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Linear transformations on Grassmann product spaces Westwick, Roy 1959

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LINEAR TRANSFORMATIONS ON GRASSMANN PRODUCT SPACES by ROY WESTW1CK B.A., Iff.A., U n i v e r s i t y of B r i t i s h Columbia, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy i n the Department of Mathematics We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA 1959 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of / | ? ^ g u ^ ^ r T L ^ L < ? The U n i v e r s i t y of B r i t i s h Columbia, Vancouver Canada. Date >?$anAk-2,\J /?6Q GRADUATE STUDIES Field of Study: Linear Algebra Algebra Group Representations Real Variable Other Studies: Logic Quantum Mechanics Information Theory PUBLICATIONS 1. M. Marcus, B. N. Moyls, and R. Westwick, Some Extreme Value Results for Indefinite Hermitian Matricies, Illinois J. Math., vol. 1, (1957) pp. 449-457. 2. M. Marcus, B. N. Moyls, and R. Westwick, Some Extreme Value Results for Indefinite Hermitian Matricies II, Illinois J. Math., vol. 2, (1958) pp. 408-414. 3. M. Marcus, B. N. Moyls, and R. Westwick. Extremal Properties of Hemitian Matricies II, Canadian J. of Math., vol. XI, No. 3, 1959. M. Marcus, R. Ree H. Davis R. Restrepo P. Deuel W. Opechowski . R. E. Burgess (Eijt P»tln>rstty tff ^r i i tsh GluUuuiiia Faculty of Graduate Studies FINAL ORAL E X A M I N A T I O N FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of ROY WESTWICK B.A., University of British Columbia, 1956 M.A., University of British Columbia, 1957 IN ROOM 221, BUCHANAN BUILDING WEDNESDAY. MARCH 30th, 1960 AT 3:30 P.M. COMMITTEE IN CHARGE DEAN G. M. SHRUM: Chairman M. D. Marcus E. Morrison B. N. Moyls W. Opechowski D. C. Murdoch F. A. Kaempffer D. Derry R. E. Burgess External Examiner: O. Taussky-Todd California Institute of Technology A B S T R A C T T h e objective of this thesis is to determine the linear trans-formations of a Grassmann product space which sends the set of non-zero Grassmann product vectors (also called pure vectors) into itself. Let U be an n-dimensional vector space over a f ield F and let r be an integer such that 0 — r — n. The r-th Grassman product space wil l be denoted by Aj(U). Subspaces of A r ( U ) consisting e n -tirely of pure vectors are cal led pure subspaces. With each non-2ero pure vector of A r ( U ) we associate an r -dimensional subspace of U . By studying the set of subspaces of U corresponding to a basis set of a pure subspace of A r ( U ) we are able to deduce the form of this pure subspace. In this way we are able to classify the pure subspaces of A r ( U ) , arriving at only two essentially distinct types. We next study the maximal pure subspaces, i . e. the pure subspaces which are not contained in larger pure subspaces. They are of importance because the assumptions on the linear transform-ations under consideration imply that a maximal pure subspace is mapped into another maximal pure subspace. The form of the transformation is now almost completely determined by examining the incidence relations between pairs of maximal pure subspaces before and after the transformation is applied. Some algebraic m a n i -pulations are then