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Linear transformations of symmetric tensor spaces which preserve rank 1 Cummings, Larry 1967

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THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE F INAL ORAL EXAMINATION FOR THE DEGREE OF -DOCTOR OF PHILOSOPHY LARRY JEAN CUMMINGS B.Sc . R o o s e v e l t U n i v e r s i t y , C h i c a g o , 1961 M.Sc. DePau l U n i v e r s i t y , C h i c a g o , 1963 MONDAY, May 1st , 1967 a t 3:30 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 S . Page E x t e r n a l E xam ine r : R. C. Thompson Depar tment o f Ma thema 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 of COMMITTEE IN CHARGE Cha i rman: B. N. Moy l s W. G. Brown D. B u r e s B. N. Moy l s L. S o b r l n o R. Wes tw ick J . W h i t t a k e r S a n t a B a r b a r a R e s e a r c h S u p e r v i s o r : R. Wes tw ick LINEAR TRANSFORMATIONS OF SYMMETRIC TENSOR SPACES WHICH PRESERVE RANK 1. A b s t r a c t I f r > 1 i s an i n t e g e r t h e n U/ \ ( r ; deno te s the v e c t o r space o f r - f o l d s ymmet r i c t e n s o r s and P r [ U ] i s t h e s e t o f r a n k 1 t e n s o r s i n ^ ( r ) • ^ e t U be 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 l g e b r a i c a l l y c l o s e d f i e l d o f c h a r a c t e r -i s t i c n o t a p r ime p i f . r = p f o r some p o s i t i v e i n t e g e r k . We f i r s t d e t e r m i n e the max ima l subspaces o f r ank 1 s ymmet r i c t e n s o r s . Suppose h i s a l i n e a r mapp ing o f U ( r ) s u c h tiwtt- h ( P r [ U ] ) c P r [ U ] and k e r h fl P U [ ] = 0 . We show t h a t e v e r y such h i s i n d u c e d by a n o n - s i n g u l a r l i n e a r mapping o f U , p r o v i d e d d im U > r+1 . T h i s work p a r t i a l l y answers a q u e s t i o n r a i s e d by Ma rcu s and Newman (Ann. o f M a t h . , 7 5 , (1962) p . 6 2 . ) . GRADUATE STUDIES Theo ry o f F u n c t i o n s o f a R e a l V a r i a b l e Po int_ S e t T o p o l o g y Theory o f R i n g s Theo ry o f Numbers and A l g e b r a i c Numbers L i n e a r A l g e b r a Topo l o g y Theory o f F u n c t i o n s D. W. B r e s s l e r So C l e v e l a n d D. C. Murdoch R, Co Thompson B. No Moy l s Wo G. Brown R. C l e v e l a n d LINEAR TRANSFORMATIONS OF SYMMETRIC TENSOR SPACES WHICH PRESERVE RANK 1 by LARRY CUMMINGS B . S c , R o o s e v e l t U n i v e r s i t y , C h i c a g o , 1961. M . S c , DePau l U n i v e r s i t y , C h i c a g o , 1963. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Depar tment o f Ma thema t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t he r e q u i r e d s t a n d a r d . THE UNIVERSITY OF BRITISH COLUMBIA M a r c h , 1967. (c) Larry Cummings 1967 tn present ing th is thes i s in p a r t i a l f u l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree tiiat the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and s t u d y . I f u r ther agree that permission for extensive copying of th i s thes i s f o r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives^ It is understood that copying or pub l i ca t i on of th i s thes i s for f i n a n c i a l gain shal l not be allowed without my wr i t ten permiss ion. Depa rtrnent The Un ive r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada D a t e /jpfri /0 , tfC?l i i . S u p e r v i s o r : Dr. R. Wes tw i ck . ABSTRACT I f r > 1 i s an i n t e g e r t h e n ^ ^ r ^ deno te s the v e c t o r space o f r - f o l d s ymmet r i c t e n s o r s and P r [ U ] i s t he s e t o f r a n k 1 t e n s o r s i n U, \ . L e t U be 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d o f c h a r a c t e r i s t i c ]r n o t a p r ime p i f r = p f o r some p o s i t i v e i n t e g e r k . We f i r s t d e t e r m i n e the max ima l subspaces o f r a n k 1 s ymmetr i c t e n s o r s . Suppose h i s a l i n e a r mapping o f * J ( r ) s u c h t h a t h ( P r [ U ] ) c P [U] and k e r h n P [U] = 0 . We have shown t h a t e v e r y such h i s i n d u c e d by a n o n - s i n g u l a r l i n e a r mapping o f U , p r o v i d e d d im U > r+1 . T h i s work p a r t i a l l y answers a q u e s t i o n r a i s e d by Marcus and Newman (Ann. o f M a t h . , 75, (1962) p . 6 2 . ) . i i i . TABLE OF CONTENTS page PART I 1 §1 The Symmetr ic P r o d u c t Space 1 § 2 Some P r o p e r t i e s o f Symmetr i c P r o d u c t s 9 PART I I 18 §1 The Max ima l Pure Subspace 18 § 2 I n t e r s e c t i o n s o f Max ima l Pure Subspaces 34 PART I I I 42 §1 P r o d u c t P r e s e r v e r s 42 § 2 A s s o c i a t e Mappings 47 § 3 I nduced P r o d u c t P r e s e r v e r s 5 3 BIBLIOGRAPHY 60 i v . ACKNOWLEDGMENTS I am i n d e b t e d t o my s u p e r v i s o r , Dr . R. Wes tw i ck , f o r h i s . v a l uab le g u i d a n c e . G r a t i t u d e i s a l s o due t o Dr. B. N. M o y l s , whose e x c e l l e n t t e a c h i n g p r o v i d e d an i n t r o -d u c t i o n t o t h i s p r o b l e m and whose s u g g e s t i o n s have g r e a t l y enhanced t he m a n u s c r i p t . I am a l s o g r a t e f u l t o the U n i v e r s i t y o f B r i t i s h Co l umb i a and the N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r f i n a n c i a l s u p p o r t w h i l e w r i t i n g t h i s t h e s i s . PART I I n P a r t I we d e s c r i b e the symmetr i c p r o d u c t space and deduce some p r o p e r t i e s o f r ank 1 t e n s o r s . These r e s u l t s a r e the b a s i c t o o l s u sed i n P a r t I I where we seek max ima l subspaces o f r ank 1 t e n s o r s . P a r t I I I d i s c u s s e s l i n e a r t r a n s f o r m a t i o n s o f s ymmet r i c p r o d u c t s pace s . I n what f o l l o w s a r ank 1 s ymmet r i c t e n s o r w i l l be c a l l e d s i m p l y a s ymmetr i c p r o d u c t . §1 The Symmetr i c P r o d u c t Space I f U i s any v e c t o r space ove r a . f i e l d F we denote the r - f o l d c a r t e s i a n p r o d u c t o f U by U . L e t F (U ) denote a f r e e v e c t o r space o f U r t o g e t h e r w i t h a mapping ty : U r - F ( U r ) , where the image o f U r under ty f r e e l y g e n e r a t e s F ( U r ) . I f H i s an a r b i t r a r y v e c t o r space o ve r F and f : U r - H i s any f u n c t i o n , t h e n t h e r e i s a un i que l i n e a r mapping g . F (U ) - H such t h a t f = goty . D e f i n i t i o n 1.1 I f H i s any v e c t o r space o ve r F , l e t A^ denote the c o l l e c t i o n o f t ho se l i n e a r mapp ing s , g : F ( U r ) - H, f o r w h i c h t h e r e i s a m u l t i l i n e a r mapping f : U r -» H , such t h a t f = g0ty , and denote u A„ by A n . L e t G n . = n « k e r ! I f r\ i F ( U r ) - F ( U r ) / G n i s t he c a n o n i c a l map, F ( U r ) / G n i s the r - f o l d t e n s o r p r o d u c t o f U and the t e n s o r p r o d u c t o f any r v e c t o r s x 1 , . . . , x r o f U i s the c o s e t f t x i > • • • >*r) where f* = rioty . I n t he s e q u e l we w i l l denote F ( U r ) / G 0 by U ^ r ^ and f ( x 1 3 . . . , x r ) by x 1 ® . . . ® x r . D e f i n i t i o n 1.2 L e t A 1 be the c l a s s o f t ho se mapp ings , g , i n A Q such t h a t g = f o f where f i s b o t h m u l t i l i n e a r and s ymmet r i c . L e t Q± = n g € A k e r g . I f C : F ( U r ) - F ( U r ) / G 1 i s t he c a n o n i c a l map, F ( U r ) / G ^ i s t he r - f o l d s ymmetr i c  p r o d u c t o f U and the s ymmet r i c p r o d u c t o f any r v e c t o r s x ^ , . . . , x r o f U i s t he c o s e t a ( x 1 , . . . , x r ) where a = . I n t he s e q u e l we w i l l denote F ( U r ) / G 1 by u ( r ) and a ( x 1 , . . . , x ) by x 1 * . . . * x . The s e t o f a l l s ymmetr i c p r o d u c t s i s deno ted by P [U] . Theorem 1.3 E v e r y e l ement o f ^ ( r ) i s a f i n i t e sum o f s ymmetr i c p r o d u c t s . some P r o o f : I f t e U^j.) t h e n t i s a c o s e t g + G-^  f o r g e F ( U r ) . S i n c e t h e e l emen t s ty( X-^  , . . . , x^, a r e a b a s i s o f F ( U r ) g i s a f i n i t e sum: £ a ^ t y ( x ^ , . . . ) where each ct^ e F and ( x ^ n , . . . , x ^ ) e U r . Hence t = E a i t | i ( x i l , . . . , x ± r ) .+ G]_ = S a 1 ( i | r ( x 1 1 , . . . , x i r ) + G^) = E a± x ^ * . , . * x i r . A l s o i t f o l l o w s i m m e d i a t e l y f r o m D e f i n i t i o n 1.2 t h a t r the mapping a : U ^ ( r ) i s m u l t i l i n e a r and s ymmet r i c . 3 . Moreove r , i f x and y a r e s ymmet r i c p r o d u c t s w i t h r - l common f a c t o r s ( c o u n t i n g r e p e t i t i o n s ) t h e n x + y i s a symmetr i c p r o d u c t . Theorem 1.4 L e t H he any v e c t o r space . F o r e v e r y s ymmet r i c m u l t i l i n e a r f : U -» H t h e r e i s a un i que l i n e a r .u:;u.r h : u ^ r j - H such t h a t f = ha . P r o o f : L e t g : F ( U r ) - H be t h e un i que l i n e a r mapping f o r w h i c h f = g«ji . Any e lement o f U ( r ) m a y be w r i t t e n a +. G 1 where a e F ( U r ) . D e f i n e h : U ^ r j - H by h ( a + G-^) - g ( a ) . S i n c e g e. A 1 , whenever a-b e G-^  we have h ( a ) = h ( b ) , showing h i s w e l l - d e f i n e d . Mo reove r , h w ( x 1 , . . . , x r ) = h ( t y ( x 1 , . . . , x r ) + G 1 ) = &^(x±)...,xr) = f ( x ^ . . . , x p ) f o r any ( x ^ , . . . , x ) € U r . The mapping h i s seen t o be l i n e a r because g i s l i n e a r . I f h ' : U ( r ) - H i s any. o t h e r l i n e a r mapping such t h a t f = h»a t h e n i f a + = E a ^ ^ x ^ ^ , . . . , x ^ r ) i s any e lement o f ^ ( - r ) '• h ' ( a + G x ) = E a 1 h 6 a ( x 1 1 , . . . , x l r ) = E a ± f ( x ± 1 > • • • » x ± r ) = E a i h « j ( x i l , . . . , x i r ) = h ( a + Q 1 ) . Remark: I f U i s an n - d i m e n s i o n a l u n i t a r y v e c t o r space ove r the complex numbers , Theorem 1.5 w i l l show our d e f i n i t i o n o f a symmetr i c p r o d u c t space c o i n c i d e s w i t h t h a t o f Marcus and Newman [4 , p . 4 8 ] . T h e i r d e f i n i t i o n a r i s e s i n the f o l l o w i n g way: F o r each w & S r d e f i n e a m u l t i l i n e a r mapping f 4 . f r om U t o by f T ( u r . . . 3 u n ) = u 7 r - l ( i ) ® - " ® u 7 r - l ( r ) L e t denote the un i que l i n e a r mapping o f U V J " ' f o r w h i c h f = g > o f , w h e r e ^ i s t he mapping f r o m U r t o i f * i n D e f i n i t i o n 1.1 . The t o t a l l y s ymmetr i c o p e r a t o r on i s the mapp ing; S = 1/r.' £_ g T w h i c h can a lway s be d e f i n e d TreS. p r o v i d e d c h a r P \ r l s i . e . , c h a r F > r . I t i s r e a d i l y v e r i f i e d t h a t S o t i s a m u l t i l i n e a r and symmetr i c mapping. L e t i denote the mapping o f u ( r ) such t h a t ioa = S o f d i a g r am: These s t a t e m e n t s a r e summarized i n t he f o l l o w i n g Theorem 1 . 5 I f U i s a v e c t o r space o ve r a f i e l d F and char F > r o r i s z e r o t h e n T j ^ r ) r n g s P r o o f : L e t ^ be the un i que map ^ f o r w h i c h p o f = a and s e t j = ^ j r n g S . We p r o ve t h a t i o j i s t he i d e n t i t y map on r n g 3 and. j o i t he i d e n t i t y on U ^ r ) • I f 3 a r n g S l e t s = S ( t ) where t = ^ f c t ( w k ) i s a f i n i t e sum and w^ e U r . Then, i o j ( s ) = i o ^ ( s ) . Bu t 5 . T h e r e f o r e , i « j ( s ) = V " 0 ^ ^ = V 0 ^ ™ ^ = SW = s ' C o n v e r s e l y , i f s e U ^ r ^ t h e n s i s a f i n i t e sum E k a ( x k , . . . , x ^ ) and i ( s ) = 2 ^ 1 « a ( x k , . . . , x k ) = ^ S o t ^ , . . . , x k ) where ( x k , . . . , x k ) e U r f o r each k . T h e r e f o r e , i ( s ) = l / r - I k ^ t C ^ ^ ) , . . . , ^ ^ ) ) . A c c o r d i n g l y , J o i ( s ) = ^ o i ( s ) = 1 / r ! ^k ^ 8 ^ ^ - 1 ( 1 ) - • • ' X T - l ( r ) 5 = l / r l ^ ^ s r a ^ - l ( l ) - " ' x f - l ( r ) ) = 1 / r i Z k I V 6 S r a ( x J , . . . , x J ) = E ^ a C x ^ , . . . , x k ) = s because a i s s ymmet r i c . L e t H ] denote the v e c t o r space o f homogeneous p o l y n o m i a l s o f degree r i n n i n d e t e r m i n a t e s o ve r any f i e l d . The d i m e n s i o n o f H r [ , . . .,? ] i s ( n + £ ~ 1 ) • [ 1 , P - 6 ] . I f U 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 t he same f i e l d , p r o b a b l y the s i n g l e most i m p o r t a n t f a c t about T ^ ( r ) Theorem 1.6 u ( r ) = H r [ ? ] _, . . . , ? n ] P r o o f : Suppose u ^ , . . . , u n i s a b a s i s o f U . L e t a v e c t o r x = a , u, +...