e x i s t e n c e o f a (G,a) space f o r W . Denote by 0 , t h e s m a l l e s t subspace o f F(W) , w h i c h c o n t a i n s a l l e lemen ts o f each o f t h e f o r m s ( i ) and ( i i ) : ( i ) (WJ_, . . . , a w ± + Bw^, . . . ,wn)<f> - a ( w 1 , . . . , w ± , . . . ,wn)(J> - B ^ , . , w ^ , . ,w n )<|), where a,B e F, w i e and i e I . ( i i ) ( w r w 2 i . . . ,wn){> - a ( g ) ( w g ^ 1 ^ , W g ( 2 ) ^ ' • ^ w g ( n ) ^ where g e G, w. 6 , and i e I . Such e l e m e n t s w i l l be r e f e r r e d t o as e lemen ts o f t y p e ( i ) o r t y p e ( i i ) r e s p e c t i v e l y . L e t T be t h e q u o t i e n t space F ( W ) / Q > w e s h a l l show t h a t T i s a ( G , a ) space. - 11 -L e t T| : P (W)— ^ T be t h e n a t u r a l homomorphism. Set T = (J)TI . Then T i s a m u l t i l i n e a r and (G, a ) f u n c t i o n . Por i f c P , t h e n ( w - ^ . . . 3OM^ + pw£, . . . * w n ) t . - a(wlS. . . , w t , . . . , w n ) r -P(w1,. . .,w^ ,. . .,wn)*C = [(w1,...,awi + Pw^ ,. . . ,wn)<J>-- a ( w 1 , . . . , w i , . . . ,w n ) ( j ) -p ( w 1 , . . . 9v'±3. . . ,w n )( j) ]T| = 0 s i n c e t h e e x p r e s s i o n i n t h e square b r a c k e t s i s o f t y p e ( i ) . A g a i n i f g~e G , t h e n (w1,w2,. . . , w n r r - a ( g ) ( w g ( 1 ^ w g ( 2 ^ . . . , w g ( n ) ) r = [ ( w 1 , w 2 , . .. awn)4) - a(g)(w g^ 1j,w g^ 2j,...,w g^ nj)4)]n 0 s i n c e t h e e x p r e s s i o n i n square b r a c k e t s i s o f t y p e ( i i ) . Thus X i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n . A l s o WT = { w t | w € W } i s a s p a n n i n g s e t f o r T , s i n c e g i v e n z i n T , t h e r e e x i s t s y i n P(W) s y = E a e weA w w where A i s a s u i t a b l e subse t o f S , such t h a t z = y T 1 = ( E a w e )n = E a w (e ")n = E a w ( w ) ^ = E ^ ( w t ) wcA we A we A we A Now, i f U i s any v e c t o r space over P and f i s any m u l t i l i n e a r and ( G , a ) f u n c t i o n o f W i n t o U , t h e n by P r o p o s i t i o n 1.33 t h e r e e x i s t s a u n i q u e l i n e a r t r a n s -f o r m a t i o n f ' : P(W)- , such t h a t (j)f' = f . - 12 -C l a i m : 0 C k e r f ' . Fo r i f y = (w^,... ,am^ + Bw^, ... *w )<|) - a(w1,.. . ,wia. . .-,w )<|) - P(w^,.. . , w ^ , . . . ,wn)(J) i s an e lement o f t y p e ( i ) , t h e n s i n c e f ' i s a l i n e a r t r a n s -f o r m a t i o n we have y f ' = (w1,...,aw1 + 8w^ ,. . . ,w n)(J ) f ' - afw^ . . . , w . } . . . J w n ) ( j ) f ' - B ( w 1 , . . . , w j , . . . ,wn)4)f' = ( w 1 , . . . , a w i + B w ^ , . . . , w n ) f - a ( w 1 , . . . , w i , . . . , w n ) f - BCw^,. .. ,w'±i. . . *w n)f = 0 , s i n c e f i s m u l t i l i n e a r . A g a i n i f y = ( w ^ W g , . . . ,wn)(J> - ( w g ( i ) > w g ( 2 ) > • • • * w g ( n ) ^ i s an e lement o f t y p e ( i i ) , we have y f = ( w 1 , w 2 , . . . , w n ) ^ f ' - < J ( e ) ( w g ( i ) ' w g ( 2 ) " ' * ' w g ( n ) ^ f ' = ( w ] _ , w 2 , . . . , w n ) f - a ( g ) ( w g ( l ) , w g ( 2 ) , . . . , w g ( n ) ) f = 0 - 13 -s i n c e f i s ( G , a ) . Thus e lemen ts o f t y p e ( i ) and ( i i ) a re c o n t a i n e d i n t h e k e r f ' and s i n c e fi i s t h e subspace g e n e r a t e d by t h e e lemen ts o f t y p e ( i ) and ( i i ) , we have Q c k e r f ' , w h i c h p r o v e s t h e c l a i m . D e f i n e 7 ' : T >\J , as f o l l o w s . , I f z e T , t h e r e e x i s t s y e F ( W ) , such t h a t z = yn. . Set (z)7 = ( y ) f ' . S ince fi c k e r f , 7 i s w e l l - d e f i n e d . 7 i s a l i n e a r t r a n s f o r m a t i o n . For i f a,f3 e F, and z-^, z 2 e T , we have z 1 = y-jj] and z 2 = y 2 n f o r some y.^ and y F(W) . Then az-^ + £ z 2 = (ay-j_ + P y 2 ) r ) and t h e r e f o r e ( a z 1 + f3z2)7 = ( a y 1 + f3y 2)f= a ( z 1 ) 7 + f3(z 2 )7 , w h i c h shows t h a t 7 i s a l i n e a r t r a n s f o r m a t i o n . A l s o i f w e W , we have (w)T7 - ( w ^ r i f = (w(())f ' = wf . Thus Xf = f . Now, l e t g : T >U be any l i n e a r t r a n s f o r m a t i o n , such t h a t 1g = f . We w i l l show t h a t g = f . I n o r d e r t o s h o w t h i s , i t i s s u f f i c i e n t t o show, t h a t g = 7 on t he s p a n n i n g s e t o f WTi o f T , s i n c e g and 7 a re b o t h l i n e a r t r a n s f o r m a t i o n s . L e t wT e W , t h e n (wT)g = (w)rg = ( w ) f = ( w ) t f = (wt)7 Hence g = 7 . T h e r e f o r e T i s a ( G , a ) space o f w and by t h e l a t t e r s t a t e m e n t o f t h e t h e o r e m p r o v e d a l r e a d y i s d e t e r m i n e d - 14 -•un ique ly up t o i s o m o r p h i s m . We s h a l l deno te T , c o n s t r u c t e d as above, by P(W,G,a) and 1 c a l l i t t h e ( G , a ) space o f W d e t e r m i n e d by G and a • 2 * 5 N o t a t i o n : I f w = (w-^Wg, . . . , w n ) e W , we s h a l l denote i t s image W£ under "X. b y w^Aw 2 A. , • A w n and c a l l i t t h e ( G , a ) p r o d u c t o f p u r e p r o d u c t o f t h e v e c t o r s w ^ W g * . . . , w n . 3 Some s p e c i a l cases o f t h e (G, a ) spaces . 3.1 L e t G = [e] where e i s t h e i d e n t i t y p e r m u t a t i o n i n S •. Then a = 1 » t h e t r i v i a l c h a r a c t e r , and W i s o b v i o u s l y a G - s e t . The ( G , a ) space i n t h e case i s t h e t e n s o r p r o d u c t o f t h e v e c t o r spaces V 1,V P,...,V , w h i c h i s n . n deno ted u s u a l l y b y <S V I I f ( w - ^ W g , . . . , w n ) e W t h e n we deno te w^Aw 2A. . . Aw n by t h e u s u a l n o t a t i o n w l® w2®' * *®^n * w h i c l 1 ^ s c a l l e d t h e t e n s o r p r o d u c t o f t h e v e c t o r s w ^ , w 2 , . . . , w n . 3.2 L e t G = S n and V, = y 2 = = V . Then W i s a G - s e t . L e t o be t h e l i n e a r . cha rac te r o f G , g i v e n by <j(g) = s i g n g , f o r a l l g e G . Then P(W,G,.a) i s t h e n Grassman space w h i c h i s deno ted by AV . I f (w-^Wg, . . . 3 w n ) e ,W, we d e n o t e w-jAWgA. . . Aw n by t h e u s u a l n o t a t i o n W^AW 2A...Aw n , w h i c h i s c a l l e d t h e skew-symmet r i c p r o d u c t o r t h e c a r e t p r o d u c t o f t h e v e c t o r s w ^ , w 2 , . . . , w n . 3.3 L e t G and W , as i n 3.2, and a = 1 , t h e t r i v i a l - 15 -c h a r a c t e r . I n t h i s case P(W,G,a) i s t h e symmet r i c p r o d u c t w h i c h i s d e n o t e d by V ( n ) . I f (w -^Wg, . . . , w n ) € W t h e n we deno te w^AWgA...Aw by t he u s u a l n o t a t i o n w l ' w 2 ' * " ' * w n , w h i c h i s c a l l e d t h e symmet r i c p r o d u c t o f t h e v e c t o r s w ^ , W g , . . . , w n . We s h a l l c a l l t h e s e t h r e e k i n d s o f s p a c e s , t he c l a s s i c a l spaces . \ - 16 -CHAPTER I I I n t h i s c h a p t e r , we s h a l l s t a t e t h e p r o b l e m o f t h e t h e s i s . The concep t o f " p s e u d o - d e t e r n i i n a h t " f u n c t i o n , a d e t e r m i n a n t l i k e f u n c t i o n i s i n t r o d u c e d , and t h i s i s used i n t he R e p r e s e n t a t i o n Theorem Form I I , w h i c h i s t h e ma in r e s u l t o f t h i s c h a p t e r . 