needed i n order to display the form of the trans-formation completely. With the suitable assumptions, our results state that the transformations under consideration are induced by linear transform-ations of the vector space U , except possibly when 2r = n. When 2r = n two types of transformations are possible. This arises from the fact that the two types of m a x i m a l pure subspaces have the same dimensions, (unlike the situation when 2r + n). One type of trans-formation (those induced by linear maps of LT) does not alter the type of pure subspaces, while the other interchanges the two types. ABSTRACT Let D be an n-dimensional v e c t o r space over an r . 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 l e t A U denote the [ J -dimensional space spanned by a l l Grassmarm products X , A X 0 A . . . A X , x. e U . Let T be a l i n e a r t r a n s f o r m a t i o n 1 2 r ' 1 r of A U i n t o i t s e l f such t h a t (*) i f x - ^ X g ^ ... A X r i s not zero then T ( X ^ A X ^ , A ... A X p ) i s a pure non-zero v e c t o r of grade r , t h a t i s , of the form U-^A TJ^ A . . . / \ U R , U^e U . Theorem 1 . I f T s a t i s f i e s (*) and n = 2 r then there i s a l i n e a r t r a n s f o r m a t i o n A of U i n t o U , such t h a t T i s the r t h compound of A. Theorem 2 . I f n = 2 r then there i s a l i n e a r r t r a n s f o r m a t i o n of A U , which we s h a l l denote by J , s a t i s f y i n g (*) such t h a t J i s not an r t h compound. Moreover, i f T i s any t r a n s f o r m a t i o n s a t i s f y i n g (*) then e i t h e r T or TJ i s an r t h compound. These r e s u l t s are s i m i l a r t o those of Wei-Liang Chow (Annals of Math., v. 50 ( 1 9 4 9 ) , 5 2 - 6 7 ) . We r e p l a c e Chow's c o n d i t i o n t h a t T be a one-to-one adjacence p r e s e r v i n g t r a n s f o r m a t i o n mapping the set of non-zero pure v e c t o r s of grade r onto i t s e l f by the c o n d i t i o n (*). TABLE OP CONTENTS CHAPTER PAGE I DEFINITIONS AND INTRODUCTORY REMARKS . . . . 1 I I PAIRWISE ADJACENT SUBSPACES 5 r I I I THE PURE SUBSPACES OF A U 10 IV THE ONE-TO-ONE C-TRANSEORMATIONS 16 V THE MAIN RESULTS 31 i i ACKNOWLEDGEMENT The author wishes to acknowledge the generous a s s i s t a n c e g i v e n "by Dr. Marvin Marcus i n the p r e p a r a t i o n of t h i s paper. We are a l s o p leased to acknowledge the f i n a n c i a l support of the N a t i o n a l Research C o u n c i l . i i i CHAPTER I DEFINITIONS AND INTRODUCTORY REMARKS The object of this paper i s to determine the linear transformation of a Grassmann product space which sends the set of non-zero pure vectors into i t s e l f . If U i s an n-dimensional vector space over a f i e l d F and r i s an integer, r o t r t i i then a Grassmann product space A U i s defined as follows, r If r = o then A U = P. I f U r ^ n then l e t G r(U) denote the space of a l l alternating r-multilinear functions on U to F. That is,, i f we Gr(U) and x^ e^ U, i = 1,2,. .. ,r then w ( x a Q ) * • • • > xa.