+a u o f U c o r r e s p o n d t o t he l i n e a r homogeneous l i n n c p o l y n o m i a l i ( x ) = a i ? x + * ' ' + a n ^ n " ^ ( x i ' * ' * > x r ) € *J s e t f ( x l 3 . . . 3 x r ) = i ( x 1 ) . . . . i ( x r ) . I t i s e a s i l y v e r i f i e d t h a t f i s m u l t i l i n e a r and s ymmet r i c . A c c o r d i n g t o Theorem 1.4 t h e r e i s a l i n e a r h : u ( r ) ~* H r ^ l * ' * * 3^n) s u c n t h a t f = hpa • L e t G he t he s e t o f t h e s e r - t u p l e s (a,,...,a ) o f i n t e g e r s such t h a t 1 _< '<_.... <_ a r <_ n . The images f ( u ^ , . . . , u & . . ) = w h e r e (a 1 , . , . - *a r ) e G ^ , n ^ f orm the n a t u r a l "bas i s o f H r [ ^ , . . . - , ^ n ] ' E v e r y b a s i s e l ement i s such an image s i n c e Ca rd G = ( n + ^ - ^") , [5 , Theorem 4 .2, p.9] . A c c o r d i n g l y , i f u denote s t he p r o d u c t a ( u > , . . ) U ) where a = ( a 1 , . . . , a T , ) € G r n t h e n t he images h ( u a ) a r e a b a s i s o f H [?•,,...,? ] • T h e r e f o r e , h i s on to and r ^1 ^n dim 1 d im H r . . . , ? n ] . On t h e o t h e r hand t h e - p r o d u c t s u , a € G . span U ( r ) : I n v i e w o f Theorem 1 .3 we need o n l y p r o ve symmetr i c p r o d u c t s depend on the u . F o r v e c t o r s x . = E CL . u . % a i x j j 1 1 ^ » • • j 2* j a ( X l , • • • 3 x r ) = E a e S a . . . a u a r , n 1 r where S i s t he s e t o f i n t e g e r r - t u p l e s w i t h e n t r i e s i n r , n e { l , . . . , n ) . S i n c e a i s s ymmet r i c t he sum may be r e s t r i c t e d t o G r n . C o n s e q u e n t l y , d im U ( r ) _< Ca rd G r n . 7. We c o n c l u d e t h a t d im U, , = ( n + T ~ 1 ) and h i s { r ) v r a v e c t o r space i s o m o r p h i s m o f u ( r ) 8 1 1 ( 1 ^ [ i ^ , . . . , § r ] w h i c h " p r e s e r v e s p r o d u c t s " i n t h e sense t h a t the image o f a symmetr i c p r o d u c t i n ^ ( r ) i s a lway s a p r o d u c t o f homogeneous l i n e a r p o l y n o m i a l s . F o r f u t u r e r e f e r e n c e we s t a t e : C o r o l l a r y 1.7 I f U i s an n - d i m e n s i o n a l v e c t o r space t h e n TT ,n + r - 1\ dxm U , r ) = ( r ) . L e t U be 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 . C o r o l l a r y 1.8 I f x ^ , . . . , x r a r e any v e c t o r s i n U t h e n x-^*. . . * x r = 0 i f and o n l y i f x^ = 0 f o r some i . P r o o f : L e t - £ ^ = 1 c t ^ u^ where € F , i = l , . . . , r and u ^ , . . . , u n . i s a b a s i s o f U . Under t h e i s omo rph i sm o f Theorem 1.6 x ^ # c . . * x r c o r r e s p o n d s t o the p o l y n o m i a l ^C35^) » w h i c h must t h e n be i d e n t i c a l l y z e r o i f x - L * , , . . * x T , = 0 . S i n c e F [ ? ^ , . . . , § ] i s an i n t e g r a l domain [ 2 , p. 1 0 6 ] , f o r some k , 1 <_ k < r , we have i ( x , ) = a . §, + ...+a § = 0 . Hence a . =...=a = 0 and so • x k x n k n 1 k r k = 0 . C o n v e r s e l y , i n any r i n g (and i n p a r t i c u l a r 4 F [ § . . . 3 § r ] ) a>0 = 0 f o r e v e r y e l ement a . L e t U be 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 . C o r o l l a r y 1,9 I f x ^ , . . . , x r , y-^ •>...,y a re n o n - z e r o v e c t o r s 8. o f U t h e n x 1 * . . . * x r = y-^*.. . * y r i f and o n l y i f t h e r e i s a 7r i n S r and s c a l a r s Xj_ i n F such t h a t x i = X^y^-^) f o r e v e r y i = l , . . . , r ; w h i l e Xj_ = 1 .. P r o o f : Under the i s o m o r p h i s m o f Theorem 1.6 b o t h x ^ * . . . * x r and y 1 * . . . * y r c o r r e s p o n d t o a p r o d u c t o f r l i n e a r homogeneous p o l y n o m i a l s . I n the G a u s s i a n domain F t ? ^ . . . , ? ] l i n e a r p o l y n o m i a l s a r e p r i m e s and i t s u n i t s a r e i n F , [ 2 , p.127] Hence, t h e r e a r e \ i i n F such t h a t x i = \^ 7T^^ f o r each i = l , . . . , r . Were ^ i - i ^ £ 1 t h e n X l # . . . * x r = ( i r T _ 1 \ i ) y 1 * . . .#y r so t h a t (1 - T T ^ = 1 X 1 ) y 1 * . . . * y r = 0 w h i c h I m p l i e s y - ]_ * . . . * y r = 0 , c o n t r a d i c t i n g C o r o l l a r y 1.8. The c o n v e r s e i s i m m e d i a t e . / L e t U be any v e c t o r space . /' / / / Theorem 1.10 I f A i s a l i n e a r map o f U t h e r e i s a un i que . y l i n e a r map, A ( r ) ' o f , U ( r ) s u c h t h a t A ^ r ^ ( x 1 * . . . * x r ) = A x 1 # . . . * A x r f o r e v e r y x.^*.. „ * x r g P r [U] . P r o o f : D e f i n e f : - u ( r ) "by f ( x i > • • • > x r ) = A x 1 * . . „ * A x r . S i n c e f i s m u l t i l i n e a r and s y m m e t r i c , Theorem 1.4 en su re s t he e x i s t e n c e and u n i q u e n e s s o f A ( r ) • I f A i s a l i n e a r mapping o f U t h e mapping A, >. o f ^ ( r ) i s s a i d t o be i n d u c e d by A . To show a l i n e a r mapping h o f U ( r ) i s i n d u c e d by a mapping A o f U i t i s s u f f i c i e n t t o show h ( x ) = A, f o r e v e r y x i n P„ [U] 9. s i n c e P r [ U ] c o n t a i n s a b a s i s o f u ( r ) • §2 Some P r o p e r t i e s o f Symmetr i c P r o d u c t s • I n t h i s s e c t i o n we o b t a i n some r e s u l t s w h i c h w i l l be u sed i n P a r t s I I and I I I . L e t U be any v e c t o r space . Theorem 1.11 I f x 1 * . . . * x r + y 1 * . . . * y r = z 1 * . . . * z r e P r [ U ] and x e U t h e n x * x 1 * . . . * x r + x * y 1 * . . „ * y r = x * z 1 * . . . * z r e P r + 1 [ U ] P r o o f : D e f i n e f x U r - u ( r + 1 ) f o r any (v 1,...,v r) i n U r by : f (.v1,. . . ,v r) = x * v 1 * . . . * v r C l e a r l y , f i s m u l t i l i n e a r and s ymmet r i c . By Theorem 1.4 t h e r e i s a l i n e a r h : U ^ r ^ - U ^ r + 1 ) such t h a t f = b*y . Suppose x 1 * . . . * x r + y 1 * . . . * y r = z1*...*zT € P r [ U ] . Then h ( x 1 # . . . * x r ) + h ( y 1 * . . , * y r ) = h ( z ] _ * . . . * z r ) and so h » a ( x 1 J . . . , x r ) + b a ( y 1 , . . . , y r ) = *XJ{Z19 . . . , z p ) . T h e r e f o r e , x * x 1 * . . , * x r + x * y 1 * . . . * y r = x * z 1 * . o . * z r . D e f i n i t i o n 1.12 I f S i s a s ub se t o f any v e c t o r space U t h e n L ( S ) w i l l denote the subspace o f U spanned by t he v e c t o r s i n S . L e t x = x ^ * . . . * x r e P [U] . By t h e n o t a t i o n U ( x ) we mean t h e subspace L(x^t...,x ) . The 1 - d i m e n s i o n a l subspaces L ( x ^ ) ,, i = l , . . . , r , a r e c a l l e d t h e f a c t o r s o f x . A 1 - d i m e n s i o n a l subspace L ( u ) i s a f a c t o r o f X o f 10. m u l t i p l i c i t y k i f x = u * . . . • * u * x i t + i * « • • * x r where L ( u ) ^ L ( x i ) , i = k + l , . . . , r . Thus , c o u n t i n g m u l t i p l i c i t i e s , e v e r y n o n - z e r o e lement o f P r [ U ] ha s r f a c t o r s . Theorem 1.-13 L e t U be 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 . I f (1) x * x 1 * . . . * x r + x * y 1 * . . . * y r = z e P r £ u ] t h e n x i s a f a c t o r o f z and i f z = x * z , * . . . * z t h e n 1 r x 1*...*x r + y 1 * . . . * y r = z 1 * . . . * z r e P [U] . P r o o f : D e f i n e U 0 b y U = L(-x)©U Q and l e t P be t he p r o j e c t i o n map on UQ a l o n g L ( x ) . L e t ^ ( r ) be t h e mapp ing o f u ( r ) d e f i n e d i n Theorem 1.10. Suppose z = z i * * o « * z r + 1 • A p p l y i n g P ^ t o ( l ) we g e t P ( r ) ( z ) = P ( z 1 ) * . . . * P ( z r + 1 j = 0 . C o r o l l a r y 1.8 i m p l i e s P('Z^) = 0 f o r some i , say i = z r + 1 - Hence L ( x ) i s a f a c t o r o f z . S i n c e F[§-L> • • • > ? r l i s a n i n t e g r a l domain, Theorem 1.6 y i e l d s x 1 * . . . * x r + y 1 * e „ 0 * y r = z 1 * . . . * z r . L e t U be any v e c t o r space . Theorem 1.14 I f x , y , and x+y a r e n o n - z e r o e l emen t s o f P r [ U ] t h e n dim U ( x ) and dim U(x)nU(y) d i f f e r by a t most one. P r o o f : Suppose t he c o n t r a r y , so t h a t dim U ( x ) n U ( y ) + l < d i rnU(x) . I f x =• x ^ * . . „ # x r there., a r e - a t ' L e a s t ' two i ndependent , x^ , 11. say x 1 and such t h a t U ( y ) $ L^x-^Xg) i s d i r e c t . L e t f : U - U he a l i n e a r mapping such t h a t f | U ( y ) . i s t he i d e n t i t y map and e i t h e r f ( x - L ) o r f ( x 2 ) i s z e r o . I f f ( r ) • U ( r ) ~* ^ ( r ) l s m a P P i n § d e f i n e d i n Theorem 1.10, t h e n f ( r ) ( x ) = 0 and so ; , - ; ' 7 = f ( . r ) ( y ) = f ( r ) ( z ) ^ • f ( z 1 ) * . . V * f ( z r ) where x '+ y '= z" ± = ' z j * ; . , * z . I f z i ^ U-(-y) f o r some i t h e n f can be cho sen so t h a t we a l s o have f(z^) = 0 f o r t h i s i . F o r ZJL i U ( y ) i m p l i e s one o f L ( z 1 , x 1 ) n U(y) o r L(z±,x2) n U (y i s z e r o . . i Now f o r such an f , f ^ ^ z ) = 0, a c o n t r a d i c t i o n s i n c e y =j= 0 . T h e r e f o r e , z^ e U ( y ) and by t he c h o i c e o f f we have fCz^) = z^ f o r e v e r y i l = l , . . . , r . Hence y = and so x = 0 , a c o n t r a d i c t i o n and the theorem f o l l o w s . C o r o l l a r y 1.15 I f x , y , x + y a r e n o n - z e r o e l emen t s o f P [U]_ w h i l e dim U ( x ) = 1 and .d im U (y ) = 2 t h e n U ( x ) c U (y ) . C o r o l l a r y 1.16 I f x , y , x + y a r e n o n - z e r o e l ement s o f P r [ U ] and dim U (x ) = dim U ( y ) = 2 t h e n U (x ) n U(y) ^ 0 . 12. Theorem l l 17 L e t x and y be n o n - z e r o e l emen t s o f P rt.U]> where U 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 i n f i n i t e f i e l d P . I f L ( x , y ) e -P r [U ] and dim U(y") > 2 t h e n x and y have a common f a c t o r . P r o o f ; S i n c e dim U ( y ) > 2 we may suppose a r e i n d e p e n d e n t v e c t o r s , where y = y 1 * « . . * y r . Complete Jl>Y2>y-$ t o a b a s i s u 1 , . . . , u n 6 f: U , where Y± - n± f o r i = 1,2,3.. v By a s s u m p t i o n , f o r e v e r y x i n F t h e r e i s a p r o d u c t z ( \ ) = z 1 ( \ ) * . . . * z r ( \ ) such t h a t x + Xy = z ( x ) . F o r each i - l , . „ , , n l e t U = L ( u i ) ® V^ ^ and p 1 : U - be a p r o j e c t i o n where V\ = L ( u 1 5 . . . . . . , u n ) . We have P ( r ) ( x ) = P ( r ) ( z ( 0 ) f o r i = 1 ,2,3, ; i . e . , (1) . p 1 ( x 1 ) * . . . * p : L ( x r ) = P i ( z 1 ( x ) ) * . . . * p 1 ( z r ( x ) ) . I f p 1 ( X j ) = 0 f o r some i , 1 < i < 3 , and some j , l-,<_ j _< r , t h e n L ( x . ) = L ( y . ) i s a common f a c t o r o f x and y . Assume x and y have no common f a c t o r s . Then, i n p a r t i c u l a r , p 1 ( x . ) ^ 0 f o r a l l i = 1,2,3 and 1 ; \ J j = l , . . . , r . The v e c t o r s • z 1 ( x ) , . . . , z ( x ) may be chosen 1 1 so t h a t p ( x j ) = P ( z - j ( x ) ) f ° r a 1 1 3 = l , . . . , r . T h e r e f o r e i f z , ( x ) = l £ = 1 a . . k ( x ) u k and x . = E £ = 1 , t h e n (2) a j k ^ ^ = ^ j k ^ = 1 > • • ° J R > = 2 , . . . , r . A p p l y i n g C o r o l l a r y 1.9 t o ( l ) when i = 2 we have a 7T i n S r and s c a l a r s c ^ ( x ) f o r w h i c h 13-P (Zj (x)) = C j ( \ ) P ( x T ( j ) ) 0' = • Thus , (3) . a J k (x) = C j (x) P T ( j ) k k = 1 , 3 , . . . , n . I f f o r some j , a . v ( \ ) = 0 f o r a l l k ^  2 t h e n L ( z j ) = L ( y 2 ) I s a common f a c t o r o f y and z(x) , hence a f a c t o r o f .x . So we may assume f o r each j = l , . . . , r t h e r e i s a k = l , 3 * . . . * n such t h a t ^ 0 • I f k ^  1 f o r a l l j t h e n (2) and (3) i m p l y (4) . c j ( x ) = p j k P T ( j ) k f o r e a c h = l , . . . a r . Now i f t h e r e i s a j f o r w h i c h k = 1 t h e n ^Tr(o)k = 0 * k = 3*...j»n . A p p l y i n g C o r o l l a r y 1.9 t o ( l ) when i = 3 we have : P 5 ( z j ( 0 ) « dj(x) p 5 ( X u j ( j ) ) f o r some s c a l a r d.(x) and UJ i n S . I n p a r t i c u l a r , ( 5 ) . «. 