1 Th roughou t t h i s and t h e f o l l o w i n g c h a p t e r s , we s h a l l assume, u n l e s s o t h e r w i s e s t a t e d , t h e f o l l o w i n g : G I s any p e r m u t a t i o n group on I = { l , 2 , . . . , n } . F an a r b i t r a r y f i e l d , o any l i n e a r c h a r a c t e r o f G . v\ a f i n i t e d i m e n s i o n e d v e c t o r space o v e r F , f o r each 1 i e I . W = V^xVgXv . x V n ( c a r t e s i a r i : . p r o d u c t ' i s . a. G -se t . F(W) i s t h e f r e e space ove r F , g e n e r a t e d by t h e s e t ¥ . P ( ¥ , G , a ) i s t h e ( G , a ) space o f ¥ d e t e r m i n e d by G and <j. <|) i s t h e mapp ing o f ¥ i n t o F ( ¥ ) , as d e f i n e d i n 1.2, Chapte r I . TI i s t h e n a t u r a l homomorphism o f F ( ¥ ) i n t o P ( ¥ , G , a ) X = (J) Tl • N o t e : We s h a l l see i n Remark 3.2, t h a t we may t a k e V l = V 2 * ' * = V n = V ( s a v ) ' From t h e n o n , ¥ = V x V x . . . x V ( n c o p i e s ) . - 17 -2. S ta temen t o f t h e Prob lem Given an a r b i t r a r y e lement ( V ^ V 2 J -• • J V R ) e W , f i n d a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t v-jAVgA. . . A v n = 0 . The answer t o t h i s p r o b l e m i s known i n t h e case o f t h e c l a s s i c a l spaces , d e f i n e d i n ~}} Chapter I and i s t h e f o l l o w i n g : ( i ) v-f&VQ® • = 0 l f a n d o n l y i f v i = 0 f o r some i , 1 _< i _< n . ( i i ) v-^AVgA. . . A v n = 0 i f and o n l y i f v-^Vg, ••• > v n a r e l i n e a r l y dependen t . ( i i i ) v i ' v 2 ' - , , ' v n = 0 i f a n d o n l y i f v i = 0 f o r s o m e 1 1 _< i _< n . See [1],[2],[3],[4] f o r r e f e r e n c e . 2.1 Remark: ( i i i ) i s p r o v e d i n [ l ] , under t he r e s t r i c t i o n , t h a t V i s an n - d i m e n s i o n a l u n i t a r y space. L e t P(W,G,a) be t he ( G , a ) space o f W . L e t U be any f i n i t e d i m e n s i o n a l v e c t o r space over F o f d im m , where m = max ' { ' d i m . V \ |: 1 _< i'_C n;;} u.- Le t ( • ,/,•,.•>•! ., W' = UxUx. . . x U ( n c o p i e s ) . Then we have t h e f o l l o w i n g 3. Theorem (Embedd ing ) : There e x i s t s an i s o m o r p h i s m ( v e c t o r spaces) o f P(W,G,a) i n t o P ( W ' , G , a ) , and t h i s i somorph i sm - 18 c a r r i e s i n t o W ' V , where ' i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n o f w' i n t o P ( W ' , G , a ) . P r o o f : For each i e I , l e t f ^ ; V^- >U be any n o n -s i n g u l a r l i n e a r t r a n s f o r m a t i o n , s a t i s f y i n g f ^ = f g ( j _ ) a l l g e G . T h i s i s p o s s i b l e , s i n c e d im U = m'= max d i m V±3 1 X.' i <_ n and V-^  = V g ( i ) f o r a l l i € I and g e "G Then d e f i n e f : W W' , by s e t t i n g ( w 1 , w 2 , . . . , w n ) f = ( w 1 f 1 , w 2 f 2 , . . . , w n f n ) • I n t h e f o l l o w i n g d i a g r a m , c o n s i d e r t h e mapp ing f%' t W P(W,G,a) - > P ( W ' , G , a ) . T h i s i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n . For i f a ,8 e F , t h e n f o r any i e I , ( w 1 , . . . , a w 1 + f 3 w £ , . . . , w n ) f t ' = (wJf1,. . . , ( a W j + e w ^ f ± , . . . >wn fIi) x'' = ( w n f 1 9 . . . , a w 1 f i + B w ^ f A , . . . , w n f n ) T ' , s i n c e f i i s l i n e a r = a ( w x f 1 , . . . , v t ± f ± , . . . ^ w n f n ) X ' + B ( w , f , w ' f . , w f yX' s i n c e %' i s m u l t i -1 1 1 1 n n l i n e a r . 19 -+ p(w15w2,... 3wj,... *w n ) rc ' , which shows that f t ' i s m u l t i l i n e a r . Again i f g € G , then (w1,w2,.. . ,w n)fi:' = (w 1f 1,w 2f 2,. . . > w n f n ) t ' = ( u p U g , . . . 3<un)*t' j where u i = w i f i f o r each ±, e I = c r ( g ) ( u g ( 1 ) , u g ( 2 ) , . . . , u g ( n ) ) t ' , since-*' i s (G,a) function = °(s)(wg(l)'fg(l)^g(2)fg(2)- ' ' > wg(n) fg(n) - a ( g H w g ( i ) f p W g ( 2 ) f 2 , . . . , w g ( n ) f n ) t / , since f .=f ,. s for a l l i - e I . 1 g ( i ) = a ( g ) ( w g ( l ) ^ w g ( 2 ) - - - w g ( n ) ) f e , : / ' which shows that f t ' i s a (G,a) function. Therefore 5 "by the universal mapping property, there exists a unique l i n e a r transformation fx* of P(W,G,a) into P(W,G,a) , such that T f r T = f%' . Now for each i s I , U = ^ i ^ i © (direct sum), where U' i s "a complement of \ % f 4 i n U . Define t U * v"i , as follows: If u g U , then u = v . f 1 + uf (uniquely), where v. e V. and e U J . Set ug^ = . C l e a r l y g^ i s a l i n e a r t r a n s f o r m a t i o n and a l s o f ^ g i = i , t h e i d e n t i t y map o f D e f i n e g : W'- W , by s e t t i n g ( u - ^ U g , . . . > u n ) g = ( u - j g ^ U g g g , . . . i . u n g n ) . Then f g = i t h e i d e n t i t y map o f W . For i f (W-pWg, . . . >W ) 6 W , t h e n (w1,Wg,. . . , w n ) f g = ( w ^ f - p W g f g , . • . . > w n f n ) S o ( w ^ g ^ W g f g g g , . . . ^ ^ ) = (W-^Wg, Now I n t h e f o l l o w i n g d i a g r a m P(W,G,a) P ( W ' , G , a ) gT can be e a s i l y v e r i f i e d as f t ' i n t h e p r e v i o u s d i a g r a m t o be m u l t i l i n e a r and ( G 5 a ) . T h e r e f o r e by t h e u n i v e r s a l mapp ing p r o p e r t y , t h e r e e x i s t s a u n i q u e l i n e a r t r a n s f o r m a t i o n ~g% o f P ( W ' , G , a ) , such t h a t r€~&% = g*t • We w i l l show now, t h a t fx' "gT! = 1 , t h e i d e n t i t y map on P(W,G,a) . L e t (w-^WgA. . . Aw ) e P(W , G , a ) , t h e n ( w ^ W g A . . . A w n ) f T Tr g t : = ( w - ^ W g , . . . , w n ) X frT g t = (w^Wg,. . . 3 w n ) f t / g ^ - 2 1 -= ( w ^ w g , . . . , w n ) f g ^ r = ( w 1 , w 2 , . . . ,w n ) * r ; Thus Y%r g*Cr= i i d e n t i t y map on W . Bu t s i n c e W*C i s a s p a n n i n g s e t o f P(W,G,a) and fV gf i s a l i n e a r t r a n s f o r m a t i o n , we have f°t' g°tf = i t h e i d e n t i t y map on P(W,G,a) . Hence f i r ' i s an i s o m o r p h i s m o f P(W,G,o) i n t o P(w',G,a) . A l s o i f ( w 1 , w 2 , . . . , w N ) T e WX , t h e n ( w -^Wg, . . . , w n ) t f T' = ( w 1 , w 2 , . . . , w n ) f T ' = ( w 1 f 1 , w 2 f w . , . . . , w n f n ) - C/ e W ' T • T h e r e f o r e f%' c a r r i e s W*tf i n t o W'T' . 3.1 Remark; I n Theorem 3* we may t a k e U t o be any v e c t o r space over P o f d i m e n s i o n e x c e e d i n g max { d im | I e I ] 3« 2 Remark; I n v i e w o f t h e Embedding Theorem 3j we s h a l l assume, t o make t h e p r o o f s s i m p l e r , b u t w i t h no r e s t r i c t i o n on t h e g e n e r a l i t y o f t h e P r o b l e m , t h a t = V 2 = . . . = V = V(say). So W becomes V x V x . . . x V ( n c o p i e s ) . We s h a l l a l s o assmume t h r o u g h o u t , t h a t d im V = m and br1>y2>' • - >7K) 1 s a b a s i s o f v . 4. Some D e f i n i t i o n s . (1) (w-^Wg,. . . ,w n ) ( j ) e F(W) i s c a l l e d a (G ,a) e l e m e n t , i f and o n l y i f t h e r e e x i s t s g € G , such t h a t a ( g ) + 1 - 2 2 and w. jW , . \ a re dependent f o r a l l i € I ; i . e . a ( g ) ^ 1 and d i m < £ w i * w g ( i ) a w g 2 ( i ) * « • • J w g o r d g-1 .)> = 1 f o r a l l i s I . ( 2 ) (w -^Wg, . . . ,w n ) ( j ) and ( w ^ , w 2 , . . . ,w )^<J) i n F(W) a re s a i d t o be G - r e l a t e d , i f and o n l y i f , t h e r e e x i s t s g e G , such t h a t w^ = w g ( j _ ) f °r a ± l I e I . ( 3 ) ( w 1 , w 2 , . . . ,wn)(J) e F(W) i s s a i d t o s a t i s f y t h e p r o p e r t y P , i f and o n l y i f , f o r each i € I , I > 2 , e i t h e r w. e ( w. | 1 < j < i ] o r w. i s i n d e p e n d e n t o f t h e s e t { w. | 1 < j < i ) . 1 J ( 4 ) ( w ^ , w 2 , . . . , w n ) € W i s c a l l e d a t r i v i a l e l e m e n t , i f and o n l y i f w^ = 0 f o r some i € I . O the rw ise i t i s a n o n - t r i v i a l e l e m e n t . 5. Theorem ( R e p r e s e n t a t i o n Theorem Form I ) ; I f (w -^ ,w 2 , . . . ,w ) e W i s a n o n - t r i v i a l ' e l e m e n t , t h e n ( w - ^ , w 2 , . . . ,w n ) ( j ) can be k w r i t t e n i n t h e f o r m ( w - ^ , w 2 , . . . ,wn)<J) = w + Z c i ^ i > ^ o r some p o s i t i v e i n t e g e r k , where w e n , c^ e F , T^ € F(W). For each i , 1 <_ 1 <_ k, T^ ' s a t i s f i e s t h e p r o p e r t y P and i f i | j , t h e n T. and- T . a re n o t G - r e l a t e d . J. J P r o o f t L e t { y i 3 y 2 J * ' ' , y m ^ 1 ° e a " b a s i s o f v • F o r e a c n m i i € I , l e t w, = E t>. .ry. , and s e t A. = { j | 1 < j < m , b . . 