(r)^ = wCx-p . . . , x r ) where (-1) = -1 depending on whether a i s an even or odd permutation of l , 2 , . . . , r . Then r A U i s defined to be the dual space of G r(U). If x^e U, i = 1,2, r . . . , r , then the element f = x-^  A X 2 A . . . A x r of A U defined by f(w) = w(x-,,...,x ) for a l l w £ G (U) i s called the grassmann product of x-^ , . . . , x . The vectors X-^A X 2 A . . . A X R of A u are also called pure vectors. We note the following facts, (see [5] ). If [x-p . . . , x n } i s a basis of U then [ x ± A . . . A x i j l ^ i - L < . . . < i r ^ n j 1 r r r i s a basis of A u a^ cL s 0 d i m A U = f j . If x 1 A . . . A x p = y 1 A . . . A y r jt o then <x-j^,... , x r > = < j ± , . . . ,y r ) , where < ^ x - ^ > x r denotes the subspace of U spanned by ^ x ^ , . . . , x r ^ 1 2 A l s o X-^A . . . A X = 0 i f a n d o n l y i f d i m < x 1 } . . . , x r > ^ r - l . U n l e s s o t h e r w i s e i n d i c a t e d , we a d o p t t h e c o n v e n t i o n t h a t i f r U-, ^ U i s a s u b s p a c e t h e n A U-, i s g e n e r a t e d b y a l l x-, A . . . A X 1 1 r /k\ 1 r where x^etT^. I f d i m U-^ = k t h e n A i s a I r j - d i m e n s i o n a l r s u b s p a c e o f A ^• I f A:IT—>TJ i s a l i n e a r t r a n s f o r m a t i o n t h e n t h e r t h compound o f A, d e n o t e d b y C ( A ) , i s t h a t l i n e a r t r a n s f o r m a t i o n r o f A U s u c h t h a t C r ( A ) (x-^/\ ... A x r ) = AX-^A . . . A A x r . I f A i s n o n - s i n g u l a r t h e n C r ( A ) s e n d s t h e s e t o f n o n - z e r o p u r e v e c t o r s i n t o i t s e l f . I f 2 r £ n t h e n u n d e r t h e a s s u m p t i o n s o f Theorems 4.3 o r 5*6 we p r o v e t h a t e v e r y t r a n s f o r m a t i o n we a r e c o n s i d e r i n g i s o f t h e f o r m a . C r ( A ) w h e r e a # 0 and A i s n o n -s i n g u l a r . I n Theorems 4.6 a n d 4.7 we d e t e r m i n e t h e t r a n s f o r m a t i o n s f o r t h e c a s e 2 r = n . The r e s u l t s we o b t a i n a r e s i m i l a r t o t h o s e o b t a i n e d b y W e i - L i a n g Chow i n [1] . He p r o v e s t h a t any o n e - t o - o n e a d j a c e n c e p r e s e r v i n g t r a n s f o r m a t i o n o f t h e G r a s s m a n n s p a c e o f a l l t h e [ r ] o f S n, ( n - l > r > o ) , o n t o i t s e l f i s a t r a n s f o r m a t i o n o f t h e b a s i c g r o u p o f t h e s p a c e . S n i s a p r o j e c t i v e s p a c e o f d i m e n s i o n n o v e r a n a r b i t r a r y g r o u n d f i e l d K . [rj d e n o t e s a n e l e m e n t o f d i m e n s i o n r o f S . Two e l e m e n t s [r3„ a n d [ r ] v a r e a d j a c e n t n a b u i f t h e y a r e ( r - l ) - i n c i d e n t . A n a d j a c e n c e p r e s e r v i n g t r a n s f o r m a t i o n o f t h e G r a s s m a n n s p a c e o f a l l t h e E r D i s a t r a n s f o r m a t i o n w h i c h maps p a i r s o f a d j a c e n t L r 3 i n t o p a i r s o f a d j a c e n t C r ] . I f we w e r e t o d e f i n e a d j a c e n c y s i m i l a r l y , t h a t 3 i s , two non-zero grassmann p r o d u c t s X-^A ... A x r and y-^  A ... A y r are a d j a c e n t i f <x x,...,x r> O <7i»•••»y r> n a s dimension r - l then by lemma 3.1 the t r a n s f o r m a t i o n s we are c o n s i d e r i n g are adjacence p r e s e r v i n g t r a n s f o r m a t i o n s i f we r e s t r i c t e d them t o the non-zero pure v e c t o r s . With Chow's r e s u l