2 (x) = d J ( x )P ( B ( J ) 2 • S i n c e P ^ . ^ , = 0 , ctj k(x) = 0 whenever k = 3 , . . . , n . I f P w ( j )2 = 0 t h e n z j (x ) = a j i ( x ) y i s o L(z.(x)) = I ' ( y P ) wou ld he a common f a c t o r o f y and z(x) , hence a f a c t o r o f x . T h e r e f o r e , P w ( j )2 ^ ^ a n d i m p l i e s (6) d . u ) = ^l\i)2 • F o r any \ i n F , t h e r e f o r e , t he c o o r d i n a t e s o f a s e t o f r v e c t o r s z 1 (x ) , . . . , z „ (x ) such t h a t x+xy = z ^ ( x ) * . . . * z r (x ) 14. a r e among t h e e l emen t s o f t he f i n i t e . s e t c o n s i s t i n g o f t he n r s c a l a r s and p r o d u c t s o f t he se i n t he f o r m : A c c o r d i n g l y , t h e r e a r e f i n i t e l y many s e t s o f v e c t o r s z ^ , . . . , z . f o r w h i c h x + ^y = z ^ * , . . * z r . Bu t F i s i n f i n i t e . Hence t h e r e a re d i s t i n c t s c a l a r s X , x' such t h a t x + Xy = x + x ' y w h i c h i m p l i e s y = 0 . T h i s c o n t r a d i c t i o n c o m p l e t e s the p r o o f . Theorem 1.18 L e t U be 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 ove r an i n f i n i t e f i e l d F and x , y n o n - z e r o e l emen t s o f P r [ U ] . I f l».(x,y) <= P r [ U ] w h i l e dim U (x ) = dim U ( y ) = 2 b u t U ( x ) ^ U ( y ) t h e n x and y have a common f a c t o r . P r o o f : A c c o r d i n g t o C o r o l l a r y 1.16, U ( x ) n U (y ) ^ 0 . Hence dim U(x ) n U ( y ) = 1 . L e t U ( x ) n U ( y ) = L ( u ) . whenever x and y have a common f a c t o r , t h a t f a c t o r must be L ( u ) . F i r s t , we assume L ( u ) i s a f a c t o r o f x , say , b u t n o t y and show y and z have a common f a c t o r where x + y '= z e P r [ U ] (Any f a c t o r o f y and z i s a l s o a f a c t o r o f x . ) S e c o n d l y , we suppose L ( u ) i s n o t a f a c t o r o f e i t h e r x o r y and use the a s s u m p t i o n t h a t L ( x , y ) c P [U] t o o b t a i n a c o n t r a d i c t i o n . Assume L ( x ^ ) = L ( u ) and U ( x ) = L(x^,Xg) where x = x 1 * . . . * x r . I f y = y-]_-*<> • o * y r t h e n t h e r e i s i , 1 <. i <. r , such t h a t y^ ^ U (x ) , o t h e r w i s e U (x ) = U ( y ) , c o n t r a r y t o a s s u m p t i o n . Suppose y^ ^ U (x ) . 15. Then U ( y ) = L (x^ ,y . j J and we may comp le te t he Independent v e c t o r s x - ^ x ^ y ^ t o a b a s i s u ^ , . . . ^ o f U . F o r each i = l , . . . , n l e t U = L ( u i ) ^ V i and p 1 : U -* be a p r o j e c t i o n where = 11(0^ , . . . , u ^ , . . . ,u ) . Then, f o r b o t h i = 1 and I = 2 p ^ x 1 ) = 0 b u t p 1 ( y . j ) £ 0 f o r a l l j = l , . . . , r . I f x + y = z e P r [ U ] v e c t o r s z .^ , . , . , z r may be chosen so t h a t z = z 1 * . . . * z r and C o r o l l a r y 1.9 y i e l d s p x ( y j ) = p x ( z j ) f o r each j = l , . „ . , r . S i n c e y^ = p 1 ( y 1 ) we must have =. a x 1 + y^ i n o r d e r t h a t p ( z ^ = p ( y . ^ . Then, p z-j^  = p ( z^ ) and C o r o l l a r y 1.9 i m p l i e s , i n p a r t i c u l a r , t h a t 2/ \ = X p (yjjJ f ° r some X i n F and some k, .1 X k <_ r . P Bu t y j = p (y-j) ;.for ,anyn j = 1,.. . , r , j T h e r e f o r e , L ( z 1 ) = L ( y k ) i s a common f a c t o r o f y and z . I f L ( u ) i s n o t a f a c t o r o f e i t h e r x o r y l e t U (x ) = L ( x 1 , u ) , U ( y ) = L ( y - L , u ) and comp le te t he i n d e p e n d e n t v e c t o r s x 1 , y 1 , u t o a b a s i s . S i n c e x + L ( y ) c P [U] , f o r each X i n - F t h e r e i s a z(x) e P r [ U ] such t h a t o 1 p x + Xy = z(x) • S i n c e p. (y-^) = 0 and p (x^) = x i f o r a l l p, \ ( x ) =.P/^(z(x)) , and we may choose v e c t o r s s«- - )Zr(\) such t h a t C o r o l l a r y 1.9 y i e l d s P~(z i(x)) = x.. X ^ 1 ^  » o ^  2 J o Til l £ X° S JT OX"S j i f z i(x) = a ± ( x ) x 1 + ^±(\)y1 + Y i ( \ ) u and = a^x^ + b^u 16. t h e n a ^ ( \ ) = Yj_ ( X). = t> i i = .1,... , r . S i n c e p 1 ( x 1 ) = 0 and p 1 ( y ^ ) = y.j_ f o r a l l i , P ( r ) ( x y ) = X P ( r ) ( y ) = P ( r ) ( z ( x ) ) , and C o r o l l a r y 1.9 i m p l i e s t h e r e a r e s c a l a r s u ^ x ) and a p e r m u t a t i o n r i n S r such t h a t : P 1 ( z 1 ( \ ) ) = X ^ ( x ) y T ^ j and p 1 ( z i (x ) ) = " i ( x ) y ^ j _ . ) i = 2 , . . . , r . T h e r e f o r e , ^(x) = X y 1 ( x ) o ^ 1 ^ Y l(x) = ^ ( X ) ^ - ) and. $ i(x) = ^iCxJc^jL) Y i ( x ) = i i i ( x ) d 7 r ^ i ^ i = 2 , . . . , r where y^ = c ^ + d^.u j = l , . . . , r . S i n c e L(u) ji ^(v^) > Q± ^ 0 f o r a 1 1 i = l , . . . , r . T h e r e f o r e , the r a t i o b ^ / p ^ x) = d T ( i ) / 6 . r ( i ) i s d e f i n e d f o r each i = l , . . . , r . Hence, f o r any X i n F and a l l i = l , . . . , r 63^(\) c an a,ssume a t most f i n i t e l y many d i f f e r e n t v a l u e s i n F . Because o f (1.) we c o n c l u d e t h e r e a r e a t most a f i n i t e number o f v e c t o r s z 1 , . . . , z such t h a t x + Xy" = z 1 * . . . * z r . B u t F i s i n f i n i t e . Hence t h e r e a r e d i s t i n c t s c a l a r s X , X ' such t h a t (x - x ' ) y = 0 , c o n t r a d i c t i n g the a s s u m p t i o n t h a t y was n o n - z e r o . 17. Theorem 1.19 L e t U be 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 and x , y n o n - z e r o e l emen t s o f P [U] . I f x+y = z e P r [ U ] t h e n U ( z ) c L ( U ( x ) , U ( y ) ) . P r o o f : L e t u ^ , . . . , ^ be a b a s i s o f U ( x ) s uch t h a t x^ = u ^ . I f z ^ e U (x ) f o r a l l i , where z = z ^ * . . . * z , t h e n we a r e done. O t h e r w i s e , wheneveruu^, z. . , u£ , . . z^a id^an : ' l ndependentdse t i t may be c o m p l e t e d t o a b a s i s u ^ , . . . , u o f U . L e t p x denote the mapping o f U d e f i n e d by p x ( u . ) = (1 - 6, , ) u . . Then, P ( r . ) ( x ) = 0 and P ( r ) ( y ) = P ( r ) ( z ) 5 i . e . , 1 1 1 1 P ( y x ) * . . . * P ( y r ) = P ( z ^ * . . . * p ( z y ) . S i n c e z± was i n c l u d e d i n t he b a s i s and canno t be u ]_ , P X ( Z ^ ) = z ^ • C o r o l l a r y 1.9 i m p l i e s - p X ( y j ) = \ z ^ f o r some j , 1 <_ j <_ r , • arid \ e F . C o n s e q u e n t l y , y^ = ax^ + f o r some a e P . S i n c e X ^ 0 we see t h a t e L ( U ( x ) , U ( y ) ) . 18. PART I I § 1 . The Max ima l Pure Subspaces . I n t h i s p a r t , ou r c o n c e r n w i l l be t o f i n d the s ub -s p a c e s o f ^ ( r ) w h i c h c o n t a i n o n l y p r o d u c t s . D e f i n i t i o n 2.1 A subspace P o f u ( r ) i s pu re i f P c P r [ U ] . Examp le s : The z e r o subspace and subspaces g e n e r a t e d by a s i n g l e s ymmetr i c p r o d u c t a r e pu re subspaces o f ^ ( r ) • -*-f dim U > 2 t h e n u ( r ) i s p u r e : e . g . , i f u ^ u ^ u ^ a re i n d e p e n d e n t v e c t o r s o f U t h e n u^ f. Ug + u^ * u-^ 1 Pg'flJ] . The sum o f two p r o d u c t s i n ^ ( r ) w h i c h have r - l common f a c t o r s i s a p r o d u c t . Hence a n y ' s e t [x-^*. „ » * x r _ 1 * v : v € V} , where V i s a subspace o f U and x-^, . . . , x r _ - L n o n - z e r o v e c t o r s , f o rms a pu re subspace w h i c h we w i l l denote X j ^ * . . . * x 1 * V . C l e a r l y ; , ... . :x„ x ^ * . . , * x r _ 1 * V c x ^ *„ . . * x r _ - j _ *U . A c c o r d i n g l y , we seek o n l y \, t ho se pu re subspaces w h i c h a r e n o t c o n t a i n e d i n a l a r g e r pu re s ub space ; i . e . , t he max ima l pu re subspaces o f U . D e f i n i t i o n 2.2 F o r any n o n - z e r o v e c t o r s x ^ , . . . , x ^ t he pure subspace x ^ * . . . * x r ^ * U i s c a l l e d a t y p e 1 subspace . 19. Theorem 2.3 I f d im U = n , any t y p e 1 subspace i s an n -d i m e n s i o n a l subspace o f u ( r ) • P r o o f : I f u 1 , . . . , u n i s a b a s i s o f U t h e n t he p r o d u c t s . . * x r _ i * u i > i = 1,.-. . , n , f o rm a b a s i s o f x ^ * . . . * x - ^ U . The f o l l o w i n g d e f i n i t i o n and lemma a r e u s e f u l i n showing t h a t e v e r y t y p e 1 i s max ima l pu re when; d im IT i s - n o t . 2 D e f i n i t i o n 2. k A s e t o f v e c t o r s , , . 3 , i n a v e c t o r space U i s a f l a t o f U i f t h e r e i s a v e c t o r x and a .subspace S such t h a t 3 = x+S . I n p a r t i c u l a r , e v e r y subspace o f U i s a f l a t o f U . 3 i s a p r o p e r f l a t i f 3 = x + S and S i s a p r o p e r subspace o f U . Lemma 2.5 I f U 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 ve r a f i e l d F and ' a,ny c o l l e c t i o n o f p r o p e r f l a t s i n U such t h a t C a r d & < C a r d F t h e r e I s a n o n - z e r o x £ \j& . P r o o f : I f d im U = 1 , U ha s no p r o p e r f l a t s and i n t h i s ca se the theorem i s t r i v i a l l y t r u e . The p r o o f i s by i n d u c t i o n on the d i m e n s i o n o f U . L e t x-^,...,x n be a b a s i s o f U and denote the n-1 d i m e n s i o n a l subspace L ( x ^ . . . , x a - 2 , x n _ 1 + x x n ) by S x . Then Ca rd (S^ : ^ e F ] = Ca rd F . ( I f = S / t h e n , i n p a r t i c u l a r , x n _ 1 + \xn = c i 1 x 1 + - • •+a n _ 2 x n - 2 + a n - l ^ x n - l + x ' x n ^ f o r s c a l a r s a 1 } . . . , a i n F . T h e r e f o r e , = 0 f o r a l l i = l , . . . , n - 2 and a n _ ] _ = 1 w h i c h i m p l i e s \ - \ ' . ) 20. S i n c e Ca rd & < Ca rd F ,' t h e r e e x i s t s u i n F such t h a t S 4 & . Because any f l a t £ i n & i s p r o p e r , dim 5 n S _ n-2 . Thus , each J n S i s a p r o p e r f l a t o f S . ( I f 3 = x + S, G = y + T a r e f l a t s and c e 3 A G t h e n 3 = e + S , G = e + T . T h e r e f o r e , 3 n G = c + S Q T , i f 3; n G ^ 0, and dim gt n G = dim S n T .) By i n d u c t i o n , t h e r e i s x e S such t h a t x k niS' where = {3 fl S s 3 e fi) . C o n s e q u e n t l y x ^ (jj© as w e l l . A subspace o f ^ f Y ) 1 S a max ima l pu re subspace i f no o t h e r pu re subspace c o n t a i n s i t . I f d im U = 1 t h e n u ( r ) ha s no p r o p e r subspaces b u t ^ ( T } i t s e l f i s t ype 1 and p u r e . Where d im U = 2 we s h a l l see (Theorem 2.7) t h a t U ( r ) i s a g a i n pu re p r o v i d e d t h a t F i s an a l g e b r a i c a l l y c l o s e d f i e l d . I n t h i s ca se i t f o l l o w s t h a t no t ype 1 subspace can be max ima l p u r e . When d im U > 2 we have t h e f o l l o w i n g r e s u l t : Theorem 2.6 L e t U be 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 ve r an i n f i n i t e f i e l d . I f d im U > 2 t h e n any subspace o f t y p e 1 i s " . .a , max ima l pu re subspace o f ^ ( r ) • P r o o f : We have seen t h a t e v e r y t y p e 1 subspace I s p u r e . * I f M = x ^ * . . . * x r _ 1 # U i s a t y p e 1 subspace and y an e l ement o f U'(r) such t h a t L(Msy) c P r [U] we w i l l show y e M . I n p a r t i c u l a r , y w i l l be a p r o d u c t y-^*.. . * y r . Our a s s u m p t i o n i s t h a t x - ^ . . . # x r _ 1 * u + X y 1 # . . . * y r e P r [ U ] , f o r e v e r y u e U and any s c a l a r \ . We show f i r s t t h a t 21. x 1 * . . . * x r _ 1 and y must have a- common f a c t o r . I f e i t h e r d im L ( x 1 , . . . , x > 2 o r dim U ( y ) > 2 choose a n o n - z e r o v e c t o r u 1 L ( y i ) f o r any i = l , . . . , r . (Lemma 2.5)= Theorem 1.17 i m p l i e s x 1 * . . . * x r 1 * u and y must have a common f a c t o r . Hence, by ou r c h o i c e o f u , x 1 * . . . * x r _ 1 and y must have a common f a c t o r . I f d im L ( x 1 , . . . , x ^ ) = 2 , Lemma 2.5 :'.v:i->X.ler, .c y i e l d s a v e c t o r v 1 L ( y i ) f o r any i = l , . . . , r such t h a t a t the same t i m e , v £ L ( x 1 , . . . >x-r_i ) . I f x y = x 1 * . . . * x r t h e n L ( x v , y ) c P [U] . S i n c e d im U ( x v ) > 2 , Theorem 1.17 a g a i n i m p l i e s x ^ * o . . * x r ^ and y have a common f a c t o r . I f d im L ( x 1 , . . . , x r __ 1 ) = 1 and d im U ( y ) = 2 we may choose a v e c t o r w 1 L (x -^ , . . . , x r _ 1 ) sueh t h a t w 1 U ( y ) as w e l l . I f x w = x 1 * . . . * x r _ 1 * w t h e n L ( x w , y ) c P [U] b u t U^x^) 4= U ( y ) . I n t h i s c a s e , Theorem 1.18 i m p l i e s x and y have a common f a c t o r « F i n a l l y assume b o t h d im L ( x ^ , . . . , x 1 ) and d im U (y ) a r e 1. I f L ( x 1 ) = L ( y 1 ) we a r e f i n i s h e d . A c c o r d i n g l y , suppose d im L ( x ^ , y 1 ) = 2 and choose a v e c t o r z 4 L(x-j_,y^) , I f x z = x 1 * . . . * x r _ 1 * z t h e n -X + y e P r [U] and C o r o l l a r y 1.15 y i e l d s U ( y ) c U ( x g ) . Hence y^ e L(x1>z) , c o n t r a d i c t i n g ou r c h o i c e o f z . The p r o o f w i l l be c o m p l e t e d by i n d u c t i o n on r . 