4 0 } . S ince ( w , , w p , . . . , w ) i s a n o n - t r i v i a l l , j i. c. n e l e m e n t , A^ 4 a f o r a n v i € I . L e t S = A - ^ x A g X . . . x A n , - 23 -t he c a r t e s i a n p r o d u c t . I f s e $ and s = ( s 1 , s 2 , . . . , s n ) , l e t b g = b ^ g b 2 , s, , ' , b n , s ' C l e a r l y b g + 0 f o r any S € S . D e f i n e a r e l a t i o n " ~ " on S as f o l l o w s : I f s = ( s p S g j . - . j S j j ) and t = ( t - ^ t 2 > . . . j t n ) a re i n S, t h e n s~t i f and o n l y i f t h e r e e x i s t s g e G , such t h a t ( t 1 , t 2 , . . . , t n ) = (s g ( i ) ^ 3 g ( 2 ) ^ ' ^s g ( n ) ^ i , e ' * i = s g ( i ) for a l l i s I . C l e a r l y " ~ " i s an e q u i v a l e n c e r e l a t i o n on S . I f s c S , l e t A ( s ) deno te t h e e q u i v a l e n c e c l a s s c o n t a i n i n g s . Then { A ( s ) | s e S } i s t h e s e t o f t h e e q u i v a l e n c e c l a s s e s . L e t E be a s e t c o n s i s t i n g o f a r e p r e s e n t a t i v e o f each o f t h e e q u i v a l e n c e c l a s s A ( s ) . Now (wl5w2J...,wn)<i> = ( ^ b ^ j y j , ^ *2}3yy--£l\,fj)$> S l e A l 1 ^ S 2 € A 2 2 2 Z „ V s y s )(J) s n€A, 1 , s l S l s 0€Ao 3 2 S2 S b n s y s s neA n n ' s n s n to-, a ( y s , • S to s y s J ° 1 * A 1 1 1 S 2 e A 2 2 2 E b n s y s s n c A n n > S n s n - 24 + -S b,. [(y_ , Z b 2 y s 1 e A 1 1 , s l a l s 2 e A 2 ^ , s2 s2 0S ^ A b n , s y s „ ' s n e A n n n s E A b2,s ( ys ' y s S A V s y s M 2€A 2 ' 2 °1 °2 s n € A n n n + + + + 2 b, E b , [(y , y y s ' Z b n s y s ^ s n - l s n € A n n ' s n s n S b n Q ( ys ' y q " • • ' y s >ys M s neA n n > 3 n s l s2 s n - l s n S l e A l 1 S 2 e A 2 2 s n e A n n ( y Q ....,y. )q> "1 s2 s n since each of the terms within the square brackers i s i n 0, <, we have (w1,w2,...,wn) = «)0 + S. b, I b 2 ... I b s 1eA 1 1 , s l 3 2eA 2 2>s2 " ' s ^ i ^ n ' s n s l fc2 s n - 2 5 -i e l = t» + E b ( y , y . . , y )<j) , 0 seS s s l 2 s n where UD i s t h e sum o f t h e t e r m s w i t h i n t h e square b r a c k e t s and i s i n 0 . Now i f t e A ( s ) , t h e n t ~ s , w h i c h i m p l i e s t h a t t h e r e e x i s t s g e G such t h a t ( t - ^ t g * . . . » t n ) = ( 8 g ( ' l ) , 8 g ( 2 ) " " , B g ( n ) ) s Z± = S g ( i ) f ° r a 1 1 1 € 1 o r t g ^ i ^ i ) = s i f ° r a H i € I . Hence ( y + * y f 3 . . . * y t )4> = [ ( y f * y t > . . . * y + )4> t l z2 z n z l z2 z n a ( g _ 1 ) ( y t , y t 3 . . . * y t _ ) )4>] -1 + o ( g " ) ( y + , y t * . . . * y t n \<h t g " 1 ( l ) t g " 1 ( 2 ) V ^ n j W = K t ^ g " 1 ) + a ( g ~ 1 ) ( y q , y , . . . , y q s l s2 s n where i u ( t , g " ) i s e q u a l t o t h e t e r m w i t h i n t h e square brackets, which b e i n g o f t y p e ( i i ) i s i n fi . T h e r e f o r e (w n 5 w 0 , . . . ,w )<j) = ui + E E b. u ^ t ^ g " 1 ) 1 ^ n ° seE t e A ( s ) z + E ( E o ( g " 1 ) b + ( y , y , . . . , y _ )$ ) . ssE t e A ( s ) x s . l s2 s n S a t m + E E b. ^ ( t ^ g " 1 ) = UD, and E a ( g " " 1 ) b . = b ' 0 ssS t e A ( s ) Z ' t e A ( s ) Z S - 26 -C l e a r l y UJ € Q . Then (w-^ Wg,. . . ,vn)ty •= u>+ £ l ' s ( y s , y s »'"»y s ^ Now i f s = ( s ^ S g , . . . , s n ) and t = ( t - ^ t g , . . . , t n ) a re i n S 3 t h e n ( y g ,y ,...,y )<j) and (y t ,y t ,...,yt )<j) 1 2 n 1 2 , n a re G - r e l a t e d i f and o n l y i f t h e r e e x i s t s g e G , such t h a t ( y . 3 y . 3 y )<t>=(y_ ,y *--->ya )4> 5 *1 Z2 z n s g ( l ) S g ( 2 ) s g ( n ) i . e , 1 i f and o n l y i f ( y t ,y ,...,yt.) ~ 1 2 n = ( y G 5 y Q ,...»y. s i n c e (j) i s o n e - o n e j i . e . i f S g ( l ) S g ( 2 ) S g ( n ) and o n l y i f y . = y „ f o r a l l i e I ; i . e . i f and t i S g ( i ) o n l y i f t i = s g ( i ) f o r a H i e I , s i n c e y ' s a re t h e b a s i s e l e m e n t s ; i . e . i f and o n l y i f t ~ s ; i . e . i f and only i f t € A ( s ) . Thus i f s and t e E and s + t , t h e n ( y , y ,...,y<l )4> and ( y . ,y. ,y. )<|) a re n o t s l s 2 s n z l z2 z n G - r e l a t e d . A l s o f o r each se S , and i n p a r t i c u l a r f o r each s e E 3 ( y „ 3 y 5 . . . , y _ s a t i s f i e s t h e p r o p e r t y s l s 2 s n (w^Wg,. . . >wn)4> = UJ + E b g T g , s s E where T = (y„ 5y_ 5...,y )(J) has the required form. s D l s2 s n - 27 -Compu ta t i on o f t h e C o - e f f i c i e n t s . I n t h i s s e c t i o n , we s h a l l i n v e s t i g a t e t h e c o - e f f i c i e n t s t>' , s e E , a p p e a r i n g i n t h e R e p r e s e n t a t i o n Theorem 5. s C o n s i d e r t h e f o l l o w i m g nxm m a t r i x M = / b l , l b l , 2 b 2 , l b 2 , 2 'l,m \ '2,m b n , l b n , 2 n,m where w. T* = 2 b ' .y,- , f o r i = l , 2 , . . . , n . 1 j = l 1 , J J We s h a l l c a l l M t h e m a t r i x o f c o - e f f i c i e n t s o f t h e e lement ( w ^ , w 2 , . . . , w n ) e W . For each s = ( s 1 , s 2 , . . . , s n ) i n S , we d e f i n e an n x n matrix M„ y o b t a i n e d f r o m M as b-1,8! 2, S l 1 ^  S / b 2 ,s , l , s n ' 2 , s n n?s^ n , s 2 n , s. n and a subset H o f G as s H g = { g | g e G , a ( g ) = 1 and s ± = s g ^ Clearly H_ i s a subgroup o f G . s _ , . \ f o r a l l i e I - 28 -6.1 P r o p o s i t i o n : I f s = (a^,s2>...,sn) and t = ( t ^ t g , . . . , t ) a re i n S and s~t t h e n H a n d . H, . a r e c o n j u g a t e . I n f a c t i f t ± = ^g^ j_) f 0 r S O m e S e G a 1 1 1 ^ I , t h e n H t = g " 1 H. g . : -P r o o f : S ince s~t , we have t ± = S g ^ ) f ° r some g e G and a l l i e I . Now i f h € H t , t h e n o ( h ) = r and ' ^ 1 = ^ h ( i ) ^ o r a ^"^ i G I , b u t s i n c e t ^ = s g ( i ) > w e have a ( h ) = 1 and s g ( i ) = s g h ( i ) f o r a 1 1 i G I i . e . a ( h ) = 1 and s^ = s g h g - l ( i ) f 0 1 " a 1 1 i G I . A l s o s i n c e a ( g h g - 1 ) = a ( g ) a ( h ) . . a ( g ) " 1 . '= .1 , we have ghg" " 1 e H g ', i . e . h e g'^gg • Thus H t c g _ 1 H s g and s i m i l a r l y one can show g~"4 l s g c H t . T h e r e f o r e H t = g_ 1 H g g . 6.2 P r o p o s i t i o n s L e t s = ( s ^ S g , . . . , 8 ^ ) and t = ( t ^ , t 2 , . . . , t n ) be i n S . Suppose s~ t and l e t g , h e G such t h a t t . = s i.\ and t . = s, , . \ f o r a l l i e I . I f i g ( i ) l h ( i ) (y«. >y0 * « . . * y e H is n o t a ( G , a ) e l e m e n t , t h e n s l s 2 s n o ( g ) = a ( h ) . P r o o f . S ince t ± = s g ( l ) = s h ( i ) f Q r a l l i e x ^ w e h a v e s g h - i ( i ) = s i f o r a 1 1 1 € 1 3 w h i c h i m P l i e s y S i = y " g h - i ( i ) for a l l i £ I . Bu t s i n c e ( y , y , . . . , y )§ i s n o t a s l s 2 s n ( G , a ) e l e m e n t , we must have a(gh"" '") = 1 i .e . . a ( g ) = a ( h ) , - PQ „ For each s e S , we have a s s o c i a t e d a m a t r i x M o b t a i n e d f r o m M • s and a subgroup H_ o f G i n 6. C o n s i d e r t h e c o s e t s d e c o m p o s i t i o n o f G w i t h r e s p e c t t o H g and l e t G g be a s e t o f r e p r e s e n t a t i v e s o f t h e s e c o s e t s . A l s o l e t 1 = { M e j s € S } . s D e f i n e D : A ^ F , as D(M ) = Z a ( h " 1 ) b-, b 2 . . . b n s h e G s 1 ^ s h ( l ) 2 j S h ( 2 ) n ' s h ( n ) We must show t h a t D i s w e l l - d e f i n e d j i . e . , i t i s i n d e p e n d e n t o f t h e c h o i c e o f t h e r e p r e s e n t a t i v e s o f G w i t h r e s p e c t t o H •. I f h and h ' a re two r e p r e s e n t -a t i v e s o f t h e same c o s e t , t h e n hh ' " " 1 " e H , w h i c h i m p l i e s s t h a t a(hh '~" ' " ) = 1 and s^ = s h h ' ~ l ( i ) * " o r 1 e I J i . e . , o ( h ) = a ( h ' ) and s h ( i ) = s h ' ( i ) a ^ i 6 1 • -1 T h e r e f o r e a ( h ) b n _ b 0 • „ . . . b , „ b~ _ . . . b h ' ( l ) 2 > V ( 2 ) ' " n > V ( n ) Thus i f G' i s any o t h e r s e t o f t h e r e p r e s e n t a t i v e s o f t h e c o s e t s o f G w i t h r e s p e c t t o H , t h e n Z oih"1) b-. b 0 - c . . . b h s G g 1 , S h ( l ) u-'sh(2) n ' s h ( n ) = Z a f h ' " 1 ) bp . . . b a h-'sC^ J - ^ s h ' ( l ) ^ V ( 2 ) n 5 ° h ' ( n ) w h i c h shows t h a t D i s w e l l d e f i n e d . - 30 -4 D e f i n i t i o n : The f u n c t i o n D on ^ i n t o F , as d e f i n e d i n 6.3 i s c a l l e d a p s e u d o - d e t e r m i n a n t f u n c t i o n . 