t i t would be s u f f i c i e n t t o show t h a t our t r a n s f o r m a t i o n s on the non-zero pure v e c t o r s are one-to-one and onto r a t h e r than j u s t i n t o . However, i t doesn't seem p o s s i b l e t o proceed i n t h i s d i r e c t i o n . Even i f we assume the t r a n s f o r m a t i o n s are one-to-one i t i s n o t c l e a r t h a t the s e t o f non-zero pure v e c t o r s i s mapped onto i t s e l f . A l s o , the example a t the end of c h a p t e r IV shows t h e r e e x i s t s i n g u l a r t r a n s f o r m a t i o n s f o r some v e c t o r spaces over non-a l g e b r a i c a l l y c l o s e d f i e l d s which send the s e t of non-zero pure v e c t o r s i n t o a p r o p e r subset o f i t s e l f . The Grassmann a l g e b r a of IT, denoted by A^, i s d e f i n e d o l n t o be A U © A u © • • • © A U. We d e f i n e m u l t i p l i c a t i o n r t as f o l l o w s . I f w = X - , A . . . A X „ e A u and v = A . . . A y + e Au 1 r r+t 1 v then w A v = x-j^  A • . • A x p A y-^ A . • • A y^ e A U i f r+t ^  n and w A v = o i f r + t > n . We extend l i n e a r l y so t h a t w A v i s d e f i n e d f o r r t a l l W € A U a n d- veAu. I f (w Q,...,w n) and ( v Q , . . . , v n ) are i n AU then we d e f i n e ( w Q , . . . , w n ) A ( v Q , . . . j V ^ ) = ( Z 0 , . . . , Z n ) where J Z. = w.Av_; .. Then A U i s an a l g e b r a under t h i s " i = l 1 J " 1 m u l t i p l i c a t i o n . In c h a p t e r I I we s t a t e and prove some p r e l i m i n a r y lemmas on types o f s e t s of p a i r w i s e a d j a c e n t subspaces of a v e c t o r space U. 4-We c o n s t r u c t bases f o r these subspaces which show c l e a r l y how they i n t e r s e c t one another. We are then able to determine some dimensions and i n t e r s e c t i o n p r o p e r t i e s of s p e c i a l subspaces of r A U which we need l a t e r i n the paper. Lemmas 3.8 and 3.10 c o n t a i n the main r e s u l t s of chapter I I I . Here we are mainly concerned w i t h the s t r u c t u r e and i n t e r s e c t i o n p r o p e r t i e s of r pure subspaces of A U. The main r e s u l t s of the paper are contained i n chapters IV and V. In chapter IV we determine the s t r u c t u r e of the t r a n s f o r m a t i o n s w i t h the a d d i t i o n a l assumption t h a t they be one-to-one. The method of proof leads us t o r c o n s i d e r t r a n s f o r m a t i o n s from one Grassmann product space A U r to another, say A W, which sends a s p e c i a l c l a s s of pure r r subspaces of /\ U i n t o the corresponding c l a s s in A W. We r r then show t h a t i f A W = A U then the c o n d i t i o n i s s a t i s f i e d . I n chapter V we r e p l a c e the one-to-one hypothesis by the c o n d i t i o n t h a t F be a l g e b r a i c a l l y c l o s e d . We then show t h a t the t r a n s f o r m a t i o n must be one-to-one. CHAPTER I I PAIRWISE ADJACENT SUBSPACES Let U be a f i n i t e dimensional v e c t o r space over a f i e l d P. I f and U 2 are n-dimensional subspaces of U then they are ad,jacent i f dim ( U ^ I J ^ ^ n - 1. A f i n i t e set {^,...,11 } of n-dimensional p a i r w i s e adjacent subspaces of U i s of Type I i f s dim H U. as n - 1 and of Type I I i f dim (U,+...