22. When r = 2 , M = x - ^ U , yns y^y^j and. we have j u s t shown t h a t e i t h e r L ( x 1 ) = L ( y 1 ) o r L ( x 1 ) = L ( y 2 ) . Our i n d u c t i o n h y p o t h e s i s i s t h a t f o r some r > 2 e v e r y t y p e 1 subspace o f U ( r _ l ) i s a max ima l pu re subspace . Assume t h a t t he common f a c t o r o f x 1 * . . . * x r _ 1 and y i s L ( x r _ 1 ) = L ( y r ) . Theorem 1.13 i m p l i e s L f x - j * . . „ * x r _ 2 * U , y i * • • • * y r _ i ) c P r [ ^ f U ] . By i n d u c t i o n , x 1 * - . . , . * x r _ 2 * U i s a max ima l pu re subspace o f U ( r - l ) and t h e r e f o r e y i * » « » * y r _ i e ' x i * * • • * x r _ 2 * U * S i n c e L ( x r - l ) = L(yT) i t f o l l o w s by Theorem 1.11 t h a t y e M . /• To o b t a i n o t h e r max ima l pu re s ub space s , i t i s h e l p f u l t o c o n s i d e r t he i s o m o r p h i s m o f u ( r ) and H r [ § 1 , . . . , § n J where n = d im U , (Theorem 1 .6). I f we choose two d i s t i n c t i n d e t e r m i n a t e s and %^ t h e n H r [ ? i > § j ] c a n be c o n s i d e r e d as an ( r + l ) - d i m e n s i o n a l subspace o f H I > [ § 1 , . . . , § n l . Mo reove r , i f t h e s e p o l y n o m i a l domains a r e t a k e n o v e r an a l g e b r a i c a l l y c l o s e d f i e l d t h e n any homogeneous p o l y n o m i a l i n j u s t two i n d e t e r m i n a t e s can be w r i t t e n as a p r o d u c t o f l i n e a r f a c t o r s [6, Theorem 10.8, p .29]. I f u ^ , . . . , u i s a b a s i s o f U the i s o m o r p h i s m o f Theorem 1.6 I m p l i e s t h a t f o r any p a i r ( u . , u . ) o f d i s t i n c t b a s i s e l emen t s t he s e t [ s ^ * . v . * s r | s ^ ' e L ( u ^ , u . f o rms a pu re subspace o f U ( r ) • T h i s s u gge s t s the f o l l o w i n g theorem: Theorem 2.7 L e t If be 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 23. an a l g e b r a i c a l l y c l o s e d f i e l d . I f S i s any 2 - d i m e n s i o n a l subspace o f U t h e n = { s 1 * . . . * s r | s^ € S} i s a pu re subspace o f u ( r ) • P r o o f : We have a l r e a d y n o t e d t h a t i f u^ and u^ a r e d i s t i n c t e l emen t s o f a b a s i s f o r U t h e n L ( u i * u j ) ( r ) c p r [ U ] • I f u,v a r e i n d e p e n d e n t v e c t o r s i n S choose a b a s i s o f U c o n t a i n i n g u and v . W i t h t h i s b a s i s f o r U , t h e i s o m o r p h i s m o f Theorem 1.6 y i e l d s s ( r ) c P r [ U ] . To v e r i f y t h a t S ( r j i s I ndeed a subspace we, must show t h a t i f x , y e S ^ r j - and x + y = z = z 1 * . . . * z r t h e n z ^ e S ; i = l , . . . , r . . Bu t t h i s i s immed ia te consequence o f Theorem 1.19 s i n c e L ( U ( x ) , U(yi)) « S . D e f i n i t i o n 2.8 L e t U be 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 l g e b r a i c a l l y c l o s e d f i e l d . I f S i s a 2 - d i m e n s i o n a l subspace o f U t h e n s ( r ) i s c a l l e d a t ype r subspace o f U ^ r j . A l s o , f o r each ::x e U we denote t he p r o d u c t x * . . . * x by * ( r ) • Remark: C o r o l l a r y 1.7 i m p l i e s any t y p e r subspace has d i m e n s i o n r + 1 . Theorem 2.9 L e t U be 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 l g e b r a i c a l l y c l o s e d f i e l d and S a 2 - d i m e n s i o n a l subspace o f U . Then ^{^T) 1 S a max ima l pu re subspace o f ^ ( r ) • P r o o f : I f d im U = 2 t h e n U := S and i s max ima l 24. p u r e . Assume t h e n t h a t d im U > 2 . Suppose t h a t L ( S ^ r j , y ) i s pu re f o r some y e . Then y must he i n P r [ U ] ; say, y = y 1 * . . . * y r . We must show t h a t each y i e S , I f d im U ( y ) =,2 and u i s any v e c t o r i n S t h e n u ( r ) + y € p r t u ] • C o r o l l a r y 1.15 i m p l i e s u € U ( y ) . Hence S = U ( y ) and y± e S , i = l , . . . , r . I f d im U ( y ) > 2 an a p p l i c a t i o n o f Lemma 2.. 5 "to t he space S shows t h a t we c an p i c k i n S a v e c t o r u w h i c h i s n o t c o n t a i n e d i n any o f the f a c t o r s o f y w h i c h a r e t h e m s e l v e s i n S j i . e . , u £ S , u | ^(y^) > i = l , . . . , r . Then L ( u ( r ) , y ) c P r [ U ] . Theorem 1.17 i m p l i e s t h a t u ( r ) and y have a common f a c t o r , w h i c h i s i m p o s s i b l e . (Theorem 1.17 i s a p p l i c a b l e s i n c e an a l g e b r a i c a l l y c l o s e d f i e l d i s i n f i n i t e . ) The f a c t ' t h a t any ^fr^ 1 S a pu re subspace i m m e d i a t e l y s u gge s t s t h a t pu re subspaces o f u ( r ) m a y be o b t a i n e d by c o n s i d e r i n g s e t s o f p r o d u c t s w i t h a c e r t a i n number o f common f a c t o r s such t h a t t he r e m a i n i n g f a c t o r s o f ea«h p r o d u c t l i e i n t he same 2 - d i m e n s i o n a l subspace o f U ; e . g . , i f x 1 , . . . , x r _ k a r e n o n - z e r o v e c t o r s o f U and S i s a 2 - d i m e n s i o n a l subspace o f U t h e n [x^o. . * x r _ k * s 1 * . . . # s k | s i e S, i = l , . . . , k ] f o r k > 1 s h o u l d be a pu re subspace . A n a t u r a l n o t a t i o n n f o r such a s e t i s x - , * . . . * x _ , * S , , > . 2 5 . Theorem 2.10 L e t U be a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e : o v e r an a l g e b r a i c a l l y c l o s e d f i e l d . I f 1 < k < r l e t S be a 2 - d i m e n s i o n a l subspace o f U and x ^ , . . . , x k any n o n -z e r o v e c t o r s o f U . Then, x i * < ' • • * x r _ k * S ( k ) ^ s a P u r e subspace o f u ( r ) • P r o o f : Suppose x = x 1 * . . . * x r j c * s i * » • • * s j c a n Q y = x 1 # . ..'*Xjj_^*t 1*.. , * t k a r e i n x i *«> • » * x r _ k * s ( k : ) ' Theorem 2.7 i m p l i e s t h e r e a r e v e c t o r s w ^ , . . . , w k o f U such tha.t - t l ) - s 1 * . . . * s k + t - j _ * . . . * t k = w 1 * . . . * w k i n U ^ k ^ . Theorem 1.19 i m p l i e s w i € S , i = l , . . . , k . A p p l y i n g Theorem 1.11 tsa(L-.)yexafctlyjiie»-kc folime.&nwe o b t a i n . x + y = X j * , , • * x r _ k * w 1 * « • € x i * * * * * x r « k * S ( k ) * Subspaces o f t h e f o r m x ^ * . . . * x r k * ^ ( k ) a r e n o ^ n e c e s s a r i l y max ima l pu re s ub space s ; e . g . , i n tf(^) w e have xl*£>(2) c S ^ ^ whenever x ^ i s a v e c t o r i n a 2 - d i m e n s i o n a l subspace S o f U . We a v o i d t h i s s i t u a t i o n by t h e f o l l o w i n g d e f i n i t i o n : D e f i n i t i o n 2.11 Assume U l 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 l g e b r a i c a l l y c l o s e d f i e l d . L e t k , r be Ln^; i n t e g e r s s a t i s f y i n g 1 < k < r and x ^ . . . ^ k any n o n - z e r o v e c t o r s i n U . I f S i s a 2 - d i m e n s i o n a l subspace o f U such t h a t x i | S } i = l , . . . , r - k , t h e n the pu re subspace 26, i s c a l l e d a t y p e k subspace o f U I f d im U = 1,2 t h e r e a r e no subspaces o f t ype k, 1 < k < r . O t h e r w i s e , we have Theorem 2.12 L e t U be 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 l g e b r a i c a l l y c l o s e d f i e l d P . I f 1 < k < r and d im U > 2 t h e n e v e r y subspace o f t y p e k is< a max ima l pu re subspace o f u ( r ) -P r o o f : L e t M be a subspace o f t y p e k and assume L (M ,y ) c P r [ U ] f o r some y e U ^ r ^ . Then, i n p a r t i c u l a r , y € P p [ U ] . L e t y = y 1 * . . . * y r . I f M = x i * » • • * x r _ k ; * s ( k ) we may choose i n d e p e n d e n t v e c t o r s s , t . i n S such t h a t s , t 1 L ( y i ) f o r any i = 1 , — , r . I f x = x 1 * . . . * x r _ k * s * t ^ t h e n x + L ( y ) c P [U] s i n c e x + L ( y ) c L (M , y ) . By d e f i n i t i o n , no x ^ i s i n S . C o n s e q u e n t l y , dim U ( x ) > 2 . S i n c e P i s a l g e b r a i c a l l y c l o s e d i t i s i n f i n i t e and Theorem 1.17 s t a t e s x and y must have a common f a c t o r , say L ( x r _ k ) = L ( y r ) . Theorem 1.13 i m p l i e s .'..( L ( x 1 * . . . * x r _ k _ 1 * S ( k ) , y ^ . ^ y ^ ) c P ^ U ] . I f r - k = 1 Theorem 2.9 i m p l i e s y 1 * » » » * y r _ 1 e S ^ ^ j a By i n d u c t i o n on r - k , x ^ # . . • * x r _ k _ i * ^ ( k ) 1 subspace o f U/ , \ , so y 1 * . „ . * y n e x , * . i s a max ima l pu re nd hence y € M . *x. r - k - l * S ( k ) . T h e r e f o r e , y e M . 27. Our pu rpo se i n t h e r e m a i n i n g p o r t i o n o f §1 i s t o d i s c o v e r when t h e o n l y max ima l pu re subspaces o f u ( r ) a r e t h o s e o f t y p e s l , . . . , r . Theorem 2.13 L e t U be 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 l g e b r a i c a l l y c l o s e d f i e l d F . Assume c h a r F i s k n o t a;: p r ime $3 such t h a t r = p f o r some p o s i t i v e i n t e g e r k . I f P i s any pu re subspace o f ^ ( r ) a n < * ^ i m P > 1 t h e n P c o n t a i n s a v e c t o r x such t h a t d im U ( x ) > 1 . P r o o f : Assume t h a t P i s a pu re subspace o f ^ ' T } such t h a t d im U ( x ) = 1 f o r e v e r y x € P b u t f i s n o t o f t he f o r m L (U/ >) f o r some u i n U ; i . e . , U c o n t a i n s a t l e a s t two i n d e p e n d e n t v e c t o r s , say u and v , such t h a t u ( r ) a n c * v ( r ) a r e i n P ' C o m p l e t i n g u and v t o a b a s i s o f U , u ^ y and v ( r ) w i l l be p a r t o f a b a s i s o f u ( r ) b y Theorem 1.6, hence i n d e p e n d e n t . Since $> i s a pu re subspace t h e r e i s a w i n U such t h a t u ( r ) + v ( r ) = w ( r ) * Theorem 1.19 i m p l i e s w e L ( u , v ) . L e t w = au + bv where a,b e F . S i n c e u ( r ) a n ( * v ( r ) a r e * i n d e p e n d e n t n e i t h e r a = 0 no r b = 0 . By i n d u c t i o n , w ( r ) = a ? u < ; r ) ^ ^ ^ S i n c e t he p r o d u c t s u ( r _ f c ) * v ( k ) , k = 0 , . . . , r a r e p a r t o f a b a s i s o f U ^ r ^ we have : 28. a = b = 1 ( k ) a r " V = 0 k = l , . . . , r - l . ,.1 S i n c e a and b a r e n o n - z e r o , a ~ b ^ 0 and (^) o l = 0 f o r k = l , . . . , r - l . Hence F has p r ime c h a r a c t e r i s t i c p and p | ( £ ) f o r e v e r y k = l , . . . , r - l . I n p a r t i c u l a r , p | r . L e t k be the h i g h e s t power o f p ' ' k k d i v i d i n g r and s e t r = p q . I f q ^ l t hen p < r and P I ( pk ) • 0 n t n e ° t h e r h and , ( g) = q<>s where s i s t he i n t e g e r 1 I P J ^ " M . . Bu t p S« s s i n c e p^ d i v i d e s I I m-1 p k _ m k k p q-m i f and o n l y I f i t d i v i d e s p -m whenever 1 <. X <_ k . k T h e r e f o r e , p >J; ( g ) • T h i s c o n t r a d i c t i o n c o m p l e t e s t he „> p r o o f . Remark: I f F i s an a l g e b r a i c a l l y c l o s e d f i e l d o f p r ime c h a r a c t e r i s t i c p and r = p f o r some p o s i t i v e i n t e g e r k t h e n Theorem 2.13 i s f a l s e . I n t h i s ca se .. A = { x ^ r ^ | x e U ) i s a pu re subspace o f U : 1/r I f x ^ e A t h e n ct*(r) = ( a - x ) f ^ r ) f o r any a e F because F i s a l g e b r a i c a l l y c l o s e d . I f x ( r ) , y ( r ) e A t h e n x ( r ) + y ( r ) = (x + y ) ( r ) k because p | ( m ) f o r each m = l , . . . , r - l . 2 9 . D e f i n l t i d n o 2 . 1 4 I f P i s a pu re subspace o f u ^ and u € U l e t P n , ( u ) = {t e 'P : L ( u ) i s a f a c t o r o f t ' o f . m u l t i p l i c i t y a} . I f P_(u) = P t h e n L ( u ) w i l l he c a l l e d a f a c t o r o f P o f  m u l t i p l i c i t y a . P (u ) w i l l s i m p l y denote {t e P : L ( u ) i s a f a c t o r o f t ] . I f L ( u 1 ) , . . . , L ( u k ) a r e t he f a c t o r s o f a pu re subspace P w i t h m u l t i p l i c i t i e s c ^ , . . . , a k t he s t a t e m e n t " P ha s m f a c t o r s " w i l l a lway s mean a^+. . »+0^. = m . Any P (u) i s a subspace o f P . Any pu re subspace can have a t most r f a c t o r s . I f P has r f a c t o r s t h e n P i s 1 - d i m e n s i o n a l . Any max ima l pu re subspace o f * ^ r j w i t h e x a c t l y r - l f a c t o r s i s e v i d e n t l y a subspace o f t y p e 1 . A t the o t h e r ext reme we have : Theorem 2 . 