5 P r o p o s i t i o n : F o l l o w i n g t h e n o t a t i o n o f t h e R e p r e s e n t a t i o n Theorem Form I , i f s € E , t h e n b g = D ( M g ) , where b ' i s t h e c o - e f f i c i e n t o f ( y , y , . . . , y _ . F u r t h e r s l s2 s n i f ( y g > y g f . » y B )<t> i s n o t a ( G , a ) e lement and 1 2 b n t e A ( s ) , t h e n D(M^) = a ( g ) D ( M g ) , where g e G , such t h a t t i = s g ^ i ) f o r a l - L i € I . o o f : I f s = ( s , ) e S and g e G , we s h a l l w r i t e S g = ( s g ( D j S g ( 2 ) - - - s g ( n ) ) ' The e q u i v a l e n c e c l a s s o f S , c o n t a i n i n g s € S i s A ( s ) = [ t j t e S and t ~ s } = { s | s e S and g c G } . Bu t i f Sg and s^ a re i n A ( s ) and i f g and h a re i n t h e same c o s e t o f G w i t h r e s p e c t t o H_ , t h e n s s /4\ ~ s, , . \ f o r a l l i € I ; i . e . , s „ = s, . g ( i j h ( i ; g h T h e r e f o r e A ( s ) = { s_ | s _ e S and g e G_ } . g g s l o w b ' = Z a ( g - 1 ) b . , where g e G such t h a t t = s_ b t s A ( s ) z g b y e q u a t i o n ( 1 ) (see R e p r e s e n t a t i o n Theorem Form I ) = Z a ( g ~ 1 ) b , t b 2 . . . . b . t € A ( s ) 1> Z1 d i % 2 n > z n - 31 = s ^ S ° < « - ) i 3 g ( l ) b 2 , s g ( 2 ) - ' - V s g ( n ) • But b , „ b c o • - . " b ^ • = 0 , i f and o n l y i f 1 , 8 g ( l ) 2 ^ S g ( 2 ) n ' s g ( n ) b . , = 0 f o r some i e I ,•if and o n l y i f s I A i , s g ( I ) S t 1 ' * A i f o r some i ; i f and o n l y i f s_ 4 S . g T h e r e f o r e b ' = E a ( g _ 1 ) b , b 0 e , geG s g ( l ) C , b g ( 2 ) ; " u n , s g ( n ) = D(Mg) , which p r o v e s t h e f i r s t a s s e r t i o n . S ince t e A(s) we have t = s_ f o r some g e G_ . Then g s D(M.) = E aih"1) b , +. b« + • ..t>_ + > where G, t h € G t l j t h ( l ) 2 ' * h ( 2 ) n ^ h ( n ) . Z i s a set o f r e p r e s e n t a t i v e s o f H^ i n G . S ince t i = s g ( j _ ) f o r a l l i e I } we have * h ( i ) = s g h ( i ) ^ o r a 1 1 i € I • Hence D(M +.) = E aCh" 1) b , e b 0 o . . . b , -(2) t h € G t ^S g h ( l ) 2 > s g h ( 2 ) n > s g h ( n ) By P r o p o s i t i o n 6 .1, we have H g = gH^g""1" , and t h e r e f o r e [G : G s ] = [G : G t ] . C l a i m ; G' = { gh | h e G + } i s a s e t o f r e p r e s e n t a t i v e s of H i n G . s Clearly | G ^ | = [G : H i . A l l t h a t i s n e c e s s a r y i s t o show t h a t two d i f f e r e n t r e p r e s e n t a t i v e s o f G^ g i v e r i s e to two d i f f e r e n t r e p r e s e n t a t i v e s o f G' . So i f gh and s - 32 -g h ' s h o u l d b e l o n g t o t h e same c o s e t o f G w i t h r e s p e c t t o H , t h e n ( g h ) ( g h / ) " 1 € H , w h i c h i m p l i e s S S h h #-1 - ~~1 € g~ H g = H t b y P r o p o s i t i o n 6.1; i . e . h and h ' must b e l o n g t o t h e same c o s e t o f G w i t h r e s p e c t t o H > w h i c h w o u l d c o n t r a d i c t t h e s u p p o s i t i o n t h a t h and h ' b e l o n g t o G^ . T h e r e f o r e (2) becomes D(M ) = Z ^ " X - > \ s b-2 B .-..b n s. x Z h € G t aig"1) 1 , s g h ( l ) d)3&i(2) n ' s g h ( n ) i-=- S oik"1) b , • h 0 a . ..b a U " 1 ) k€G s l i S k ( l ) 2 > s k ( 2 ) n ' s k ( n ) where k = gh = — - ~ n D ( M S ) * b y t h e d e f i n i t i o n o f D i n 6.3 a ( g " ) = o ( g ) D(M ) » w h i c h comp le tes t h e p r o o f . 7. R e p r e s e n t a t i o n Theorem Form I I ; . I f (w-^Wg* • • • * w n ) I s a n o n - t r i v i a l e lement o f W , t h e n ( w -^Wg, . . . ,,wn)<|) can be w r i t t e n i n t h e f o r m ( w 1 , w 2 , . . . ,w n )( f ) = u) + I D ( M s ) ( y g ^ , y g ^ . . . , y g )(J) -(3) where UJ € ft , and f o r each s € E , ( y >y >.•.>y )<j) s l s2 s n s a t i s f i e s t h e p r o p e r t y P and i f s , t e E , s 4 " t , t h e n (y, 5 y c j . . . > y a )$ and ( y . , y , . . . , y . )<J) a re n o t & . l s2 s n x l r2 t n G - r e l a t e d . - 33 -We s h a l l c a l l (4), a r e p r e s e n t a t i o n o f (w-^Wg* • . .w n ) ( j ) w i t h r e s p e c t t o t h e b a s i s [ y ^ y g j • • • > y m } o f V . 7.1 Remark: I f E' i s any o t h e r s e t o f r e p r e s e n t a t i v e s o f t h e e l e m e n t , t h e n D(Mg) and D ( M g / ) a re r e l a t e d by D(Mg/) = a ( g ) D(Mg) , where s ' = s g f o r some g e G . 7.2 Remark: I f s = (s-^Sg, ... ,sn) e S , t h e n ( y s > y s * • • • * y s i s a ( G , a ) e lement i f and o n l y i f , t h e r e e x i s t s g e G such t h a t o-(g) 4 1 and s = s . g P r o o f : Immedia te f r o m t h e d e f i n i t i o n o f a ( G , a ) e lement and t h e f a c t s t h a t ( y , y ,. . . ,y )§ s a t i s f i e s t h e p r o p e r t y s l s2 s n P and y - ^ y g * • • • > y m i s a 1 1 i n d e p e n d e n t s e t o f v e c t o r s . e q u i v a l e n c e c l a s s e s , t h e n Is a n o t h e r r e p r e s e n t a t i o n . By P r o p o s i t i o n 6.5* i f s ' e A(s) and ( y , y , . . . > y . )<|> i s n o t a ( G , a ) s 1 s 2 s n - 34 -CHAPTER I I I I n t h i s c h a p t e r , we c o n t i n u e our s t u d y o f t h e p r o b l e m . The main r e s u l t s a r e Lemma 2 . 1 and Theorem 2 . 3 . Then:, we d e t e r m i n e a b a s i s o f t h e (G,a) space P(W,G,o) . We s h a l l conc lude t h i s c h a p t e r , by s t u d y i n g t h e p r o b l e m , i n case when G can be w r i t t e n as a d i r e c t p r o d u c t o f i t s s u b g r o u p s ; i . e . , G = G-j® G 2 & . . . & G n ( d i r e c t p r o d u c t ) , such t h a t t h i s d e c o m p o s i t i o n i s " d i s j o i n t " . 1. . C o n s t r u c t i o n o f a m u l t i l i n e a r and ( G , q ) f u n c t i o n . L e t ( v - ^ , v 2 , • • . , v n ) be a n o n - t r i v i a l e lement o f W . I f f o r each i e I , v . = § a . .y . .. , c o n s i d e r t h e s e t s A. x x , j j x and t h e s e t S = A ! X A p X . . . x A n , as d e f i n e d i n t he R e p r e s e n t a t i o n Theorem Form I . I f s e S , d e f i n e f ; w » F , as f o l l o w s . s / I f ( w 1 > w 2 , . . . , w n ) € W and = ^ b i , j y j ' • f o r i = l , 2 , . . . , n , t h e n s e t ( w , , w p , . . . , w ) f = s a ( g "1 ) b , bp . . . b . 1 2 n 5 geG Q 1 , 8 g ( l ) 2 , S g ( 2 ) n ' S g ( n ) . where G i s a s e t o f r e p r e s e n t a t i v e s o f t h e c o s e t s o f H s s i n G . One can e a s i l y show t h a t f i s a w e l l - d e f i n e d ' ^ s f u n c t i o n , i . e . i n d e p e n d e n t o f t h e c h o i c e o f G g . Then we - 35 -have t h e f o l l o w i n g 1.1 Lemma; f i s a m u l t i l i n e a r and (G,a) f u n c t i o n . P r o o f : I f a , p e F , t h e n f o r any i e I , i f w., = 2 b . . y . i J = 1 i , J J and w' = § b j . y . we have ( w 1 3 . . . , a w , + B w ! , . . . , w ) f j = = S a (g~ )b-, „ b 0 „ . . . ( a b . _ +Bb.' „ ) • • •*>„•«, ^ ^ S g ( l ) 2 ' S g ( 2 ) ^ s g ( i ) ^ s g ( i ) n ' S g ( n ) = a S a ( g " ) b n „ b 0 „ . . . b . „ • • • ' b „ „ g £ G s ^S g ( l ) 2 ' s g ( 2 ) ^ s g ( i ) n > s g ( n ) + P E a ( g ~ 1 ) b , „ b 0 _ . . . b . ' e 0 g € G s ^ ( l ) 2 > S g ( 2 ) ^ s g ( i ) n ' s g ( n ) = a ?B(wia.. • J W ^ . . . » w n ) f . s + B f s ( w 1 , . - . . , w ^ , . . . , , w n ) f s , w h i c h shows f i s m u l t i l i n e a r . A g a i n , i f h e 0 , t h e n ( w h ( 1 ) > » h ( 2 ) » • • • . » h ( n ) ) f s - E a ( g - X ( l ) , s 6 ( 1 )b h ( ^ ) , B g ( 2 ) - - - b h ( n ) 5 s g ( n ) -s = S a(g~"*")b, „ b 0 „ . " - ' " b ^ „ g€G< 5 1 ^ S g h " l ( l ) 2 ^ s g h - l ( 2 ) n ' s g h ~ l ( n ) = S °1±L2 TO b . . . b keG^ a ( h ) l a S k ( l ) < ^ s k(2) n ' s k ( n ) where k = g h " 1 , and G' = { g h " 1 | g e G c } i s a n o t h e r S S set of r e p r e s e n t a t i v e s o f t h e c o s e t s o f G w i t h r e s p e c t to H f l . Hence ( w h ( l ) , w h ( 2 ) , . . . , w h ( n ) f g = — L . S a ( k "1 ) b 1 b p c . . . b a ( h ) k s G ! X ' s k ( l ) ^ s k ( 2 ) n ' b k ( n ) - 36 = " 7 ^ ( w l ' w 2 ' " " w n > f s a ( h ) w h i c h shows f i s : ( G , a ) . 