+U ) 6 n + 1. i - 1 1 1 3 I f {U-p...,^} i s of Type I and dim (U-^+.-.+U ) = n+s-1, or i f [u.^...,^] i s of Type I I and s dim r\ U^ = n-s+1, then |u^,...,U ] i s an independent s e t . i = l Otherwise i t i s dependent. Lemma 2.1 Let (U-pU 2,... ,U g] be a f i n i t e set of n-dimensional s p a i r w i s e adjacent subspaces of U. Then dim r\ U^ n-s+1. Pro o f : The proof i s by i n d u c t i o n on s. Suppose the lemma i s v a l i d f o r s = k and l e t U 1 , . . . , U J j. +-^ be p a i r w i s e adjacent n-dimensional subspaces of U. Then k k dim r\ U. ^ n-k+l. Let TS1 = C\ U. . Then i - 1 i - 1 dim (U' + U k + 1 ) ^ dim + U k + 1 ) ^ n+1. Therefore dim ( U ' n u k l ) = dim U' + dim U f e + 1 - dim (U» + U k + 1 ) ^ n - k+1 + n - (n+1) = n - (k+1) + 1. 5 6 F i n a l l y , the hypothesis c o n t a i n s the statement of the lemma when s = 2. Lemma 2.2 Any f i n i t e s et of n-dimensional p a i r w i s e adjacent  subspaces of TJ i s e i t h e r of Type I o r of Type I I . P r o o f : Let {TJ-^  ,U2,... ,U g ^ be a f i n i t e s et of n-dimensional p a i r w i s e adjacent subspaces of U. I f = = ... = U g then [ IT, ,U0,... ,U e ] i s both of Type I and of Type I I . Suppose on the other hand t h a t U-^  ^  JJ^. We note t h a t : ( i ) I f U1r\U± # V2nVl± then U ie,U 1+U 2, t h i s f o l l o w s from the p a i r w i s e adjacency assumption. Therefore i f TJ-^ +uV) then HjOlT^ = U^nu^- a n d both are equal to U,nUa. ( i i ) I f U^+U2 f o r some i then f o r any j we cannot have a l l the three i n c l u s i o n s , U.QTJ 1 +U 2 U.QTJ 1 +TJ. The c o n t r a r y i m p l i e s £ ( U 1 + U 2 ) 0 ( ^ + ^ ) 0 ( 1 ^ 2 + ^ ) = ^ o l T g which i s i m p o s s i b l e s i n c e dim U. > dim (U-,nUp). Therefore one of the i n c l u s i o n s does not h o l d and combining t h i s w i t h ( i ) we get t h a t U-jn'OV) — ^y I f U i c u 1 + U 2 f o r i = l , 2 , . . . , s then U1+U2+... +U gS i^+Ug. Therefore dim ( U ^ u ^ . . . +U g) ^ dim (I^+T^) = n+1. On the other hand, i f f o r some i , U^U-^+tT^ then by ( i i ) s s U , n U 0 G n U. and t h e r e f o r e dim f) U. ^ n - l . 1 d j = l 3 j = l 3 7 Lemma 2.3 Let |U-pU2,... ,Ug } be a (Type I set of n-dimensional  p a i r w i s e adjacent subspaces of U. Then there e x i s t v e c t o r s x l ' * • • > xn-l ' yl»* * * ' y s ^ U s u c h t h a t u i = Kxi> : ' ) xn-1 > y± > • Furthermore, ( U-pU2,... ,U g ] i s independent i f and o n l y i f (x 1,...,x^_ 1,y 1,...,y ] i s independent. s Proof: Since dim f~\ U. ^ n-1 we can choose a b a s i s i = l s ( x ^ , . . . , x n _ ^ ^ o f an (n-1)-dimensional subspace of f\ U p The ex i s t e n c e now of the i s c l e a r . The second statement f o l l o w s a t once from U1+U2+.'. .+US = < x 1,... »x n <_ 1,y 1,... ,y s > . Lemma 2.4 Let {U-pU 2,•••>U S } be a Type I I independent set of n-dimensional p a i r w i s e adjacent subspaces of U. Then there  e x i s t v e c t o r s x-^,... »xn+-^ i n U such t h a t = <^  X p . .. >xjL_2 , xi+i»* * * ' x n + l ^ } 1 = l»2>«««»s. g P r o o f : Since dim f\ U. * n-s+1, n-s+1 ^  0. i = l 1 Therefore, by lemma 2.1 and f o r j = l, 2 , . . . , s s dim H U. ^ n-s+2 ^ 1. i = l 1 i*0 Therefore f o r each j = l , 2 , . . . , s