1 5 L e t U be 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d F . I f c h a r F i s n o t a k p r ime p sueh t h a t r = p f o r some p o s i t i v e i n t e g e r k t h e n t he o n l y max ima l pu re subspaces o f ^ ( T ) w i t h o u t f a c t o r s a r e t ho se o f t y p e r . P r o o f : L e t P be a max ima l pu re subspace o f u ( r ) w i t h o u t f a c t o r s . We f i r s t show t h a t P c anno t c o n t a i n an x such t h a t d im U ( x ) > 2 . Assume, t o the c o n t r a r y , t h a t x = x 1 * . . . * x r i s such an e l ement o f P . Then each P , ( x i ) i s a p r o p e r subspace o f P , s i n c e P ha s no f a c t o r s . F o r e v e r y y e P , L ( x , y ) c P c P [U] . Theorem 1 . 1 7 i m p l i e s y 30. has a f a c t o r i n common w i t h x . T h e r e f o r e , P = U j j ^ P - (Xj_) > c o n t r a d i c t i n g Lemma 2.5. Thus , f o r any n o n - z e r o y e P , e i t h e r d im U (y ) = 1 o r d im U ( y ) = 2 . Theorem 2.13, howeve r , i m p l i e s t h a t P c o n t a i n s a t l e a s t one x f o r w h i c h d im U ( x ) = 2 . We s h a l l show t h a t f o r any y e P , U ( y ) c U ( x ) . T h i s i m p l i e s P c S ^ r j where S = U ( x ) . E q u a l i t y w i l l h o l d s i n c e P was assumed t o he max ima l . I f d im U ( y ) = 1, C o r o l l a r y 1.15 i m p l i e s U ( y ) c U ( x ) s i n c e x + y € P [U] i f y € P . Hence we may r e s t r i c t ou r a t t e n t i o n t o t ho se y € P such t h a t dim U ( y ) = 2 . I f d im U (y ) = 2 and x , y have no common f a c t o r s , Theorem 1.18 y i e l d s U ( x ) = U ( y ) . I t r e m a i n s t o c o n s i d e r the ca se when dim U(x ) = d im U ( y ) = 2 , U ( x ) ^ U ( y ) , b u t x , y have common f a c t o r s . C o r o l l a r y 1.16 i m p l i e s dim U ( x ) n U ( y ) = 1 . Hence x , y canno t have two i n d e p e n d e n t f a c t o r s . L e t U ( x ) n U ( y ) = L ( u ) and x = X l * . . . * x k * u ( r _ k ) y = y 1 * . . . * y m * u ( r _ m ) o < k,m < r - i where we suppose L ( u ) i s d i s t i n c t f r o m e v e r y L ( x i ) and L ( y n - ) j i = l , . . . , k , j = l , . . . , m . S i n c e 31. dim U ( x ) = dim U (y ) = 2 , k,m > 0 . I f k ••< m and x = x x * - • • * x k * u ( m _ k ) > f = y i * - - - * y m t n e n L ( x ^ y ) <= p r [ u ] i m p l i e s L ( x , y ) c P m [ U ] by Theorem 1.13. S i n c e k ^ m , dim U ( x ) = 2 . I f d im U ( y ) = 2 , Theorem 1.18 i m p l i e s x and y have common f a c t o r , w h i c h i s i m p o s s i b l e . . Hence k ^ m. S i m i l a r l y , one shows k < m . T h e r e f o r e , k = m . I f dim U ( x ) = 2 t h e n L ( x , y ) c P ^ t ^ l and s i n c e x and y have no common f a c t o r s we have a l r e a d y seen t h a t U (y ) c U ( x ) , i m p l y i n g U ( y ) c U ( x ) . Hence we may assume d im U ( x ) = dim U (y ) = 1 . S i n c e P ha s no f a c t o r s , P ~ B(u) ^ 0 . I f d im U ( t ) = 1 and t e P ~ P ( u ) t h e n x + t e P r [ U ] , so U ( t ) c U ( x ) by C o r o l l a r y I .15. S i m i l a r l y , U ( t ) c U ( y ) . T h e r e f o r e , U ( t ) = L ( u ) $ i . e . , t h e r e i s an a i n F such t h a t t = a u ( ~ ) € p ( u ) • Thus d im U ( t ) = 2 f o r e v e r y t € P ~ B (u ) . To c omp le t e the p r o o f we show P ~ P (u ) c o u l d n o t c o n t a i n an e l ement t h a t has a f a c t o r i n common w i t h b o t h x and y . Assume, t o t he c o n t r a r y , ; t h a t P ~ P ( u ) does c o n t a i n such an e l e m e n t , say t = t ^ * . . . * t r . S i n c e dim U (x ) = 1 and dim U(.y;) = 1 b u t u i s n o t a f a c t o r o f t , L ( x ^ ) must be the common f a c t o r o f x and t , L(y^<) the common f a c t o r o f y and t . We may assume t h a t x-^y-^u a re . i n d e p e n d e n t and U ( t ) i s L ( x 1 , y 1 ) . L e t t = x 1 * y ^ # t ^ * . . . * t r , 32. £ = y ^ * t ^ * . . . * t r , and suppose k deno te s x - ^ k i ) * u ( r - k ) ' Then, Theorem 1.13 i m p l i e s L(x*,$) c PR_3_[U] . I f k = 1 t h e n L ( u ) c U ( t ) by C o r o l l a r y 1.15 > c o n t r a d i c t i n g u 1 L ( x 1 , y 1 ) . T h e r e f o r e , U ( £ ) = L ( x 1 , u ) . I f d im U ( i ) = 1 t h e n U ( £ ) = L ( y 1 ) and C o r o l l a r y 1.15 wou ld i m p l y L ( y 1 ) c U ( x ) , a g a i n a c o n t r a d i c t i o n . T h e r e f o r e , d im U(x') = d im U(^ ) = 2 . I f U ( x ) V U ( £ ) Theorem l . l H r i m p l l e s x" and £ have a common f a c t o r , n e c e s s a r i l y L ( x ^ ) . I f y deno te s y l ( k - l ) * u ( r - k ) w e f i n a y a I i ^ ^ a l s o have a common f a c t o r , n e c e s s a r i l y L ( y ^ ) . R e p e a t i n g the argument k-1 time.s we see t h a t r = 2k and t = ^j.^J^llk) 3 f o r s o m e n o n - z e r o a i n F . S i n c e x + t e P [U] Theorem 1.13 i m p l i e s u ( k ) + ^ l ( k ) e * k ^ ^ f P r some n o n - z e r o s c a l a r \ . S i m i l a r l y , u ( k ) + ^ ^ ( k ) € ^k^-^ ^ o r s o m e non - z e r o s c a l a r \i . L e t u ( . k ) . + X y l ( k ) = Z ! » . . . t z k and u ( f c ) + - w ^ , . ^ . Theorem 1.19 i m p l i e s z ^ e L ( u , y 1 ) and w^ e L(u,x-^) f o r e v e r y i = l , . . . , k . I f dim, L( z ^ , r ..'., z k ) : = ! 2 ' t h e n L ( z 1 , . . . , z k ) = U ( y ) and U ( x + t ) = L ( x 1 , U ( y ) ) ha s 3 d i m e n s i o n s . But t h e n Theorem 1.17 i m p l i e s x + t and y + t must have a common f a c t o r w h i c h i s i m p o s s i b l e . T h e r e f o r e , dim. L ( z 1 , -z^ . ) = 1.. S i m i l a r l y , d im L ( w 1 ,w f c) =. 1. We may assume z-^ = au +• py-^ and w^ = 6u + YX^^ where a , P , Y » & a r e non - z e r o s c a l a r s . C o n s e q u e n t l y , t h e r e a r e n o n - z e r o s c a l a r s X , |j such t h a t x + t = X x ] _ ^ c ) * ( a u + ^ y ^ ) ^ ) and 33-y + t = n y i ( k ) * ( Y U + 6 x i ) ( k ) • Hence, d im U(x+t ) = d im U(y+t)=2. F u r t h e r m o r e , U (x+t ) ^ U(y+t ) . Theorem 1.18 i m p l i e s x+t and y+t must have a common f a c t o r w h i c h i s i m p o s s i b l e . Thus we have shown P ~ P (u ) c o n t a i n s some e l ement w i t h e i t h e r no f a c t o r i n common w i t h x , o r no f a c t o r i n common w i t h y . L e t t be an e lement o f P ~ p ( u ) w i t h no f a c t o r i n common w i t h x . We have shown t h a t dim U ( t ) = 2 . Theorem 1.18 i m p l i e s U ( x ) = U ( t ) . I f t ha s no f a c t o r i n common w i t h y t h e n U (y ) = U ( t ) as w e l l and we a r e f i n i s h e d . O t h e r w i s e , l e t t = t^*,. . . * t and suppose L ( t ^ ) = L ( y 1 ) i s a common f a c t o r o f t and y . Then, L ( y 1 ) = L ( y ) c U ( t ) = U ( x ) . Bu t U (y ) c U ( x ) i m p l i e s U ( y ) c U ( x ) and t h i s c o m p l e t e s the p r o o f . Theorem 2.16 I f U 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d F and 1 < k < r t h e n any. max ima l pu re subspace o f u ^ r ) ^wi th e x a c t l y r - k f a c t o r s ( c o u n t i n g r e p e t i t i o n s ) i s t ype k p r o v i d e d the c h a r a c t e r i s t i c : o f F i s n o t a p r ime p f o r w h i c h r = p where k i s a p o s i t i v e i n t e g e r . P r o o f : I f L ( x . ^ ) , . . . , L ( x r _ k ) a r e t he f a c t o r s o f such a subspace P l e t Q = {u^*...*^ | x ^ * . . . * x r k * u ] _ * • • • * u k e p ) • Theorem 1.13 i m p l i e s Q i s a pu re subspace o f U . Mo reove r , Theorem 1.11 i m p l i e s Q i s a max ima l pu re subspace because P i s . S i n c e P ha s e x a c t l y r - k f a c t o r s , Q ha s 3 4 . no f a c t o r s . Theorem 2.15 i m p l i e s Q = f o r some 2 - d i m e n s i o n a l subspace S o f U . T h e r e f o r e , P i s a t ype k subspace o f u ( r ) > 1 < k < r . Remark; Comb in ing t he r e s u l t s o f Theorems 2.6, 2.15 and 2.16, we have shown t h a t when U 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 i^ -o v e r a f i e l d o f c h a r a c t e r i s t i c n o t a p r ime p , i f r = p f o r some p o s i t i v e i n t e g e r k , t h e n the max ima l pure subspaces o f u ( r ) a r e p r e c i s e l y t ho se o f t y p e s l , . . . , r ; p r o v i d e d P i s a l g e b r a i c a l l y c l o s e d . §2. I n t e r s e c t i o n s o f Max ima l Pure Subspaces . D e f i n i t i o n 2.17 Two t ype 1 subspaces x 1 * . . . * x r 1 * U and y, * . . . * y , . -, *U a r e a d j a c e n t i f x , * . . . * x , and y - , * . . . * y -, " ± r - l — l r - l "1 " r - l have e x a c t l y r - 2 common f a c t o r s c o u n t i n g m u l t i p l i c i t i e s . Remark: When r=2 any two d i s t i n e t t y p e 1 subspaces a r e a d j a c e n t and , i n g e n e r a l any two a d j a c e n t subspaces a r e d i s t i n c t . Throughout §2 we w i l l a lway s assume the subspaces under d i s c u s s i o n a r e d i s t i n c t and t h a t P i s an a l g e b r a i c a l l y c l o s e d f i e l d . Theorem 2.18 Two t ype 1 subspaces M,N a re a d j a c e n t i f and o n l y i f d im MflN = 1 . O t h e r w i s e , dim MflN = 0 . P r o o f : L e t M = x 1 * . . . * x r j*U and N = y 1 * . . . * y 2.*u b e t y p e 1 s ubspaces and assume MHN ^ 0 . I f t e MDN t h e n t = x 1 * . . . * x r _ 1 * u = y x * . . . * y r _ 1 * v 3 5 . f o r some u and v i n U . I f t i s n o t z e r o Theorem 1.13 i m p l i e s L ( u ) ^ L.(v) s i n c e M and N a r e d i s t i n c t , and C o r o l l a r y 1.9 shows x 1 * . . . * x r _ 1 and ' y ^ * * • • * y r _ i have e x a c t l y r - 2 common f a c t o r s . C o n v e r s e l y , i f M.N a r e a d j a c e n t l e t L ( x i ) = L ( y i ) ; i = l , . . . , r - 2 . Then M x ^ ) £ L ( y r _ i ) and i f t = x 1 * . . . * x r _ 1 * y ^ c 1 h w e e h a y 4 ; " L ( t ) c ^ l n N , I f s i s any n o n -z e r o e l ement o f MDN t h e n Theorem 1.13 i m p l i e s s e L ( t ) . T h e r e f o r e , MHN = L ( t ) . Theorem 2.19 L e t M = x 1 * . . . * x r 1 * U be a subspace o f t ype 1 and N = y-^*.. • * y r _ k * S ^ ^ be a subspace o f t y p e k where 1 < k < r . L e t x denote x - , * . . . * x „ , and y denote 1 r - l 17 y1*»"*yr.1 • Then, (a ) d im MC1N = 2 i f and o n l y i f x and y have e x a c t l y r - k common f a c t o r s w h i l e t he r e m a i n i n g k-1 f a c t o r s o f x a r e i n S . (b) dim MOT = 1 i f and o n l y i f x and y have e x a e t l y r - k - 1 common f a c t o r s w h i l e t he r e m a i n i n g k f a c t o r s o f x a r e i n .. S '.. ( c ) d im MfiN = 0 i f any o n l y i f e i t h e r (1) x and y have f ewe r t h a n r - k - 1 common f a c t o r s . (2) x and y have a t l e a s t r - k - 1 common f a c t o r s b u t a t l e a s t one o f the r e m a i n i n g f a c t o r s i s n o t i n S . 36. P r o o f : (a ) I f d im MAN = 2 l e t \,t2 be i n d e p e n d e n t e l ement s o f t he i n t e r s e c t i o n . There e x i s t v e c t o r s u - ^u^ i n U and s ^ , . . . , s k , * s ^ , . . . i n S such t h a t t x = - - * X T _ 1 * X X 1 = y - ! * ' • - * y r - k * s l * ' * ' * s k t 2 = x x * . . . * x r _ 1 * u 2 = y±*. . • * y r _ j t * s { * « • - * s k • (1) Assume b o t h u^ and Ug a r e n o t i n S . S i n c e t - ^ t g a r e i n d e p e n d e n t so a r e u ] _ J u 2 • Were k = r - l C o r o l l a r y 1.9 wou ld i m p l y L ( u ^ ) = L ( y 1 ) = L ( u g ) , a c o n t r a d i c t i o n . O t h e r w i s e , C o r o l l a r y 1.9 i m p l i e s L ( u 1 ) = L ( y 1 ) and L ( u g ) = L ( y g ) , say. Then Theorem 1.13 i m p l i e s x l * " ' * x r - l = X y 2 * * * ' * y r - k * s l * * * - * s k A c c o r d i n g t o D e f i n i t i o n 2.10, y^ 1 S f o r any i = l , . . . , r - k . From C o r o l l a r y 1.9 and Theorem 1.13 i t f o l l o w s t h a t L ( y 1 ) = L(y^)} c o n t r a d i c t i n g t he i ndependence o f u-^  and u^ • T h e r e f o r e , e i t h e r u-^  e S o r Ug e S . I f u-j^  c S t h e n L(u- L ) ^ L ( y i ) f o r any I = l , . . . , r - k . Then C o r o l l a r y 1.9 and (1) i m p l y each f a c t o r o f y i s a f a c t o r o f x ; i . e . , x and y have r - k common f a c t o r s . M o r e o v e r , t he r e m a i n i n g f a c t o r s o f x a r e i n S . The r e s u l t i s t he same i f u^ e S . C o n v e r s e l y , i f x and y have r - k common f a c t o r s w h i l e t h e r e m a i n i n g f a c t o r s o f x a r e i n S t h e n 37. x-^*. . .