2. S o l u t i o n t o t h e P rob lem We now come t o our main r e s u l t . We need t h e f o l l o w i n g lemma,, w h i c h i s a l s o a s p e c i a l case o f our p r o b l e m . 2» 1 Lemma^ L e t (vjL>v2>'' * * v n ^ € ^ such t h a t (v2.'v29'' ' , v n ^ s a t i s f i e s t h e p r o p e r t y P . Then a n e c e s s a r y and s u f f i c i e n t c o n d i t o n f o r v-jAVgA.. . A v n = 0 i s t h a t (v-^Vg,... , v )<|) i s a ( G 5 a ) e l e m e n t . P r o o f ; ( S u f f i c i e n c y ) . I f . (v^Vg, • • • ,v )<J> i s a (G 5a) e l e m e n t , t h e n t h e r e e x i s t s g e G such t h a t a ( g ) 4 1 and v i ' V g ( i ) a r e dependent ^ o r a 1 1 i e I . A l s o s i n c e ( v l ' v 2 * ' * * * v n ^ s a t i s f i e s t h e p r o p e r t y P , we have v i = vg ( - j _ ) f o r a x l i e I • Hence ( v - ^ V g * . . . >vn)<|) = L ( v r v 2 , . . . ,vn)4> - a ( g ) ( v g ( 1 ) , v g ( 2 ) , . . . , v g ( n ) )4>] + a ( g ) ( v g ( 1 ) , v g ( 2 ) , . . . , v g ( n ) ) ( | ) = u> + a ( g ) ( v 1 , v 2 , . . . ,vn)(j) , where UJ i s t h e t e r m w i t h i n t h e square b r a c k e t s . S ince a ( g ) 4 1 > w e have ( v 1 , v 2 , . . . , v n ) 4 ) = — € n . l - a ( g ) Hence v-jAVgA.. . A.v = 0. ( N e c e s s i t y ) . Suppose f a l s e . ~ 37 -Choose ctJL = 1 and d e f i n e a i i n d u c t i v e l y as f o l l o w s ; a 2 i s t h e f i r s t i n d e x j such t h a t v j 4 v ^ J a v i s t h e f i r s t i n d e x j , ' such t h a t v . 4 any one o f v - , , v , . . . , v . I f t h e s e a re p r e c i s e l y k d i s t i n c t 1 a 2 v e c t o r s v^ , we have d e f i n e d 1 = a-^ <a2<« • • <ot^ _< n . C l e a r l y { v , , v , . . . , v } i s an i n d e p e n d e n t s e t o f 2 1c v e c t o r s . Ex tend , t h i s t o a "basis { y - ^ y g * • • • * y m } o f V , such t h a t y . = v , i = 1,2,'. . .,k _< m . a i Then f o r each i e I , i f i = otj f o r some j , m • v± = £ a i j > ^ y ^ > where a A ^ = <^  1 i f i = a j 0 i f 4 4 a j I f a j < i < a j + 1 , t h e n v i = v a , f o r s o m e o" _£ J • „ ( l i f 4 = a y I n t h i s case v . = E a . y where a . . =i ° i l = 1 l i ,< , (g i f -t + a . , And f i n a l l y i f a < i < n , t h e n v . = v^ f o r some m J 1 i f I = a,, j' < k , t h e n v , = S a , y , where a . =•) J " 1 1=1 x ^ 4 x ^ (0 i f I 4 ct . / Thus , i n e v e r y case A i i s a s i n g l e t o n , v i z . { j } i f i = a. A± = J £ j ' } i f a j < i < a j + 1 where j ' <_ j [ j ' 3 i f a k < i _< n , where j ' _< k . T h e r e f o r e S = A - ^ x A g X — x A n = { s ) s a y , where s a = j , - 38 j = 1,2,. . . , k . Thus ( v ^ v g , . . . 5v n )( j ) = ( y s , y S 2 > • • • , y s )4> and i s n o t a ( G , a ) e l e m e n t , by our a s s u m p t i o n . T h e r e f o r e by t h e Remark 7 .2, Chapter I I , g e G i m p l i e s g ( g ) = 1 o r s . 4= s , . \ f o r some i e I . 1 T g ( i ) D e f i n e f j W-s -^ P , as i n 3> by -1-( w , , w p , . . . , w ) f = S o ( h " ) b T b 2 • • • b r i c , > 1 2 n s heG e ^ ( l ) 2 ' S h ( 2 ) n ' s h ( n ) s where w. = § b. . y . f o r a l l i e I . By Lemma 1 . 1 , f_ i s m u l t i l i n e a r and ( G , a ) f u n c t i o n . T h e r e f o r e by t h e u n i v e r s a l mapp ing p r o p e r t y , t h e r e e x i s t s a un ique l i n e a r t r a n s f o r m a t i o n f \ o f P(W,G,a) i n t o P , w h i c h makes t h e f o l l o w i n g d i a g r a m W c o m m u t a t i v e ; i . e . t ? = f ( v v v 0 , . . . , v _ ) f = S a ( h "1 ) a - 1 a 2 . . . a 1 " n fa heG e ^"hCl) ^ S h ( 2 ) n ' s h ( n ) But i f h 6 G 0 and h 4 H_ , t h e n e i t h e r a ( h ) | 1 or s s s i ^ s h ( i ) f o r s o m e 1 € 1 3 X 1 ( 1 s i n c e ( y S ] , 'y s 2 * " ' , y s ^ r s :(W,G,a) Now i s n o t a ( G , a ) e l e m e n t , we have s^ 4= s h ( i ) ^ o r s o m e 1 e I - 39 Therefore s n ( i ) 4 \ since A^ = {a±} . Thus a. „ = 0 and therefore a, • . a 0 „ ...a „ = 0 i ' s h ( i ) 1 ^ s h ( l ) 2>sh(2) n' sh(n) for every h , h e G_ and h 4 H_ . Therefore (v,,v 2,... ,v )fs = Z a(h'" 1)a 1 ap s , .••• a n « , . 1 d n heG s ± , 8 h ( l ) d> sh ( 2 ) n' sh(n) = a, „ a 0 „ ...a_ „ where h € H ^ S h ( l ) 2> Sh ( 2 ) n' sh(n) s = a l , s 1 a 2 , s 2 " - a n , s n = 1 4 o • But since T f g = f g » we have ( v - ^ V g , — •»vn^^s ^ 0 a n d since ? i s a li n e a r transformation, we have s (v-^Vg,. . . ,v ^ + 0 5 i . e . , v^VgA. ..Avn 4= 0 , which i s a contradiction. 2.2 Remark: I f ( V-^,Vg, ..., v^ ) e ¥ i s a t r i v i a l element, then v-^ AVgA. . . Av n = 0 . Proof: Let. v^ = 0 for some i € I . Then (v^... 9v ± f...>v n)X = ( v i a . . . , 0 v ± , . . . * v n ) t where 0 i s the zero element of the f i e l d F ; and since X i s m u l t i l i n e a r , we have ( v x , . . . , v ± , . . . . v n ) t = 0 ( v ! , . . . , v 1 , . . . , v n ) t = 0 (v^VgA. . • Av n) = 0 . . ' In t h i s case, we s h a l l say that v-jAVgA. . . Av n = Gv t r i v i a l l y . The Main Theorem. We can now s t a t e and p r o v e t h e main r e s u l t o f t h e T h e s i s . 2.3 Theorem; Suppose ( v ^ v 2 , . . . , v ) € W i s a n o n - t r i v i a l . e l e m e n t . Let ( v v v 2 , . . . , v )<f) = u) + Z D(M_) (y , y . . , y _ )4> (4) 1 ^ n ssE • 8 s l • s2 3 n be i t s r e p r e s e n t a t i o n w i t h r e s p e c t t o some b a s i s ( y ^ , y 2 j . . . j , y m } o f V . Then a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r v ] A v 2 A . . . A v n t o be z e r o i s , t h a t f o r each s s E , e i t h e r (y_ , y e , , . . . , y _ )(|) i s a ( G , a ) e lement or S i S /-> s_ 1 ^ n D ( M „ ) i s z e r o . P r o o f ; L e t E' = { s | s e E , (y- , y . . , y_ )<j) i s n o t a — s l s 2 s n ( G a a ) e lement } • E ' may be an empty s e t . Then (4) becomes ( v 1 , v 2 , . . . , v n ) ( t ) = + D (M S ) ( y s , y f l , . . ' . , y 8 )4> S€rj—JD ± c n + 2 D(M ) ( y , y s , . . . , y . -(5) S€E j 1 2 n We s h a l l p r o v e t h e s u f f i c i e n c y f i r s t . E -E ' i s t h e i n d e x s e t t h a t s e l e c t s t h e n o n - v a n i s h i n g te rms i n t h e sum (5) . Thus ( v v v p , . . . , v )(J) = in + z D ( M ) ( y _ - , y _ . , . . . * y q )<K (6) 1 d n S€ E - E ' S s l s 2 s n Now i f s e E -E ' , t h e n ( y o , y , i . . , y _ )(|) i s a ( G , o ) • s l s 2 s n e l e m e n t . A l s o i t s a t i s f i e s t h e p r o p e r t y P . T h e r e f o r e - 41 -by t h e Lemma 2.1 y A y o A . . . A y _ = 0 . Thus on a p p l y i n g s l s2 s n r\ t o (6), we o b t a i n v-^AVgA.-.AVj^ = 0 . To p r o v e n e c e s s i t y , we assume i t t o be f a l s e ; i . e , suppose t h e r e e x i s t s s e E' , such t h a t D(M_) 4 0 s D e f i n e f s w- - ^F , as i n 1, by -1' (wvw2,...,w ) f = E a ( h " )b-, b 2 q • --K s 1 d' n 8 h€Gs 1 ^ s h ( l ) d>ah(2) n ' s h ( n ) where w. = E b . , y , , f o r a l l i e I . 1 3=1. 