there i s an x. ^  0 such t h a t u S S x . £ f\ U. and x . f] U. . A l s o c h o o s e f x , . ,... ,x„, ] t o be a J i = i " i = l s+x n+x' i*0 b a s i s of U p We a s s e r t t h a t { x ] _ > • • • » x n + i _ ] i s independent. I f not then x. = 2Z. a n-x. f o r some t such t h a t 1 ^ t ^ s. x*t Since i 4 t i m p l i e s x^ €. U^ i t f o l l o w s t h a t X ^ . G U ^ . . Therefore 8 s x^ e / l which i s c o n t r a r y to the way i n which x^ was chosen. F i n a l l y , s i n c e dim U. = n and si n c e x. e U. f o r j ^  i the r e s u l t f o l l o w s . Lemma 2.5 L e t ^ U-pIL^,... ,U g ] be an independent s e t of n-dimensional p a i r w i s e adjacent subspaces of U. Then r r r dim (/\U1+ A ^ 2 + * • ' + A u s) i§. give* 1 hy ^ r 1 ) + s ( r - l ) i f f V U 2 ' " " U s } °£ I SS£ (?) + (£i) + + ( r I s + i ) M { % , U 2 , . . . , U s } i s of T ^ I I -Proof: I f [ U-pIL^,... ,Ug ] i s of Type I then l e t x 1 , . . . , x n _ 1 , y-j_j...»y be chosen as i n lemma 2.3. Then the union of the se t s f x . A . . . A X . I l 6 i - , < .. .<i ^  n-1 j and 1 -""r 1 l x , A ...AX. A y. l < i 1 < . . . < i -,^-n-l; j = l , . . . , s 7 i s a b a s i s of i i-j^ r—1 ~" r r r A u i + AU2+..'+ A U • The number of elements i n the two se t s I f [ U^,U2»... ,Ug } i s of Type I I then l e t x]_»• • • » x n + ] _ b e chosen as i n lemma 2.4. Then s-1 \ J \ X X A .. .AX^AX.* # # > a x _ I d + 2 ^ i : L < . } J = 0 - - r - G r i s a b a s i s of A f \ ^ 2 + ' ' ' + A ^ s * ^ n * n i s set there are Lemma 2.6 Let [ U-^ , , •..,Ug ^  be a set of n-dimensional p a i r w i s e  adjacent subspaces of U. I f they are dependent then f o r some i ' A uiCAu!+ A u 2 + - - - + AUi.x-P r o o f : I f [ U - p U ^ . . , . , U g } i s of Type I l e t x-p ...> x n_i*yi»•••>y s b e chosen as i n lemma 2.3* The hypothesis i m p l i e s t h a t f o r some t > l , y^e <x^,..., x n - l ' * ' r l ' * * * '^t-1 ^  * Therefore the v e c t o r s r r r x. A • • • A X • A y^ G AU,+ AU0+...+ A U , , and the lemma f o l l o w s 1 1 " " " r - l v ~ ± Suppose { U-pU,^, . . . , U g \ i s of Type I I . We assume [ U-^  ,U 2,... >us_]_ ] i s independent and l e t x ] _ > • • • > x n + i be chosen as i n lemma 2.4 such t h a t = '' * * , x i - l ' x i + l ' * * * , x n + l ' 1*1>•••»s~l • s 3-1 By h y p o t h e s i s , dim f) U. > n-s+1 and dim n U. = n-s+2. s ^ s 1 = 1 8,1 Therefore dim f] U. = n-s+2 and fj U. = < x a,... ,x_., > = O U. • - i 1 • - i X o J.JL » X • - i X i = l 1=1 1=1 Furthermore dim ( U g H <x^,... ,x g _ ^ > ) = s-2. L e t i y ^ , . . . , y g _ 2 } be a b a s i s of U n<x, ,...,x > . Then s i s — 1 ' u s = ^ y i , * , * , y s - 2 ' x s ' * , , , x n + i > * W e n o t e f i n a l l y t h a t r v e c t o r y e A U c of the form y. A . . . A J . - A x. A . . . A X . where s H xj 1 j + l x r K i , < . . . < i . ^ s-2 and s < i . ,-,<.. .<i„ ^  n+1 can be w r i t t e n as a 1 0 r 3+1 r r sum of v e c t o r s from / \ U - p . . . , A u s _ i * This i s c l e a r s i n c e on expanding y i n terms of ^ x i A . . . A X I | K i - j ^ c . . .