*xr_-^*S c MDN . I f t e MDN t h e n t h e r e a r e v e c t o r s u i n U and s i ' - - - - > s k i n s f o r w h i c h (2) t = x 1 * . ' . .*x r *u = y-^*. . • * y r _ k * s 1 * . . . * s f c . Theorem 1.13 i m p l i e s u c S . Hence x i * ' • ' * x r _ i * S = M n w and d im MflN = 2 . (b) I f d im MAN = 1 and MflN = L ( t ) t h e n t s a t i s f i e s (2). C o r o l l a r y 1.9 i m p l i e s x and y must have a t l e a s t r - k - 1 common f a c t o r s . I f x and y had r - k common f a c t o r s t h e n x 1 # . . . * x r _ 1 * S c MAN , c o n t r a d i c t i n g dim MAN'= 1 . C o n v e r s e l y , i f LCx^) = L ( y i ) f o r " " i = 1 , . . . , r - k - 1 , s ay , b u t L ( y k ) " "L(x i ) f o r any i = r - k , . . . , r - l w h i l e x „ x „ - i a r e i n S t h e n x , * . . . * x „ , * y _ e MAN . I f r - k r - l 1 r - l - r - k t i s any o t h e r e l ement o f MDN i t f o l l o w s f r o m ( 2 ) and C o r o l l a r y 1.9 t h a t L ( u ) = L ( y r _ k ) , s i n c e L'(y fc) ^ L ( x i ) f o r any i = r - k , . . . , r - l and y ^ , 1 S . Hence r —K t = \ x n * . . . * x „ -j * y v f o r some A i n F . T h e r e f o r e , j_ r —x r —K. dim MAN = 1 . ( c ) Assume dim MHN = 0 and suppose ( l ) I s f a l s e ; I . e . , x and y have a t l e a s t r - k - 1 common f a c t o r s . We must show a t l e a s t one o f the r e m a i n i n g k f a c t o r s o f x i s n o t i n S . B u t i f a l l k were i n S t h e n (b) wou ld i m p l y dim MflN = 1 , c o n t r a r y t o a s s u m p t i o n . C o n v e r s e l y , i f x and y have f e w e r t h a n r - k - 1 common 3 8 . f a c t o r s t h e n no u i n U can s a t i s f y (2) and i f t h e y have e x a c t l y r - k - 1 common f a c t o r s h u t some r e m a i n i n g f a c t o r o f x i s n o t i n S t h e n (2) i s a g a i n i m p o s s i b l e . T h e r e f o r e , MflN = 0 . Theorem 2.20 I f M = x.^*. . . * x r _ 1 * U i s a subspace o f t ype 1 and N = s ( r ) a subspace o f t y p e r t h e n d im MflN = 2 i f and o n l y i f x i € S f o r e v e r y i = l , . . . , r - l . O t h e r w i s e , MDN = 0 . P r o o f : F o r any t e MflN t h e r e a r e v e c t o r s s ^ , . . . , s i n S and u i n U such t h a t ( l ) t = x , * . . . * x , * u = s, * . . . * s x I r - i l r I f MDN ={= 0 t h e n ( l ) and C o r o l l a r y 1.9 i m p l y x . e S f o r a l l i = l , . . . , r - l . C o n v e r s e l y , i f each x ^ e S t h e n x ^ * . . .*xr_-^*S> c MDN; and f o r any t e MflN ( l ) and C o r o l l a r y 1.9 i m p l y u e S . Hence x 1 * . . . * x r _ 1 * S = MDN and so dim MDN = 2 . Theorem 2.21 L e t M - x, #...*x_ -^S/, x , • 1 r - K (k ) and N = y, * . . . * y *T/ \ be subspaces o f t y p e s ' k and m "'I w r - m {m) t . . . :_ . , •_ oT' v ' . y , . t y L . . . . r e s p e c t i v e l y where 1 < k <_ m < r . L e t x denote X T * . . . * X ^ , and y denote y . * . . . * y I f p i s t he j. r —ix. x r—m number o f f a c t o r s x and y have i n common t h e n 0 <_ p £ r -m and dim MHN = 1 i f and o n l y i f e i t h e r r - p = k+m o r r - p < k+m and dim SfiT = 1 . I n b o t h c a s e s t he r e m a i n i n g f a c t o r s o f 39. x , i f any , must be i n T w h i l e t he r e m a i n i n g f a c t o r s o f y , i f any , a r e i n S . O t h e r w i s e , MAN = 0 . P r o o f : I f t € MT1M t h e r e a r e v e c t o r s s ^ . - . ^ ^ . i n S and t - , , . . . , t i n T such t h a t 1 m. (1) t = x x * . . • * x r _ k * s 1 * . . . * s k = y 1 * . . . * y r _ m * t 1 * . . . * t m . We may assume L ( x ^ ) = L ( y i ) , i = l , . . . , p a r e the common • f a c t o r s o f x and y . Theorem 1.13 i m p l i e s ( 2 ) V l * - * X r - l c * V - , - * s k = . y p + l * 7 : , # y r . - f f i * V : ; ' ^ m ' L e t x = x p + 1 * . . . * x r _ k and y = y p + ] _ * . . . * y r _ m . I f MDN ^ 0 and r - p ^ k+m t h e n r - p < k+m . Fo r , i f r - p > k+m t h e n x ha s more f a c t o r s t h a n t , * . . . * t 1 m Hence C o r o l l a r y 1.9 wou ld i m p l y x and y have a common f a c t o r . I f SDT = 0 t h e n (2) and C o r o l l a r y 1.9 i m p l y r - p = k+m . I f dim SP IT = 1 t h e n x x , a r e i n T p+1 r - k and y . , . . . , y „ m a r e i n 3 .r^d 1^ S -° p+1 ' " r -m ' P r . i . When S = T ( l ) and C o r o l l a r y 1.9 i m p l y k = m . S i n c e x i , y i | S=T f o r e v e r y i = l , . . , f r - k , we have x l * - ' -*xr_}£ - y i * - " ' * y r - k ' c o n t r a d i c t i n g t he s t a n d i n g a s s u m p t i o n t h a t M ^ N . C o n v e r s e l y , i f r - p = k+m and the f a c t o r s o f x a r e i n T w h i l e t ho se o f y a re i n S t h e n x p + l * " * • * x r - - k * y p + l * ' ' ' * y r - m s a t , i s f i e s (2) and any o t h e r e lement o f MDN must be a s c a l a r m u l t i p l e o f i t . On the 40. o t h e r h a n d , i f r - p < k+m and SOT = L ( u ) t h e n ^ ' , * x r-k* y .p+l?* * - * y r - m * u ( s ) s a t i s f i e s (1) where s = ,,;p + k + m - r and any o t h e r e l ement o f MflN i s a s c a l a r m u l t i p l e o f i t . Thus , i n b o t h c a s e s d im MDN = 1 . Theorem 2.22 I f M = x i * • • • * x r _ k * s ( k ) i s a subspace o f t y p e k where 1 < k < r and N = T ( r ) i s a subspace o f t ype r t h e n Dim MflN = 1 i f and o n l y i f d im SnT = l and x± e T f o r e v e r y i = 1, . . . , r - k . O t h e r w i s e , MflN = 0 . P r o o f : I f t e MHN t h e r e a r e v e c t o r s s^,...,s^ i n S and t-^,. . . , t y i n T such t h a t ( 1 ) t — X 1 * X •, Mr S-t * • • • * S-, = tn * . . . t • v 1 r - k l . k 1 r I f MDN ^ 0 t h e n ( l ) and C o r o l l a r y 1.9 i m p l y , x± e T f o r i = l , . . . , r - k . Hence S ^ T , f o r D e f i n i t i o n 2.10 s t a t e s x 1 | S ; and ( l ) t o g e t h e r w i t h C o r o l l a r y 1.9 a l s o shows SOT ^ 0 . C o n v e r s e l y , i f SDT = L ( u ) and x± e T f o r i = l , . . . , r - k t h e n x i * • • • * x r _ k * u ( k ) € M n N a n d a n y ^ s a t i s f y i n g ( l ) must be a s c a l a r m u l t i p l e o f i t . Theorem 2.23 I f M = and N = T ( r ) a r e sub spaces o f \ t ype r t h e n d im MDN = 1 i f and o n l y i f d im SOT == 1 . O t h e r w i s e , MflN = 0 . P r o o f : S i n c e M and N a re d i s t i n c t , S ^ T and i f x 1 * . . . * x r i s i n MT)N t h e n x± e SDT ; MAN ^ 0 t h e n SOT ^ 0 , hence dim SnT SflT = L ( u ) t h e n L ( U ( r ) ) <=_ MfW and i f t = X u ( r ) f o r some X i n F . Hence so dim MDN = 1 . i = 1 , . . . , r . I f ' 1 . C o n v e r s e l y , t e MDN t h e n 42. PART I I I § 1 . P r o d u c t P r e s e r v e r s . D e f i n i t i o n 3.1 L e t U be any v e c t o r space . A l i n e a r mapping h o f u ( r ) i s a p r o d u c t p r e s e r v e r i f (1 ) h ( P r [ U ] ) c P r [ U ] (2) k e r h 0 P r [ U ] = 0 Remark: N o t i c e t h a t (2) does n o t e x c l u d e the p o s s i b i l i t y t h a t h may be s i n g u l a r . I n P a r t I I I we d i s c u s s the s t r u c t u r e o f p r o d u c t p r e s e r v e r s on U ( r ) > where U 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 . None o f ou r c o n c l u s i o n s i n v o l v e s t he f i e l d c h a r a c t e r i s t i c e x c e p t the f i n a l t heo rem. There we assume the f i e l d i s a l g e b r a i c a l l y c l o s e d and o f c h a r a c t e r i s t i c n o t a p r ime p i f k r = p f o r some p o s i t i v e i n t e g e r k i n o r d e r t o u t i l i z e t he r e s u l t s o f P a r t I I , § 1 ; i . e . , t he o n l y max ima l pure subspaces a r e p r e c i s e l y t ho se o f t y p e s l , . . . , r . W i t h t h e s e a s s umpt i on s the f i n a l theorem shows e v e r y p r o d u c t p r e s e r v e r i s i n d u c e d by a 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 o f U whenever dim U > r+1 . • Examp le s : (1) I f dim U = 1 t h e n dim = 1 . I f t he f i e l d 43-i s a l g e b r a i c a l l y c l o s e d t h e n any l i n e a r mapping o f U ( r ) i s i n d u c e d by a s u i t a b l e s c a l a r m u l t i p l e o f t he i d e n t i t y mapping o f U . (2) I f d im U = 2 and the f i e l d i s a l g e b r a i c a l l y c l o s e d , Theorem 2.7 i m p l i e s u ( r ) = • C o n s e q u e n t l y , any n o n - s i n g u l a r l i n e a r mapping o f u ( r ) w i l l be a p r o d u c t p r e s e r v e r . I n p a r t i c u l a r , assume r = 2 and u^,Ug a r e a b a s i s o f U . L e t h be some p r o d u c t p r e s e r v e r on U(2) such t h a t h ( u 1 * u 1 ) = u-^Ug . S i n c e h i s n o n - s i n g u l a r i t c o u l d n o t be i n d u c e d by a s i n g u l a r mapping o f U . I f A i s any n o n - s i n g u l a r mapping o f U t h e n h = ^(2) y i e l d s A ( u 1 ) * A ( u 1 ) = u 1 * u 2 . B u t C o r o l l a r y 1.9 i m p l i e s L (A (u^ ) ) •= L (u - L ) =-L(ug) w h i c h i s i m p o s s i b l e . Thus , h i s a p r o d u c t p r e s e r v e r on U ^ r j , b u t i s n o t i n d u c e d . Theorem 3»2 L e t U be any v e c t o r space . I f P i s a pu re subspace and h a p r o d u c t p r e s e r v e r o f ^ ( r ) t h e n h ( P ) i s a pure subspace and h|p i s a monomorphism. P r o o f : S i n c e h ( P [U] ) c P r [ U ] and P c P [U] we have h ( P ) c P_,[U] . i f h | P / x ) = 0 t h e n x e k e r h n P^[U] = 0 . Theorem 3-3 I f U 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d t h e n images o f a d j a c e n t subspaces under a p r o d u c t p r e s e r v e r on u ( r ) a r e d i s t i n c t . P r o o f : L e t h be a p r o d u c t p r e s e r v e r on U/ \ . I f 44. M = x 1 * . . . * x r _ 1 * U i s any t y p e 1 subspace t h e r e i s a l i n e a r mapping A o f U on to h(M) d e f i n e d by A( ..•; A (u ) = h ( x ^ * . . . * x ^ _ ^ * u ) f o r each u i n U . S i n c e h i s a p r o d u c t p r e s e r v e r , C o r o l l a r y 1.8 i m p l i e s A i s a mondmorphism. L e t \ = x 1 * . . . * x r _ 2 * y i * U f o r i = 1,2 be a d j a c e n t subspaces o f j and .h a p r o d u c t p r e s e r v e r such t h a t h ( M 1 ) = h ( M 2 ) . F o r the mappings k± : U -»'h^M^) j u s t d e f i n e d , h ( M 1 ) = h(Mg) i m p l i e s A g ^ l i s a n o n - s i n g u l a r mapping o f U . S i n c e the f i e l d i s assumed a l g e b r a i c a l l y c l o s e d A ^ A ^ ha s a n o n - z e r o e i g e n v e c t o r , say v . I f X i s t he c o r r e s p o n d i n g e i g e n v a l u e t h e n A^v = XA^v . S i n c e h i s a p r o d u c t p r e s e r v e r C o r o l l a r y 1.8 i m p l i e s y 1 = \ y 2 , c o n t r a d i c t i n the a s s u m p t i o n t h a t M-^  and M 2 a r e a d j a c e n t . C o r o l l a r y 3 . 4 L e t U be 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d and h a p r o d u c t p r e s e r v e r on u ( r ) • I f a r e a d j a c e n t and h ( M ) , h(N) a r e t ype 1 subspaces t h e n h ( M ) , h ( N ) a r e a d j a c e n t . P r oo f s S i n c e d im MHN = 1 , 0 < d im h(M) n h(.N) . Theorem 3 . 3 s t a t e s t h a t h(M) and h (N) a r e d i s t i n c t and Theorem 2.18 shows t h e y must be a d j a c e n t . Lemma 3•5 L e t n be a p o s i t i v e i n t e g e r . C o n s i d e r any c o l l e c t i o n X 1 , . . . , X f c o f n - s e t s s a t i s f y i n g k > n+1 and Ca rd X. fl X . = n - 1 , i f i ^ j . There i s an ( n - l ) - s e t , ; say ¥ , such t h a t ¥ c X^ f o r 1 <_ i <_ k . 45. P r o o f : L e t X.^  = { a ^ . . . a&n_j_*a} , Xg = {a^, . . . , a n _^ , b } and , i n a n t i c i p a t i o n , s e t W = {a -^ . . . , a ^ Then a ^ a ^ a r e d i s t i n c t , 0 < i < n . S i n c e k > n+1 t h e r e a r e a t l e a s t 3 s e t s . Assume W X^ f o r some j = l , . . . , k . S i n c e C a r d X j n = n-1 , a e X^ . S i m i l a r l y , b e X j . Hence Ca rd X j n ¥ = n - 2 . I f i ^ j , W ± X^ and ¥ $ ,Xj t h e n X^ and X j c anno t have t he same n-2 e l emen t s i n common w i t h W . Thus t h e r e a r e a t most ( n ~ o ) = n - 1 such X . . Bu t XX t h e r e a r e a t l e a s t n s e t s d i s t i n c t f r o m X.