1 , J J Then f i s m u l t i l i n e a r and (G,o) > b y Lemma 1.1 and s t h e r e f o r e by t h e u n i v e r s a l mapp ing p r o p e r t y , t h e r e e x i s t s a u n i q u e l i n e a r t r a n s f o r m a t i o n fQ t P(W,G,a) : — - ^ F , w h i c h makes t h e f o l l o w i n g d i a g r a m P(W,G ja ) c o m m u t a t i v e ; i . e . t f „ = f ' s s Now s i n c e i n (5)» f o r each s € E-E' , ( y , y 3--->Yc )i> s l s2 s n i s a ( G , a ) e l e m e n t , and i t a l s o s a t i s f i e s t h e p r o p e r t y P , t h e r e f o r e , by Lemma 2.1, y Ay_ A . . . A y = 0 . Hence, s l s2 s n on a p p l y i n g -n, t o (5), we o b t a i n 0 = E D(M ) ( y , y , . . . , y a )T , and w h i c h under ? s l'" s2 3 n - 42 -becomes Z D(M_) ( y R ,y„ , . . . , y _ Ytf. = 0 . 1 S i n c e seE ' 3 s l s 2 s n s 'K? = f „ s t h i s becomes s s = D ( M 8 ) ( y ,y ,...,y ) f , = 0 i (7) ssE i d n Now we c a l c u l a t e each t e r m o f t h i s sum. F i r s t , we choose s e E' , f o r w h i c h D ( M g ) 4 0 - We know such .an s e x i s t s by our a s s u m p t i o n . Then (y« >y„ * . - . * y c ) f a = £ aO*"1) c o c • • • c s « , v s l s 2 s n S heG_ s l , f l h ( l ) s 2 * s h ( 2 ) V s h ( n ) where c " I ^ h ( i ) ' 1 i f s . = s h ( i ) o i f s ± 4 s h ( i ) f o r a l l i e I . And s i n c e s e E' , (ya , y a * . . . * y f l )<j) i s n o t a ( G , a ) e lement we have s l s 2 s n 1 i f h e H c c . . . c ~ \ by t h e s l ' ° h ( l ) s 2 ' s h ( 2 ) " V s h ( n ) 1.0 i f ' h 4 H g Remark 7 . 2 , Chapter I I . T h e r e f o r e ( y g > y S o * • • • * y s )f s = 1 • — • - ( 8 ) . Next for.any t e E' and t 4 s , we have f o r e v e r y h e G,„ CD z ± ^ s h ( i ) ^ 0 r s o m e i e I . T h e r e f o r e ( y t 3 y + 9 - . - » y f ) f „ = £ a ( h ~1 ) c. c. . . . c. e -r l t 2 ^ 3 V s h ( l ) V s h ( 2 ) V " h ( n ) = 0 „ (9) But (8) and (9) c o n t r a d i c t (7)> and t h i s >completes t h e p r o o f . Thus I f ( v p V g , • • • » v ) 6 ¥ and i s a t r i v i a l e l e m e n t , t h e n v-^AVgA• • • A v n = 0 t r i v i a l l y by t h e Remark 2.2, and i f ( v p V g j • • • , v n ) i s n o t a t r i v i a l e l e m e n t , t h e n we a p p l y Theorem 2.3 t o d e t e r m i n e a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t v 1 A v 2 A . - . A v n = 0 . B a s i s o f P ( W , G , g ) . L e t { y - ^ y g * • • • * y m3 be a b a s i s o f V . L e t A = { l , 2 , . . . , m } and S = A x A x . . . x A , t h e c a r t e s i a n p r o d u c t o f t h e n - c o p i e s o f A . D e f i n e an e q u i v a l e n c e r e l a t i o n ~ on S as i n t h e p r o o f o f t h e R e p r e s e n t a t i o n Theorem Form I, 5, Chap te r I I , i . e . i f s = ( s ^ , S g , . . . , s n ) and t = ( t - ^ t g , . . . , t ) a re i n S , t h e n s~t i f • and o n l y i f t h e r e e x i s t s g 6 G , such t h a t t = s g , where s g = ( S g ( l ) ' S g ( 2 ) ' * * • * S g ( n ) ) * L e t A ( s ) b e t h e e c l u i v -a l e n c e c l a s s c o n t a i n i n g s e- S and l e t - E be a s e t o f . r e p r e s e n t a t i v e s o f t he e q u i v a l e n c e c l a s s e s . L e t E' =' { s | s e E- , ( y , y , . . . , y o . i s n o t a s l s2 °n ( G , a ) e l e m e n t . } By t h e Remark 7.2, Chapter "II, i f ,s € S y t h e n ( y , y ^ . . . ^ y )<j) i s n o t a ( G , a ) e lement i f and s l s2 s n o n l y i f g e G and o ( g ) 4 1 i m p l i e s s . 4 s / . y f o r some i e I . . . . Hence E' = { s | s e E, • ' and . • '„• i f g e G such t h a t a ( g ) { 1 t h e n s . 4= srr(i\ f o r s o m e i e I } , x g i x i. - 44 -Then we have t h e f o l l o w i n g 3.1 Theorem: B = { y A y o A . . . A y o | s = ( s , ,a0,. . . , s ) e E' } s l s2 s n x i s a b a s i s o f t h e ( G , a ) space P(W,G,a) . P r o o f : We w i l l f i r s t show t h a t B i s a g e n e r a t i n g s e t o f t h e space P(W,G,a) . S ince Wtf i s a s p a n n i n g s e t o f P(W,G 5 a) , i t w i l l be s u f f i c i e n t t o show t h a t B g e n e r a t e s w r . L e t ' ( w 1 , w 2 , . . . , w n ) X e W f • I f ( w p W 2 J . . . , w n ) e W i s a t r i v i a l e l e m e n t , t h e n (w-^Wg* • • . *w n)TT = 0 and i s t h e r e f o r e g e n e r a t e d by t h e s e t B . So we assume t h a t ( w ^ W g , : . . , w n ) e W i s a n o n - t r i v i a l e l e m e n t . Then by (5) (see Theorem 2.3) ( w , , w ? > . . . ,w )<j) = u) + Z D(M ) ( y .,y , . . . , y 1 d n seE-E ' s s l s2 s n + Z D(M_)(y_ , y , . . . , y _ )<|) . ssE' s s l s2 s n A p p l y i n g TI , we ge t ( w , , w p , . . . , w )t = Z D(M ) ( y , y , . . . , y )V 1 ^ n seE' s s l s2 s n = Z D(M ) ( y Ay_ A. . • Ay )' seE' s s l s2 s n w h i c h shows B i s a g e n e r a t i n g s e t . We must a l s o show B i s a l i n e a r l y i n d e p e n d e n t s e t . Suppose Z a v A y o A . . . A y _ = 0 , where a € F . seE i d n - 45 -We w i s h t o show t h a t a g = 0 f o r a l l V s e E ' . L e t s e E ' be a f i x e d i n d e x . D e f i n e f_ : W * F , as i n 1 , by s e t t i n g s (w,,w0,...,wvi)f0 = £ a ( h _ 1 ) b n _ b c ••••*>., _ 1 2 n s h€G Q ^S h ( l ) 2 > s h ( 2 ) n ' s h ( n ) s where w. = § b . . y . , f o r each i e I . f i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n by- Lemma 1.1. s T h e r e f o r e by t h e u n i v e r s a l mapp ing p r o p e r t y , t h e r e e x i s t s a u n i q u e l i n e a r t r a n s f o r m a t i o n ? : P(W,G,a) * F V such t h a t tfa = f . s s s P(W,G,a) Now E a„ ( y A y a A . . . A y _ ) = 0 . i m p l i e s seE' 3 s l s2 s n 2 a _ ( y . , y Q , . . .• , y „ ) t = 0 v.•Thehcsince ,1^ ois l i n e a r , seE' s s l s2 s n S a ( y , y , . . . , y _ ftp = 0 and s i n c e t f = f , we ssE' s s l s2 s n S s s o b t a i n £ a ( y , y „ * . . . , y _ ) f = 0. (10) seE' s "1 s2 s n s B u t ( y , y > . - . * y s ) f s = 0 and ( y t , y t , . . . , y t ) f s = 0 1 2 n 1 2 n i f t e E' and t + s , by (8) and (9) (see Theorem 2.3). - 46 -T h e r e f o r e ( 1 0 ) g i v e s a_ = 0 and s i n c e s i s a r b i t r a r y , s we have a = 0 f o r a l l s e E ' . T h e r e f o r e B i s a s l i n e a r l y i n d e p e n d e n t s e t and hence i s a b a s i s . 4. D e c o m p o s i t i o n o f P(W,G,a) Le t G = G ^ G p ® . • - ® G k : : : ( 1 1 ) be a d i r e c t p r o d u c t . Por each i , l « i*Aylet I i = { a | a e I and g ( a ) + a f o r some;., g e , Gj 3 • • Then t h e d i r e c t p r o d u c t ( 1 1 ) i s s a i d to,Jbe d i s j o i n t , i f and o n l y i f i 1 =j= i 2 i m p l i e s I± n I i = Q . L e t I G = { a | a e I , g ( a ) = a . f o r a l l g e G ) . k C l e a r l y I = u I* • For each i , 0 < i < : k , l e t i = 0 1 ~ ~ I i = { a 1 i 5 a 2 i , . . . , a n ^ } and = VxVx . . . x V ( n i c o p i e s ) , L e t GQ = [ e ] where e i s t h e i d e n t i t y p e r m u t a t i o n . Then ¥ i i s a G ^ s e t f o r each i , i = 0 , l , 2 , . . . , k . I f a i s any l i n e a r c h a r a c t e r o f G , t h e n = a |G^ , . t h e r e s t r i c t i o n o f a to G i i s a l i n e a r c h a r a c t e r o f G^ . Under t h i s h y p o t h e s i s we have t h e f o l l o w i n g fa 4.1 Theorem; P(W,G,a) g (g) P ( W . , G , , a , ) ~ i=0 1 1 1 P r o o f : For each i , 0 _< i <_ k , X± ° w ± ^ P ( W i , G i , a i ) i s a m u l t i l i n e a r ; and (G -^cr .