<ir< n+1 ^  the o n l y terms x. A . . . A X. appearing w i l l have at most j £• s-2 11 x r f a c t o r s from x^,...,x ^» This completes the proof s i n c e r /\U i s spanned by v e c t o r s of the form giv e n . S CHAPTER I I I r THE PURE SUBSPACES OF A U Let U be a f i n i t e dimensional v e c t o r space over a f i e l d F. r Let C (U) denote the set of non-zero pure v e c t o r s of A U and l e t r r (Au) be the set of subspaces of A U such t h a t the non-zero v e c t o r s of each element of { ^ ( A U ) are a l l contained i n d (U). r r The elements of C(Au) are c a l l e d the pure subspaces of A U. I f Z = X ^ J A . ^ A X ^ d r ( U ) then we d e f i n e U(z)£U to be the subspace < x - ^ , x r > . Since x - j A...AX R = y-^A.. . A y r ^= 0 i m p l i e s <x 1 ?... ,x r> = <y-p ... i7Ty y U(z) i s w e l l d e f i n e d . A l s o note t h a t i f < x 1,...,x r> = <y 1,...,y r> then x- LA...AX R = a ( y 1 A . . . A y r ) f o r some aeF. Lemma 3.1 Let z 1 and z 2 be i n \ZR(^) such t h a t z-^+z^O. Then  z l + z 2 e C r ( U ) i f and on l y i f U(z^) and U ( z 2 ) are adjacent. Pr o o f : Suppose U(z^) and U ( z 2 ) are adjacent. Then s i n c e dim ( U ( z 1 ) n U ( z 2 ) ) ^  r - l there e x i s t v e c t o r s x-^,... , x r _ 1 , y 1 , y 2 such t h a t U ( z ± ) = < x±,...»xr_1,yi> , i = 1,2. Therefore z± = a ^ x-jA . . . A X ^ ^ A , a^eF, i = 1,2 and then z x + z 2 = x ^ . . . A X r _ 1 A ( a - L y 1 + a 2 y 2 ) G £ r ( U ) s i n c e z - ^ 2 ^  0 b y assumption. Suppose z-^+z2 = W-^A . . . A W P 6 | H r(U). I f W ^ A Z ^ = 0 f o r i = l , 2 , . . . , r then W ^ A Z 2 = 0 f o r i = l , 2 , . . . , r and consequently 10 11 w ietl(z.j), i = l , . . . , r ; j = 1,2. Therefore UCz-^) = <w^,..., w r > = U(z2). Suppose on the other hand t h a t W^AZ-^ ^ 0 f o r some t . Then s i n c e 0 4 w ^ z ^ = - W ^ A Z2 we have U (W ^ A Z ^ ) - U(W^.AZ2 Therefore, s i n c e dim U C W ^ A Z - ^ ) = r+1 and U(zi)£U(w.t_ z-^), i = 1,2 dim ( U ( z 1 ) n U ( z 2 ) ) ^ r-1. Lemma 3.2 Let Y e d (/\ U) and l e t z^eV, i = 1,...,s. Then { z ^ , . . . , z g ] i s independent i f and o n l y i f (U(zj),...,U(zg)} i s independent. P r o o f : I f [UCz-j^),... ,U(z g) } i s dependent then by lemma 2.6 /\ V(z±)Cj\ U ( z 1 ) + . . . + ^ U C Z J ^ ) f o r some i . Therefore z i 6 <C zi''' * > zi~i> ' I f |U(z-,) ,... ,U(z ) ] i s independent then by lemma 2.5 r x r dim ( / \ U ( z 1 ) + . . . + /\ U ( z g ) ) = s both when [ U ( z ^ ) , . . . , U ( z g ) | i s of Type I and of Type I I . Therefore dim <z^,...,z ^ = s. r We next d e f i n e two c l a s s e s of subspaces of A U and show t h a t these subspaces c o n s i s t e n t i r e l y of grassmann products. Def 3.3 0 l r ( u ) = ^ A u 1 U' i s an r+1 dimensional subspace of U.j Def 3.4 Let U-^  and U^ be subspaces of U where dim U-^  = r-1. Then V ^ u i ^ u 2 = { X 1 A * * ' A X r - l A L l | x i e U 1 , u e U 2 j . I f U 2 = U we w r i t e V ( u i ) l U 2 = 1 / ( u i ) * Def 3.5 1£(U) = | V t ui)l U2 | U l a n d U 2 a r e s u b s P a c e s o f u w i t ] l dim = r-11 . 12 Lemma 3.6 0( r(U)u V r(U)£ C ( A U ) . r r Proof: Let A U ' G $ r ( U ) and l e t z±e Cr(U)n A U' , i = 1,2 such, t h a t z-^+Zg ^ 0. Since U(z-^) and U(z2) are two r - d i m e n s i o n a l subspaces of the (r+1)-dimensional subspace IT' of U they are adjacent. Therefore z-^+Zg 6 l ~ r ( U ) . Since A U' has a b a s i s i n . r r C (TJ) i t f o l l o w s now t h a t A U ' eC ( A U). Let V ( u i ) l u p e V (U). I t i s s u f f i c i e n t t o show t h a t 1. CL r ^ ( U - j ^ l I L ^ i s a subspace of A U s i n c e by d e f i n i t i o n VCupilL^ - { 0 } Q C r ( U ) . Let z^VCUj.)