^  and Xg . A c c o r d i n g l y , t h e r e i s a s e t , say X^ , w h i c h c o n t a i n s W b u t i s d i s t i n c t f r o m b o t h X 1 and Xg . L e t X-j = [ a ^ , . . . , a n _ ^ , c } ,. R e a s o n i n g as b e f o r e we see t h a t c e X^ , f o r any X^ such t h a t W cj: X^ . Such an X . must t h e n c o n t a i n {a ,b ,c } w h i c h i m p l i e s Ca rd X - n W < n -2. J J T h i s wou ld c o n t r a d i c t Ca rd X j D X-^  = n-1 . We c o n c l u d e t h a t W c X^ f o r e v e r y i , 1 <_ i <_ k . D e f i n i t i o n 3.6 When r > 2 a c o l l e c t i o n {M^ : i e 1} o f a t l e a s t 2 d i s t i n c t t y p e 1 subspaces o f U ( r ) i s an a d j a c e n t  f a m i l y i f t h e r e a r e v e c t o r s x ^ , . . . , x r _ g , y ^ i n U such t h a t M i = x i * - - ' * x r _ * y j L * u f o r e v e r y i i n I . When r = 2 , any non-empty c o l l e c t i o n o f a t l e a s t two d i s t i n c t t ype 1 s ubspaces w i l l be c a l l e d an a d j a c e n t f a m i l y . Theorem 3.7 L e t TJ be any v e c t o r space and r > 1 an i n t e g e r . I f {M^ : i e 1} i s a c o l l e c t i o n o f p a i r - w i s e a d j a c e n t subspaces i n U ( r ) t h e n {M^ : i e I ) i s an a d j a c e n t f a m i l y whenever 46. Card I > r . P r o o f : F o r each i e I c o n s i d e r t he s e t o f o r d e r e d p a i r s , d e f i n e d by : ( L ( x ) , k ) e X^ i f L ( x ) o c c u r s k t i m e s as a f a c t o r o f M i j. e . g . , i f M-^  = x#x *y *z*u i n t h e n X1 = { ( L ( x ) , l ) , ( L ( x ) , 2 ) , L ( y ) , l ) , L ( z ) , l ) } . Each X ± i s an ( r - l ) - s e t . S i n c e {M i : i e 1} i s a p a i r - w i s e a d j a c e n t f a m i l y , .X^ 0 X j i s an ( r - 2 ) - s e t when i , j a r e d i s t i n c t e l ement s o f I . I n ca se r = 2 , {M^ : i e I ) i s t r i v i a l l y an a d j a c e n t f a m i l y . F o r r > 2 Lemma 3.5 a s s e r t s t h e r e i s an ( r - 2 ) - s e t c o n t a i n e d i n e v e r y X^ . T h e r e f o r e , {M^ : i e I ] i s an a d j a c e n t f a m i l y . C o r o l l a r y 3.8 L e t U be 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 over , an a l g e b r a i c a l l y c l o s e d f i e l d and h a p r o d u c t p r e s e r v e r o n : U ^ r ^ . I f {M^ : i e 1} i s an a d j a c e n t f a m i l y and each h ( M i ) i s a t y p e 1 subspace t h e n {(h(M^) : i € I ) i s a l s o an a d j a c e n t f a m i l y when Ca rd I > r . P r o o f : A c c o r d i n g t o C o r o l l a r y 3.4, i f i and j a r e d i s t i n c t e l emen t s o f I t h e n h(M.) and h (M. ) a r e a d j a c e n t . Hence the r e s u l t f o l l o w s f r om Theorem 3-7» D e f i n i t i o n 3.9 I f t he image o f e v e r y t ype 1 subspace i s a l s o a t y p e one subspace under a p r o d u c t p r e s e r v e r o f U ( r ) we w i l l say the mapping i s t y p e 1. Remark: I n v i e w o f Theorem 3.2, i t i s enough t o d e f i n e a t ype 1 47. mapp ing " as one f o r w h i c h images o f t y p e 1 subspaces a r e c o n t a i n e d i n t y p e 1 s ub space s . C o r o l l a r y 3.8 i s i m m e d i a t e l y a p p l i c a b l e whenever h i s a t y p e 1 mapp ing . § 2 . A s so e l a t e Mapp ing s . D e f i n i t i o n 3,10 L e t h be a t y p e 1 mapping o f ^ ( r ) - I f M - x ^ * . . . * x . r ^ * U i s a t ype 1 s ub space , chose, v e c t o r s v l * • • • j ^ r - l i n u such t h a t h(M) = y^*.. . *y ^*\J . The l i n e a r mapping A o f U d e f i n e d by Au = v i f h ( x 1 * . . . * x r _ ^ * u ) = y 1 * . . • * y r _ ^ * v w i l l be c a l l e d an a s s o c i a t e  mapp ing o f h w i t h r e s p e c t t o M . Remark; I n g e n e r a l , an a s s o c i a t e mapping may depend on b o t h h and M . A l s o , an a s s o c i a t e map o f M under h depends on t he c h o i c e o f the v e c t o r s y ^ , . . . , y ^ ^ . F o r , when y-j_*. • . * y 2.*^ = ^ i * " • - * y r _ i * ^ i * ' f o l l o w s t h a t t h e r e a r e s c a l a r s X^ i n F such t h a t y^ = X i y i r ( i ) ^ o r s o m e T € ^ r 1 a n d e a c n i = l , . . . , r - l . A c c o r d i n g l y , i f A , A ' a r e a s s o c i a t e maps o f t he same t ype 1 subspace c o r r e s p o n d i n g t o y ^ , . . . , y ^ and y l ' ' ' ' '"^T-1 > r e s P e c t i v e . l y , t h e n A = ( ' r i I i X i ) A ' • I n t h i s s e c t i o n we w i l l show t h a t i f U 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d t h e n any two a s s o c i a t e mappings o f t he same t y p e 1 mapping o f tf/ , d i f f e r by a s c a l a r m u l t i p l e . 48. Theorem 3;11 E v e r y a s s o c i a t e o f a t y p e 1 mapping i s n o n -s i n g u l a r . P r o o f : I f A i s an a s s o c i a t e mapp ing o f a t y p e 1 mapping h w i t h r e s p e c t t o x ^ * . . . * x r _ 1 * u " and Az = 0 f o r some z i n U t h e n h ( x 1 * c , * x r _ 1 * z ) = y 1 * . . . # y r _ 1 * A ( z ) = 0 . A t y p e 1 mapping i s b y d e f i n i t i o n a p r o d u c t p r e s e r v e r and so x 1 * . . . * x r _ 1 * z = 0 . D e f i n i t i o n 2.2 a s s e r t s x 1 ^ 0 , i = l , . . . , r - l . Hence, C o r o l l a r y 1.8 y i e l d s z = 0 . Theorem 5.12 Any n - d i m e n s i o n a l v e c t o r space o ve r an I n f i n i t e f i e l d F c o n t a i n s k v e c t o r s i n g e n e r a l p o s i t i o n f o r e v e r y i n t e g e r k >_ n . P r o o f : L e t u 1 , . . . , u n be any b a s i s o f U and Q^ n t he s e t o f s t r i c t l y i n c r e a s i n g sequence s , ( a - ^ , . . . , ^ ) , o f r i n t e g e r s chosen f r o m l , . . . , n . F o r each a = ( a , , . . . , a ) i n Q , l e t F = L ( u (,,...,u >) . By Lemma 2 . 5 , t h e r e i s a v e c t o r a at a n _ i u n + l i n U such t h a t u n + 1 1 U a F a f o r a e Q n _ 1 n and the v e c t o r s • " •^ u n > u n + i a r e seen t o be i n g e n e r a l p o s i t i o n . I f u i - ' ' a , ^ u k _ i a r e v e c t o r s i n g e n e r a l p o s i t i o n , f o r each a = ( a ^ , . . . , a n _ 1 ) i n Q^ . j k _ ] _ l e t F = L ( u ~ ' J . - J U ' - ) • A g a i n Lemma 2.5 i m p l i e s t h e r e I s a ° a l ° n - l v e c t o r u^ i n U such t h a t u^ 1 u a F a f o r a e Qn_i k _ i • The v e c t o r s u^, . . . , ! !^. a r e i n g e n e r a l p o s i t i o n . F o r , suppose t h e s e c o n t a i n e d a dependent n - s u b s e t : u , . . . , u , f o r some a = ( a 1 , . . . , a n ) e . Then a n ^ k s i n c e 49-u k ^ L ( u a 1 " " > % • T h e r e f o r e , { u . ^ , . . . , u a )• c C ^ , . . . , \ _ 1 3 , c o n t r a d i c t i n g t he a s s u m p t i o n t h a t t he v e c t o r s u i * * , , " ' u k _ i w e r e i n g e n e r a l p o s i t i o n . Lemma 3.13 I f U 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 ve r an i n f i n i t e f i e l d F and G a c o l l e c t i o n o f p r o p e r f l a t s i n U such t h a t C a r d C < Ca rd F t h e n U ha s a b a s i s B f o r w h i c h B n UC = 0 . P r o o f : A c c o r d i n g t o Lemma 2.5 t h e r e i s a n o n - z e r o v e c t o r u-^ 4 t)C . I f d im U = 1 we a r e done. O t h e r w i s e , L ( u ^ ) i s a p r o p e r f l a t o f U and i f C 1 = C U [ L (u^ ) } t h e n Ca rd C-j^  < Ca rd F . F o r , when Ca rd C i s i n f i n i t e , Ca rd C + 1 = Ca rd C , and when Ca rd C i s f i n i t e , Ca rd C + 1 i s f i n i t e w h i l e Ca rd F i s i n f i n i t e by a s s u m p t i o n . A p p l y i n g Lemma 2,5 "to G^ , t h e r e i s a n o n - z e r o v e c t o r u 2 1 UC 1 - I f d im U = 2 t h e n B = {u-^Ug} s a t i s f i e s B 0 UG = 0 . O t h e r w i s e , L(u^,Ug) i s a p r o p e r f l a t o f U and i f Cg = C U {L(u^,Ug)} the same argument shows t he e x i s t e n c e o f a. non-zero, u^ 1 UCp . I f dim U = 3 t h e n B = {u^,Ug,u^} s a t i s f i e s B (1 IjC = 0 . L e t d im U = n and suppose n-1 i n d e p e n d e n t v e c t o r s u, ,...,u have been d e t e r m i n e d i n t h i s manner. I f 1 n-1 C n - 1 = C n - 2 U £ L ( u i » ' • ' ' u n - l ? ) t h e n a s b e f o r e C a r d G n _ i < C a r d F -Hence t h e r e i s a n o n - z e r o v e c t o r u n £ t )C n ^ and B = { u ^ , . . . , u } 5 0 . i s a b a s i s w h i c h s a t i s f i e s B fl UC = 0 . Lemma 3 . 1 4 L e t U be 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 ve r an i n f i n i t e f i e l d F and A a n o n - s i n g u l a r l i n e a r mapping o f U . I f B i s a l i n e a r mapping o f U such t h a t L ( A ( x ) ) = L ( B ( x ) ) f o r a l l x € U e x c e p t , p o s s i b l y , f o r v e c t o r s i n a c o l l e c t i o n o f l i n e s C such t h a t Ca rd C < Ca rd F , t h e n B = \A f o r some s c a l a r X . P r o o f : I f C = £ L ( X ) : x e U and L ( A ( x ) ) £ L ( B ( x ) ) ] t h e n Ca rd C < C a r d F , by a s s u m p t i o n . By Lemma 3 - 1 3 we may choose a b a s i s e ^ . - j e d i s j o i n t f r o m UC . Then t h e r e a r e n o n -z e r o s c a l a r s such t h a t Be. = X .Ae. f o r each I n i i i I = 1 , . . . , n . L e t X = {L(u) : u = S J - I - U J L ^ , \x± ^ 0 , i = l , . . . , n ] . I f a i s a n o n - z e r o e lement o f F t h e n the mapping a - a ( E ^ = 1 e i ) shows Ca rd F £ Ca rd X . L e t L ( u ) e X ~ C . ( X ~ C i s non-empty s i n c e Ca rd C < C a r d F .) I f u = E ? = 1 u i e 1 t h e n E n n u .X .Ae . = X Ev , u.Ae. , where Bu = xAu . Because A i = l i i i i = l I i ' i s n o n - s i n g u l a r and u e X , X^ = X f o r e v e r y i = l , . . . , n . T h e r e f o r e , B = XA . Lemma 3 - l 4 g e n e r a l i z e s a r e s u l t o f M. Marcus and B. N. M o y l s , [ 3 , Lemma 2 , p . 6 2 ] . 5 1 . Theorem 3 . 1 5 L e t U be 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 ove r an a l g e b r a i c a l l y c l o s e d f i e l d F and h be a t ype 1 mapping o f U ^ r ^ . I f d im U > 2 and A , B a r e any two a s s o c i a t e maps w i t h r e s p e c t t o h t h e r e i s a n o n - z e r o s c a l a r y i n F such t h a t A = YB . P r o o f : We' f i r s t p r o ve the theorem f o r a s s o c i a t e mappings o f a d j a c e n t sub spaces . Suppose M^ = x-^*. . • * x r _ 2 * a k * u ' w n e r e k = 1 , 2 a re two a d j a c e n t sub spaces . By Lemma 3 . 1 2 we may comp le te the i n d e p e n d e n t v e c t o r s a- j^a^ t o k v e c t o r s a £ , . . . , a k i n g e n e r a l p o s i t i o n where k = Max [ n , r ] . By Lemma 2 . 5 t h e r e i s a v e c t o r , a ,... such t h a t a ^ L ( a i ) f o r i = l , . . . , k . The t y p e 1 subspace M D = x 1 * . . . * x v , 0 * a * U i s a d j a c e n t t o each M^ = x-^*. . . * x r _ g * a ^ * U where i = l , . . . , k . By C o r o l l a r y 3 - 8 , {h(M.) : i = l , . . . , k ] U {h (M o ) } i s an X cl a d j a c e n t f a m i l y o f k+1 s ub space s ; i . e . , t h e r e a r e v e c t o r s y 1 j . . . j y r _ 2 a n d b 1 , . . . , b k i b . i n U such t h a t fcCM-j.) = y-L*. . • * y r _ 2 * b i * U f o r i = l , . . . , k h ( M a ) = y 1 * . , . * y r _ 2 * b * U C o r o l l a r y 3 . 4 i m p l i e s b | L ( b ^ ) f o r a l l i = l , . . . , k . L e t A^ be an a s s o c i a t e map o f and A & an a s s o c i a t e map o f •'M .^ , By d e f i n i t i o n , f o r any u i n U , £1 h ( x 1 * . . , * x r _ p * a i * u ) = y-j^*. . . * y r _ 2 * b i * A i ( u ) f o r i = l , . . . , k h ( x 1 * . . . * x r _ 2 * a * u ) = y-L*.. . * y r _ , 2 * b * A a ( a ) -. 5 2 . I n p a r t i c u l a r , h ( x 1 * . . . * x r _ 2 * a i * a ) = y 1 * . . . * y r _ 2 * b 1 A 1 ( a ) = y±*...