^) f u n c t i o n . - 47 -k D e f i n e X' = ® 5 w > P ( W , G , a ) , as 1=0 ( w 1 , w 2 , . . . , w n ) ^ ' = ( ^ ^ ^ i * . . . ^ . . i ^ i • %' i s a m u l t i l i n e a r and ( G , a ) f u n c t i o n . For i f B,Y e F , t h e n f o r any j e l , j = a „ ^ f o r some f and i , 0 _< r _< n i \ 0 _< : i _< k , we have i=0 a l * a r 5 1 " r * 1 a n > 1 x = p ifo ( V 1 , " " W a k ' i , ' " , , w v l ) t l + Y (A) (W . , . . . , w ' . , . . . , W •I)T' So K " r 1 a k ^ L ± = p ( w x , . . . , w i , . . . , w n ) T ' + Y ( w 1 , . . . , w ± , . . . , w n ) T / ' w h i c h shows t h a t T ' i s m u l t i l i n e a r . A g a i n i f g e G , g = g 1 g 2 . . . g f c , where g ± e G ± , f o r i = l,2,...,k , t h e n ( w ^ , w 2 , . . . >™nYC' k = & CTi(Si)(wt7 , x i ' w a • s ±>--->™a , s i ^ i i=0 1 1 " g ^ l ) ' 1 a g i ( 2 ) ^ a g ± ( n ± ) ^ x 1=0 1 1 i£b " g ^ l ) ' 1 ^ ( S ) 5 1 " g ^ n ) ' 1 1 = o(g) ( W g f i ^ g ^ ' - ' - ' V n ) ^ ' w h i c h shows t h a t % ' i s ! ( G , a ) . - 48 -A l s o k i s a g e n e r a t i n g s e t o f §9 P ( W . , G ^ , a ^ ) i=0 1 1 1 I f B i " ^ s L±...h±y U1 = (s ^ s i ^ - - * s J s E ' ) al3± 1 S a 2 , i 1 1 a n , i ^ l ' 1 a 2 > 1 a n i ' 1 1 where A^ i s t h e A f o r t h e space P ( W i , G i , a i ) , t h e n B i i s a b a s i s f o r P ( W " i , G i , a i ) by Theorem 3 . 1 .: Then B = k { . P 0 { y B a / i y s „ / i " ^ i y s a - s a 1 , i , S ^ 1 , " - , S a J -L=O - , } ! o to^ 1 ct„ , 1 i d n . " 1 ^ " 2 ^ "n± fa i s a b a s i s o f ® P ( W 1 , G 1 , o 1 ) (see [4]) i=0 1 1 1 D e f i n e S / a s e t o f n t u p l e s as f o l l o w s : s s S i f and only i f f o r each i , 0 < i, < k , the . n ^ - t u p l e s (s« 4>'-'>s^, •») a r e i n , where E'. i s a ct-j^, 1 2 ^  n ^ s e t d e f i n e d s i m i l a r l y as E' i n 3 . Now l e t f : W — > U be any m u l i t l i n e a r and ( G , a ) f u n c t i o n on W i n t o U , where U i s any v e c t o r space ove r P . - 4 9 -Define f . ^ P ( W 1 , G i , o j L ) >U , by defining f on 1=0 i t s basis elements as follows: 1=0 s a v i 1 s a P , i 1 1 V , i s l s 2 sn and then extend f , l i n e a r l y to a map on 6Q P ( ¥ . , G . ,a . , ) , i=0- i i i to be denoted again by f . Clearly %'f = f and therefore P ( W i , G 1 , a 1 ) i s a ( G , o ) space of ¥ . 1=0 Hence, by Theorem 2.4, Chapter I , P ( ¥ , G , a ) i s isomorphic k to <g) P ( W 1 , G 1 , a i ) . —0 4.2 Remark; Under the hypothesis of 4, VjAVgA...Avn =0 i f and only i f .»A.,v iA-j.-.A-.v . = 0 for some i , 0 < i < Proof; Immediate from. Theorem 4.1 and 2(i) Chapter I I - 50 CHAPTER IV In t h i s chapter, we s h a l l apply our theorems to some special cases. We take p a r t i c u l a r G and a and consider special set of vectors v±>v2>''' *vn ' J o r t b e c x a s s i c a ± spaces, we rederive from our r e s u l t s the necessary and s u f f i c i e n t condition that an element of the space should be zero. We also show how our r e s u l t s lead to the known facts concerning the dimension of these c l a s s i c a l spaces. Thus we have included a l l these spaces i n a u n i f i e d approach. 1. P a r t i c u l a r i z i n g q . 1.1 Proposition: Let (v^ sv 2,...,v ) € W be a n o n - t r i v i a l element. Then we have the following: ( i ) I f a = l ( t r i v i a l Character) then v-^ AVgA. . . Av n =j= o» ( I i ) If a 4 1 and v^,v 2, •••»vn are l i n e a r l y independent, then v 1Av 2A. . . & v 4 0 . ( i i i ) I f a 4 1 and dim <[v^,Vg,...,v }> = 1 , then v^AVpA. Av = 0 . l?rocf s ( i ) Let {y-^yg** • • >ym] be' a basis of V . Then by the Representation Theorem Form I I , we have (v-,,v 0,...,v ) = ID + E B(M )(y ,'y 9...3yo )<f) • - - n seE s b l &2 s n • Since a = 1 , (y ,y *...,y i s not a (G,a) element s l s2 s n - 51 -for any s e S and In p a r t i c u l a r for any s' e E . Choose s € S , such that s i = min { j | j e A 1 ] , for any i = 1,2,...,n . Claims A ( s ) = {s} . Por i f t 6 A(s) then t~s , which implies there exists g e G such that. ( t ^ , t^s > »• »tn) = (sg(]_) * s g ( 2 ) ' * *' ' sg(n) ^  ' i.e. t. = s -i\ for a l l i e I . I f ord.(g) = k , we have i S i 1 / for any i. € I , H = s g ( D ^ ^ ( D = s g 2 ( i ) < V ( i r ± V - ^ d = s g k ( l ) ( = s i ) < t g k ( l ) ( = t . ) , which implies that s 1 = t ^ for a l l i e I ; i . e . , s = t , and t h i s proves the claim. = [ g I g € G , o(g) = 1 and Sj^ = s g ^ for a l l i e I } i n t h i s case i s H = [ g I g e G and s.' = s _ / . x for a l l i e I } . Therefore i f h e Gg and h | H g , then s^ =j= s h ( i ) f o r some i e 1 9 and since A ( s ) = {s} , hence ( s h ( l ) , s h ( 2 ) J " ** * s h ( n ) ^ S 9 w h i c h i m P l i e s t n a t s h ( i ) 4 \ for some i s I . Therefore i f v. = S a. .y. for i e I , we have a. „ = 0 and therefore x ' s h ( i ) l i S h ( l ) ^ S h ( 2 ) t U ' n' sh(n) - 52 -Hence D(M ) = E o^h"1) a , a 2 . . . a s heGq 1 ^ s h ( l ) ^ s h ( 2 ) n' sh(n) ^ S g ( l ) 2 ' S g ( 2 ) n ' S g ( n ) where g i s a r e p r e s e n t a t i v e o f t h e c o s e t H i n G . s T h e r e f o r e D(M„) = a , a ? . . . a 4 0 Hence "by Theorem 2,3 Chapter I I I , v ^ V g A . • • A v n 4 0 ( i i ) Take y^ = v^ f o r i = l , 2 , . . . , n and. e x t e n d [ y - ^ y g * . • . , y n ) t o a b a s i s [ ^i^23' • ' i 7 r r i o f V ' Since ( )<f) s a t i s f i e s t h e p r o p e r t y P and i f g e G , such t h a t c ( g ) 4 1 * t h e n g | e ( t h e i d e n t i t y p e r m u t a t i o n ) . Thus g ( i ) 4 1 ^or some i e I w h i c h i m p l i e s t h a t v^ and v g ^ i ) a r e i n d e p e n d e n t f o r some i e I . Hence ( v - ^ V g , . . . , v n ) ( j ) i s n o t a (G,a) e lement^ and t h e r e f o r e v^AVgA-.-AVj^ ^ 0 s \>y Lemma 2.1 Chapter I I I . ( i l l ) Le t y-, = v., and e x t e n d i t t o a b a s i s {y^y^s • • • * y c f V , S ince d im <{ v - L , v 2 , . . . , v n }> = 1 , we have A^ = {1} f o r a l l I e I and S = { s } , where s = ( 1 , 1 , . . . By t h e R e p r e s e n t a t i o n Theorem Form I I , ( v 1 , v 9 i . . . , v n ) | ) = a) + D(M g ) ( y - ^ y - ^ . . • rj^fy . Since a 4 1 J> (3ri *yn»• * - .y^) I s a (G,a) e l e m e n t . T h e r e f o r e by Theorem 2 .3. v ^ A V g A . . . A v n = 0 Hote that P r o p o s i t i o n 1,1 h o l d s f o r any subgroup G o f s . - 53 -2. S p e c i a l i z a t i o n t o t h e C l a s s i c a l Spaces. 2.1 Theorem: >\'"*ar; ( i ) I f P(W,G,a) = & V, * t h e n v,® v 0 & . . . (Ehr = o i = l 1 1 d n i f and o n l y i f v ^ = 0 f o r some i e I . ( i i ) I f P(W,G,a) = , t h e n v^v^A.". ,-Av = 0 i f and o n l y i f v ^ j V g v . j V ^ a re l i n e a r l y dependen t . ( i i i ) I f P(W,G,a) = v ( n ) > t h e n v l * v 2 * ' ' * ' v n = 0 i f o n l y i f v i = 0 f o r some i € I . P r o o f : ( i ) and ( i i i ) f o l l o w f r o m P r o p o s i t i o n 1.-1 ( i ) and Remark 2.2 Chapter I I I . ( i i ) ( = ^ ) . I f n o t , t h e n v-jAVgA.. . AV 4 0 hy P r o p o s i t i o n 1.1 ( i i ) , w h i c h i s a c o n t r a d i c t i o n . (%=) I f ( v ^ , V g , . . . j V n ) e W i s a t r i v i a l e l e m e n t , t h e n V 1 A V 2 A ° ' ' A v n = 0 b v t h e Remark 2.2 Chapter I I I . So we may assume v ^ 4 0 f o r a n y i e I . L e t J = { i | 1 = 1 o r v ^ i s i n d e p e n d e n t o f v-^Vg, ' ' ' v i - l ^ L e t J = { 1 = j 1 , j 2 , . . . , j ^ } ( s a y ) , where J 1 < J 2 < . . . < J r • S ince v , , v O J . . . , v „ a re l i n e a r l y d e p e n d e n t , we have x <~ n r < n . S ince f o r .any i , j , i 4 J 1 < 1)3. <. n > V^A...AV^A...AVjA...AV n = a ( i j ) v^A... AVjA...AV^A...AV n , we can w r i t e V n Av~A. . AV = C U,AU0A...AU A...AU (12) .12 n 1 2 r n . \ / - 54 -fvj , i f k = j t G J where c e F and u^ = z rj , for some j e J , i f k 4 J We s h a l l show that u-^AUgA. . . Au^ = 0 ". Clearly {u-^Ug,. . . ,u r) i s an independent set. Extend t h i s set to a basis (y^.yg^ • • • •»ym3 0 f v such that y^ = Vj , i = 1,2,. . . ,r . For each i e I , A i = ^ 3 i f i < r and A ± c ( 1,2,...,r } i f r < i _< n . Therefore i f s e E , then for any j , r < j < n , s . = s i for some i , 1 <_ i <_ r . Hence 0 ( y „ * y q ^ • • • * y q )4> i s a (G,a) element. °1 s2 s n By the Representation Theorem Form I I , (u,,U ?,.. . , u )<|) = uu + E D(M ) ( y , y , . . . , y _ )(|) -1 ^ n seE s s l s2 s n Since for each s e S , (y_ ,y_ ,...,y • )<|) i s a (G,a) s l s2 s n element, we have u^AUgA...Au n = 0 by Theorem 2.3 Chapter I I and therefore from equation (12.), we obtain V, AV~A. . . AV = 0 1 2 n Dimension of the c l a s s i c a l spaces Since our general theory gives a basis of P(W,G,a), (see 3. Chapter I I I ) i n every case, we can e a s i l y compute the dimension as follows. 