#yr_2*b*Aa(a1 h ( x x * . . . * x r _ 2 * a 2 * a ) = y 1 * . ' . . * y r _ 2 * b 2 A 2 ( a ) = y±*. . . * y r _ 2 * t ) * A a ( a 2 By C o r o l l a r y 1.11 b 1 * A 1 ( a ) = b ^ A ^ a ^ and b 2 * ^ 2 ( a ) = b * A & ( a 2 ) . , . S i n c e , i n p a r t i c u l a r , b 1 L ( b i ) f o r i = 1,2 C o r o l l a r y 1.9 w i l l a l s o i m p l y t h a t b = aA- L (a) and b = pAg(a) f o r some n o n - z e r o a,B i n P . Hence A 1 ( a ) = a - 1 p A 2 ( a ) f o r any v e c t o r a k L ( a i ) f o r i = l , . . . , k . By Lemma 2 .14 , t h e r e e x i s t s Y i n F such t h a t A-^  = Y A 2 . Now l e t M = x, * . . . * x ^ -, *U and N = y , * . . . * y , *U - - - 1 r - l J l • ' r - l be any t ype 1 s u b s p a c e s w i t h a s s o c i a t e maps A and B , r e s p e c t i v e l y . C o n s i d e r t he t ype 1 subspaces M k = x l * ° ° ° * x r - k - l * y l * * * 0 * y k * U a n d l e ' b A k denote an a s s o c i a t e map o f f o r k = l , . . . , r - 2 . We see t h a t M i s a d j a c e n t t o . M^, N i s a d j a c e n t t o M r _ 2 ». a n d M i i s a d j a c e n t t o f o r 1 = -3 . T h e r e f o r e , t h e r e a r e n o n - z e r o s c a l a r s Y-^ - • • > Y v ,_ 2 o f F such t h a t : A = Y 1 A 1  A k - 1 = Y k A k f o r k = 2 , . . . , r - 2 53. That i s , A = Y 2 _ - . - Y j . _ 2 B . §3. I n duced P r o d u c t P r e s e r v e r s . T h i s f i n a l s e c t i o n c o n t a i n s ou r ma in r e s u l t : i f U 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 l g e b r a i c a l l y c l o s e d f i e l d and d im U ^ 2 t h e n any t y p e 1 mapping o f U ( r ) i s i n d u c e d by a n o n - s i n g u l a r mapping o f U . I f we f u r t h e r e x c l u d e t h o s e c a s e s f o r w h i c h r = (Ghar F ) k , k a p o s i t i v e i n t e g e r and d im U <_ r+1 t h e n any p r o d u c t p r e s e r v e r w i l l be t ype 1, hence i n d u c e d . Theorems 3-16 and 3.17 a r e r e a l l y lemmas needed f o r t he ma in r e s u l t i n Theorem 3-18. Theorem 3-16 L e t U be 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 . I f A i s a n o n - s i n g u l a r mapping o f U and : h i s a l i n e a r mapping o f u ( r ) such t h a t f o r each n o n - z e r o x i n P j J u ] t h e r e i s a s c a l a r \(x) such t h a t h ( x ) = \ ( x ) A ^ r ^ ( x ) t h e n X ( x ) = X(y) f o r a l l x and y i n P r [ U ] . P r o o f : L e t x , ,<,..,x , , u , v be n o n - z e r o v e c t o r s o f U . I f a = Xj*...*xr_1*u .and b = x ^ * . . . * x r ^ * v t h e n a -f b = x 1 * . . . * x 1 * ( u + v ) . I f h ( a ) = \ A ^ r ^ ( a ) and h ( b ) = u A ^ r ^ ( b ) l e t h (a+b) = v A ^ r ^ ( a + b ) where ^ n / v a r e s c a l a r s . Then, h(a+b) = v A ^ r ^ ( a ) + v A ^ r ^ ( b ) because A ( r ) i s l i n e a r and h (a+b) = \A^r^(a) + u A ^ r ^ ( b ) because h i s l i n e a r . 5 4 . C o r o l l a r y 1.8 i m p l i e s v [ A ( u ) + A ( v ) ] = \ A ( u ) + uA (v ) . I f a i s a n o n - z e r o s c a l a r and u = av t h e n b = aa and h ( b ) = a \ A ^ r ^ ( a ) = a u A ^ r ^ ( a ) . Hence, X = n . Suppose u and v a r e i n d e p e n d e n t . S i n c e A i s n o n - s i n g u l a r , A (u ) and A ( v ) a r e i n d e p e n d e n t . C o n s e q u e n t l y , X = u = v . F o r any n o n - z e r o e l ement x = x 1 * . . . * x r o f P r [ U ] , i f h ( x ) = A.A^ r^(x) w r i t e X = , . . . , x r ) . I f y = y " ] _ * ' - ' * y r 1 3 a n o t h e r e l ement o f P [U] we have j u s t shown: x (x]_) • ' • > x r ) = X (x-^,. . . , x ^ _^, y^ ) = X(x1}...,xr_2,ya,y2)=...=\(y1,...,yr) Theorem 3.17 L e t U be 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 ve r an a l g e b r a i c a l l y c l o s e d f i e l d and h a t ype 1 mapping o f U^ r ) . Assume M = x -^ * . . . * x r i s a t ype 1 subspace such t h a t h(M) = A ( x ^ ) * • . .*A(x_^ -j_)*U where A i s a a s s o c i a t e map o f M . I f N ^ u ) = . . * x i * . . . * x r _ 1 * u * U , i = l , . . . , r , t h e n h ( N i ( u ) ) = A ( x 1 ) * . . .*A(x?)*. .:;*A (x r _ 1 ) *A (u ) *U . P r o o f : I f h (N^ ( u ) ) = y^*. . .*y r_-|_*U t h e n y-^*. • .*yT_,-^ and A ( x -^ ) * . . • * A ( x r _ - L ) have r-2 common f a c t o r s c o u n t i n g m u l t i p l i c i t i e s . Suppose t h e s e common f a c t o r s a r e , say , A ( x 1 ) , . . . , A ( X j ) , . . . , A ( x r _ 1 ) and l e t h ( N i ( u j ) = A ^ ^ * . . . * A ( x . ) * . . . * A ( x r _ ^ ) * v * U where the v e c t o r v 55. w i l l depend upon u . Then L ( x ^ * . . . * x r _ 1 * u ) = M f lN^u ) and t h e r e a r e n o n - z e r o s c a l a r s X ,u such t h a t h ( x 1 * . . . * x r _ 1 * u ) = x A ( x 1 ) # . . . * A ( x r _ 1 ) * A ( u ) = u A ( x 1 ) # . . . * A ( X j ) * . • • * A ( x r _ 1 ) * A ( x i ) * v . Theorem 1.13 i m p l i e s X A ( x ^ ) * A ( u ) =: u v * A ( x i ) . I f A ( x j ) = av f o r some n o n - z e r o s c a l a r a t h e n h(M) = h ( N i ( u ) ) , c o n t r a d i c t i n g Theorem 3.3 . C o r o l l a r y 1.9 i m p l i e s t h e r e a r e n o n - z e r o s c a l a r s a and S. such t h a t A ( x j ) = a A ( x i ) and A (u ) = B v .. T h e r e f o r e , h ( N i ( u ) j) =. A ( x 1 ) * . . , * A ( x ^ ) * . . . * A ( x 1 ) * A ( u ) * U . Theorem 5.18 I f U 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 ove r an a l g e b r a i c a l l y c l o s e d f i e l d , t h e n e v e r y t ype 1. mapping o f U ( r ) i s i n d u c e d by a n o n - s i n g u l a r mapping o f U , p r o v i d e d dim U^2, P r o o f : I f h i s a t y p e 1 mapping o f u ( r ) i t i s s u f f i c i e n t t o p rove t h a t h ( x 1 * . . . * x r _ _ * U ) = A ( x . ^ ) * . . . * A ( x ^ ) * U f o r any t ype 1 s ub space , M = x ^ * . . . * x , where A I s t h e c o r r e s -p o n d i n g a s s o c i a t e map: For^ any• p r o d u c t x = x ^ * . . . * x b e l o n g s t o M , so h ( x ) i s i n A ( x ] L ) * . . . * A ( x r _ 1 ) * U . I f B i s t he a s s o c i a t e , map o f h w i t h r e s p e c t t o the v e c t o r s A ( x 1 ) , . . . , A ( x ^ ^) t h e n 5 6 . h ( x ) = A ( x 1 ) # . . . * A ( x r 1 ) * B ( x r ) . Theorem 3 . 1 5 i m p l i e s t h e r e i s a n o n - z e r o s c a l a r y such t h a t B = yA , so h ( x ) = y A ( x ^ ) * . . . * A ( x r ) . Theorem 3 . 1 7 1 d emon s t r a t e s t h a t y i s i n d e p e n d e n t o f x . Hence h i s i n d u c e d by any mapping 1/r 1/r Y A , where y i s any r - f o l d r o o t o f Y • F u r t h e r m o r e , i t i s s u f f i c i e n t t o p r o ve t h a t h'(M) = A ( x 1 ) * . . . * A ( x r , _ 1 ) * U o n l y when the f a c t o r s L ( x i ) a r e d i s t i n c t ; i . e . , L ( x . ) = L ( x . ) i m p l i e s i = j f o r a l l i , j = l , . . . , r - l . F o r , i f i s any t ype 1 subspace l e t M ^ j ' - . j M ^ . be a sequence o f a d j a c e n t subspaces between and a t ype 1 subspace M .^ a l l o f whose f a c t o r s a r e d i s t i n c t . A p p l y i n g Theorem 3 - 1 7 k - 1 t i m e s we o b t a i n h ( M 1 ) = A ( x ^ )*. . . * A ( x r _2 )*U i f = x ^ * . . . *x r _-j_*U and so h w i l l be i n d u c e d . Assume t h e n t h a t M = x-^*. . . * x r _ 1 * U i s a t ype 1 subspace o f U ( r ) w i t h d i s t i n c t f a c t o r s . Then M and N = x * X g * . . . * x r _ 2 * U a r e a d j a c e n t whenever x i s a v e c t o r n o t i n L ( x ^ ) . (We can suppose dim U > 2 s i n c e when dim U = 1 our c o n c l u s i o n was o b t a i n e d i n example ( l ) , page 4 2 . ) I n C o r o l l a r y 3 . 4 i t was shown t h a t h (M) and h(N) a r e a d j a c e n t whenever h i s a p r o d u c t p r e s e r v e r . L e t h(M) = y ^ * . . . * y r _ 2 # z 1 * U , h(N.) = y]_* • • • * y r _ g * z 2 * U a n d suppose A i s a a s s o c i a t e under h . I n v i e w o f Theorem 3 - 1 5 i t f o l l o w s t h a t t h e r e a r e n o n - z e r o s c a l a r s X.,u such t h a t 57-h ( x 1 * . . . * x r _ 1 * x ) = \ y 1 * . . . * y r _ 2 * z ] L # A ( x ) = n y - j * . . . * y r _ 2 * z 2 * A ( x 1 ) . By C o r o l l a r y .1.11 \ z 1 * A ( x ) = | j z 2 * A ( x 1 ) i n S i n c e h ( M ) , h (N ) a r e a d j a c e n t , L ( z 1 ) 4 L ( z 2 ) and C o r o l l a r y 1.9 i m p l i e s t h e r e i s a n o n - z e r o s c a l a r v such t h a t z-^ = y A ( x _ ) . Renaming t he v e c t o r s y 1 , . . . , y r _ 2 , z 1 ( i f n e c e s s a r y ) we can now w r i t e h ( x x * . . . # x r - 1 * U ) = A ( x 1 ) * y 2 * . . • * y r _ 1 * u • Now assume we have shown h(M) = A ( x 1 ) * . . • * A ( x _ c ) * y i c + i * - • . * y r _ i * u f o r s o m e i n t e g e r k s a t i s f y i n g 1 <_ k _< r - 2 . Then M and N = x * x 1 * . . • * ^ ] : C + _ L * . • ' * x r _ i * U - a r e a d j a c e n t i f we choose a v e c t o r x n o t i n L ( x k + 1 ^ " I n t n i s c a s e h ( M ) , h (N ) have r - 2 common f a c t o r s and we may suppose e i t h e r (a) h (N) = A ( x x ) * . . . * A ( x k _ ; 1 ) * y * y k + 1 * . . . * y r _ 1 * U f o r some y n o t i n L ( A ( x ^ ) ) o r (b ) l i ( H ) = A ( x 1 ) * . . . * A ( x k ) * y k + 1 * . . . * y r _ 2 * y * U f o r some y n o t I n L ( y r _ ^ ) . I f ( a ) t h e n i t f o l l o w s f r o m Theorem 3-15 t h a t t h e r e a re n o n - z e r o s c a l a r s \ and u such t h a t h ( x 1 * . . . * x r _ x * x ) = X A ( x 1 ) * . . . * A ( x k ) * y k + 1 # y r _ 1 * A ( x ) = u A ^ ) * . • . * A ( x k _ x ) W k + ] _ * . . . * y r ^ A ( x k + 1 ) 5 8 . C o r o l l a r y 1.11 i m p l i e s \ A ( x k ) * A ( x ) = u y * A ( x k + 1 ) . S i n c e y -uis n o t i n L ( A ( x k ) ) , C o r o l l a r y 1.9 i m p l i e s t h e r e i s a n o n - z e r o s c a l a r r\ such t h a t A ( x k ) = T l A ( x k + i ) ' i ' e . - j A ( x k - "nXjj..^) = 0 • On l y now do we u t i l i z e the h y p o t h e s i s t h a t t h e f a c t o r s o f M a r e d i s t i n c t t o a s s e r t x k - •nXjj.+j £ 0 • B u t A i s an a s s o c i a t e map, hence (a) y i e l d s a c o n t r a d i c t i o n o f Theorem 3.11. Theorem 3.16 and (h) i m p l y : h ( x 1 * . . . * x r _ 1 * x ) m *.Y>.. f A( x^ ) Py^i* U ) * y t _ ! * A (x) = n A ( x 1 ) * . . • * A ( x k ) * y k + 1 * . • • * y r . 2 * y * A ( x k + i ^ f o r some n o n - z e r o s c a l a r s X,[i and a v e c t o r y n o t i n L ( y r - l ) ' C o r o l l a r y 1.11 i m p l i e s \ y r - 1 * A ( x ) = u y * A ( x k + 1 ) . S i n c e y i s n o t i n L ( y C o r o l l a r y 1.9 i m p l i e s t h e r e i s a n o n - z e r o s c a l a r r\ such t h a t Yr_jL - ^ ( x ^ ^ ) • Thus , r enumbe r i n g the v e c t o r s y ^ + l ' " " ' ' y r 1 ^ n e c e s s a r y ^ w e c a n w r i t e h ( M ) = A ( x 1 ) * . . • * A ( x k ) * A ( x k + 1 ) * y k + 2 * v . , ••fY-j.jj*®-' w h i c h c o m p l e t e s t he p r o o f . Theorem 3.19 L e t U be 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 ove r an a l g e b r a i c a l l y c l o s e d f i e l d n o t o f c h a r a c t e r i s t i p p when k r = p f o r some p o s i t i v e i n t e g e r k . Then, i f dim U > r+1 e v e r y p r o d u c t p r e s e r v e r o f U ( r ) i s i n d u c e d by a n o n - s i n g u l a r mapping o f . 5 9 . P r o o f : I n v i e w o f Theorem 3.18 we need o n l y show e v e r y pu re f a i t h f u l mapping o f u ( r ) i s t y p e 1 when dim U > r+1 . L e t M be a t y p e I subspace and h a p r o d u c t p r e s e r v e r o f u ( r ) • Theorem 3.2 i m p l i e s h(M) i s pure and hence c o n t a i n e d i n a max ima l pu re sub space , say P . I n P a r t I I , §1 i m p l i e s P must be a t ype k where 1 £ k <_ r . I f P i s t ype k and k > 1 t h e n d im P = k+1 . Bu t d im U = d im h(M) <_ d im P c o n t r a d i c t s d im U > r+1 . T h e r e f o r e k = 1 and h i s t y p e 1 . 60. BIBLIOGRAPHY [1] N. B o u r b a k i , Po lynomes e t F r a c t i o n s R a t i o n e l l e s , Hermann, P a r i s , 1959• [2] N. J a c o b s o n , L e c t u r e s i n A b s t r a c t A l g e b r a , Volume 1, Van N o s t r a n d , New Y o r k , 1953-[3] M. Marcus and B. N. M o y l s , L i n e a r T r a n s f o r m a t i o n s on  A l g e b r a s o f M a t r i c e s , Canad. J . Math . V o l . 11, 1959, PP. 61-66. [4] M. Marcus and M. Newman, I n e q u a l i t i e s f o r t he Permanent  F u n c t i o n , A n a l . Math . V o l . 75, 1962, pp . 47-62. [5] H. R y s e r , C o m b i n a t o r i a l M a t h e m a t i c s , Ca ru s Monograph, The M a t h e m a t i c a l A s s o c . o f A m e r i c a , 1963. [6] R. J . W a l k e r , A l g e b r a i c C u r v e s , Dover P u b l i c a t i o n s , I n c . , New Y o r k , 1962. 

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