3.1 Theorems ( i ) dim V, = mn . . i = l 1 ~ 55 -( i i ) d i m AV = ( n ) • ( i i i ) d i m V ( n ) = i***'1) • P r o o f ; d im P(W,G,a) = c a r d i n a l i t y E' , by Theorem 3.1, Chapter I I I • ( I ) I f s € S , t h e n A(s) = { s } . T h e r e f o r e E = S and s i n c e o = 1 , (y_ , y a . , y a )<|) i s n o t a ( G , a ) "1 s2 s n e lement f o r any s 6 S . T h e r e f o r e E' = E = S . n n Hence d i m J®1 V± = c a r d i n a l i t y E' = c a r d i n a l i t y S = m . ( i i ) I t i s easy t o see t h a t i f s = ( s ^ , s 2 , . . . 3 s n ) e E , t h e n ( y , y y )$ i s a ( G , a ) e l e m e n t , i f and s l s2 sn o n l y i f t h e r e e x i s t s i , j i n .1 , i | j , such t h a t T h i s means t h a t i f ( y , y , . . . , y o )(j) i s n o t i . J S-^  Sg s^ a ( G 5 a ) element, then a l l the s^'s are d i s t i n c t . Also for such an s , c a r d i n a l i t y A(s) = nJ Let S' = i s j s e S ,. &± are a l l d i s t i n c t }. Then c a r d i n a l i t y Q" - m(m - l)(m - 2)...(m - n +1) . But S c a r d i n a l i t y A(&) — c a r d i n a l i t y S'. s s E ' ' . ™ ~ • -is -w o ; _/ mfm-1)(m-2).. . (m-n+1) / m % Therefore e a r d m a x i t y E = — J — - — — ' L — L = ( ) Hence dim ^ 7 = ( ™ ) . ( I i i ) Clearly c a r d i n a l i t y E = ( ™ ) + ( | )+...+( ° ) / m+n-1 \ = < n } • - 56 -A l s o s i n c e : a = 1 , ( y 3 y . , . . . , y Q )<!> i s n o t a ( G , a ) S-j_' " S ^ ' s n e lement f o r any s = ( S g , . . . , s n ) i n E . T h e r e f o r e E = E' and hence d im = c a r d i n a l i t y E' = ("^JJ"1) -4. A s u f f i c i e n t c o n d i t i o n f o r v - j A V g A — A v n t o be z e r o . 4.1 P r o p o s i t i o n i I f ( )£efTW) i s a ( G , a ) e l e m e n t , t h e n v ^ A V g A . • . A v n = 0 . P r o o f ; ( v - ^ V g , . . . ,v n)<{) i s a ( G , a ) e lement i m p l i e s t h e r e i s g e G such t h a t a ( g ) 4= 1 and v i * v g ( i ) a r e d e P e . n d e n , f c f o r a l l i e I . L e t v g ( i ) = ^ g ( i ) v i * where X g ^ ^ € F f o r each i d . We w i l l f i r s t show t h a t V g ( i ) X g ( 2 ) ' ' ' X g ( n ) = 1 L e t g = C ^ C g . . . b e t h e d e c o m p o s i t i o n o f g i n t o d i s j o i n t c y c l e s , i n c l u d i n g a l s o t h e c y c l e s o f l e n g t h one , i f t h e r e a re any. L e t a, = dom C . , j = 1,2,. . . , k . J J Then U A . = I and A ,• fi A / = Q f o r j 4= J ' • j = l ^ J O Le t C. = ( a . lSa. 0),..,a. ) f o r j = l , 2 , . . . , k . J 0 9 x. d 3 <- j Then n^ -Hrig-K - u + n k = n I f n . = 1 for some j , t h e n Cj = ( a ^ ^ ) and v / \ = X i \ v i m p l i e s X / \ = 1 If n. > 1 , t h e n J Va.,2 = v g ( a , , l ) = X g ( a j , i ) V.,1 " 57 -V j , 3 - Vg{ay2)- Xg(ay2) v t t j,2 " X g ( a j , 2 ) ^ ^ , 1 ) ^ , 1 v a j , n ^ " v g ( a j , r i j - l ) x g ( a j , r i j - l ) v a j , h j - l = X g ( a j , n J = l ) * - ' X g ( a ; j , 2 ) X g ( a j , l ) V j , l V i = V / \ • — \ / \ V a j , l g(aj,nj) g ( a ^ r i j ) a j , r i j = Xg(aJ.,nJ)Xg(aJ,nJ»l)--'Xg(aj,2)Xg(aJ,l) V.,1 -Therefore x g ( a j , n j ) X g ( a . , n r l ) - " ^ ( ^ , 2 ) ^ ( ^ , 1 ) = 1 ; i.e. ~\\ X i \ = 1 and since j i s a r b i t r a r y , we have J TT Kfn) = 1 for j =' l,2,...,k . Therefore TT x f f(a) = TT TT K(a) = 1 ' N°W asI & ^ a ) 3 = 1 a€A. g ^ a j J ( v 1 , v 2 , . . . , v n ) f = [ ( v 1 , v 2 , . . . ,vn)<J)-a(g) ( v g ( l ) ^ v g ( 2 ) ' ' ' ' * vg( + ^ ( S ) ( v g ( 1 ) , v g ( 2 ) , . . . , v g ( n ) ) ( t ) , and since the terms within the square bracket being of type ( i i ) , i s i n fi we obtain on applying n »• v 1Av 2&...Av n = (v r l 5 v 2 , . . . , v n ) ' t ' a < g ) ( v g ( l ) ) V g ( 2 ) } - ' V g ( n ) ^ = a(g) ( X g ( l ) v r X g ( 2 ) v 2 , . . . , X g ( n ) v n r x - 58 -= X g ( a ) ( v l ' v 2 * " " v n ) and s i n c e a ( g ) 4 1 > w e have v-jAVgA.. . A v n = 0 4 . 2 Remark; One can e a s i l y see t h a t t h e c o n d i t i o n o f P r o p o s i t i o n 4 . 1 i s , however , n o t n e c e s s a r y . However, i f V i s a u n i t a r y space and G b e l o n g s t o a c e r t a i n c l a s s o f g roups G , as we s h a l l d e f i n e b e l o w , t h e n t h e c o n d i t i o n o f P r o p o s i t i o n 4 . 1 i s b o t h n e c e s s a r y and s u f f i c i e n t . 5. P a r t i c u l a r i z i n g V and G. L e t G be a subgroup o f S . I f A i s an o r b i t o f G, l e t g^ deno te t h e r e s t r i c t i o n o f g t o A . : Then g A i s a p e r m u t a t i o n o f A . L e t G A = [ g ^ | g e G } . C l e a r l y G A i s a subgroup o f , where i s t h e f u l l symmet r i c group on A . L e t G = { G | G i s a subgroup o f S n and i f A i s any o r b i t o f G t h e n G A i s c y c l i c } C l e a r l y G c o n t a i n s e v e r y c y c l i c g r o u p . As t o t h e o t h e r members of G , t h e y a re a l l a b e l i a n . L e t G £ G and ¥ = V x V x . . . x V ( n c o p i e s ) , where V i s - 59 -a u n i t a r y space o f d i m e n s i o n m . L e t a be any l i n e a r c h a r a c t e r o f G and c o n s i d e r t h e ( G , a ) space P(W,G,o) o f W . 5.1 D e f i n i t i o n s I f v = ( v - ^ V g * . . . * v n ) e W * t h e n y i s c a l l e d an i n d i c a t o r o f v i f arid o n l y i f Y : I ^ I * such t h a t YJ ; = Y-j where Y^ = Y.([f) * when and o n l y when v ^ and V j a re dependen t . L e t G y = '£ g | g e G , Y ± = Y g ^ j f o r a l l i - :€". I } . Then i t i s p r o v e d b y S t a n l e y G i l l W i l l i a m s o n i n h i s Ph.D. t h e s i s [51* t h a t v - jAVgA. . . A v n - 0 i f .and- o n l y i f S o ( g ) = 0 , f o r any i n d i c a t o r Y o f v . geG y 5.2 Theorem? W i t h G and W as I n 5 . 1 * v - L A v 2 A . . . A v n = 0 i f and o n l y i f ( v - ^ V g * . . . *v n )<j) i s a . ( G , a ) e l e m e n t . P r o o f ; ( ^ s ) I t i s a p a r t i c u l a r case o f P r o p o s i t i o n 4 . 1 . {zz}) L e t y be any i n d i c a t o r o f v .'• Then 2 a ( g ) = 0 . g S G Y T h i s i m p l i e s t h e r e e x i s t s g e G , such t h a t a ( g ) 4 1 • A l s o g e G^ i m p l i e s y^ = Yg^j_) f o r a l l i e l and t h i s i m p l i e s t h a t v ^ and v g ( i ) a r e dependent f o r a l l i e I . Hence t h e r e e x i s t s g e G * such t h a t a ( g ) 4 1 and v ^ * v g ( i ) a r e dependent f o r a l l i e I , w h i c h means t h a t ( v ^ V g j , . . . *vn)(J) i s a ( G , a ) e l e m e n t . - 60 -BIBLIOGRAPHY M a r v i n Marcus and M o r r i s Newman: I n e q u a l i t i e s f o r t h e permanent f u n c t i o n . Ann. o f Ma th . Ser 111, 75* 47-62 (1962). Hans S c h n e i d e r : Recent Advances i n M a t r i x Theory . M a d i s o n , The U n i v e r s i t y o f W i s c o n s i n P r e s s , 1964. N. B o u r b a k i : A l g e b r a M u l t i l i n e a i r e . Herman, P a r i s 1958. Chapter 3. G. D. Mostow, J . H. Sampson, J . P. Meyer : Fundamenta l s t r u c t u r e o f a l g e b r a . M c G r a w - H i l l , New York 1963. S t a n l e y G i l l W i l l i a m s o n : A c h a r a c t e r i z a t i o n o f t h e homogenous z e r o e lement i n c e r t a i n symmetry c l a s s e s o f t e n s o r s . Ph.D. T h e s i s , U n i v e r s i t y o f C a l i f o r n i a a t San ta B a r b a r a , C a l i f o r n i a , 1964. 

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