!^, Z ; L ^ 0, i = 1,2. Then z l = X 1 A ' * * A X r - l A w l ' x^eU2»w-^eU2 z 2 = 7 1 A * * , A y r - l A W 2 » y i € U i » w 2 e U 2 * Then < x x,... ,x r_ ]_ > = T^ = < j ± , . . . , y r - 1 > i m p l y i n g x 1A...AX r_ 1 = a{y-jA .. •Ay r_ 1), a^P. Therefore f o r d,b e F d z 1 + b z 2 = y xA • A y r _ 1 A (daw 1+bw 2) e 1/(^)1112. Therefore VC^) JT^e C ( A U). r Lemma 3.7 I f V e £ ( A U) M dim V = r+1 then V e Of p ( U ) u 1£(U). Proo f : Let ^ z 1 ? . . . , z r + ^ be a b a s i s of V. Then [ UCz^^) ,... , U ( z r + 1 ) } i s an independent set of r - d i m e n s i o n a l p a i r w i s e adjacent subspaces of U. I f the set i s of Type I then there e x i s t x-^,... » x r_^» y x , . . . , y r + 1 such t h a t U ( z ± ) = < x-^ . . . , x r - 1 , y i > , i = l , . . . , r + l . I f we l e t TJ1 = < x 1 ? . . . , x r - 1 > and U 2 = < y-p • •. > y r + 1 > "then V = (V(U 1)IU2. On the other hand i f the s e t i s of Type I I then t h e r e e x i s t x - p . . . , x r + 1 such t h a t UCz^) = < xi>•••> xi_i> xi+i» • • • » x r + l / > . I f we set U 1 = ( x p " M X r + 1 ) then we see r t h a t V = A u ' • 13 Lemma 3*8 Let dim U = n and l e t r ^ n be an i n t e g e r , r Let Ye C( A U) . (1) I f VeV r(U) then dim V ^  n-r+1 (2) I f dim V = r+1 and n < 2 r then V e Ot (U). (3) I f dim V ^  r+2 then Ve'V (U). Proof: Let V = <\f(^1) I U 2 e 1£(U). I f f z x,... , z t ] i s a b a s i s of V then [ U(z^) ,... ,U(z^.) ] i s an independent Type I set sin c e U-^CUCz^) f o r i = l , 2 , . . . , t . Therefore there e x i s t r - l + t independent v e c t o r s x-^,... ,x r_-^,y^,... ,y^ i n U as given by lemma 2.3. Since dim U = n we must have t h e r e f o r e r - l + t ^ n or e q u i v a l e n t l y , t ^ n-r+1. We note (2) i s an immediate consequence of (1) and lemma 3«7« F i n a l l y , suppose dim V & r+2. Let f z ^ , . . . ,z^.]be a b a s i s of V . Then { UCz-^) ,... ,U(z^.) } i s an independent s e t of r-dime n s i o n a l p a i r w i s e adjacent subspaces of U. I f the set i s of Type I I then by lemma 2.4 there can be at most r+1 elements i n i t . Therefore the set i s of Type I and so V e "U^CU). r Lemma 3*9 L e t U-^  and U 2 be subspaces of U. Then ( A u ^ n ( A U 2) = A (UjMJg). r r r Proof: C e r t a i n l y /\ (UjO U 2) Q ( A % ) 0 ( A Ug). Let dim u\ = m^ , i = l , 2 and dim (U^nU 2) = t . Let { x-p ... , x m } and [ y-p ... , y m ] be bases of U-j^  and U"2 r e s p e c t i v e l y and suppose a l s o t h a t x. = y., i = l , 2 , . . . , t . 14 The union of the sets [ x i A •»./\x^ I 1 ^  i^< ... <- i < m-^  ] and 1 r { y ± A . . .»y± | 1 * i- L< ... c i p = m 2 and i p > t ] 1 r r r i s an independent set i n Au^ + A U 2 ^ e c a u s e { x x , . . . » y t + 1 > • • • , J m ^ ^ i s ^dependent. The number of elements i n these two sets i s (r 1)"*" ( r 2 ) ~ ( r ) w n e r e ( p ) i s d e f i n e d t o be zero when t < r . r r Therefore dim ( A TJ± + A U £) =* ( m l ) + ( m 2 ) - (*J, Now s i n c e dim [ ( A u 1 ) n C A V J = r r r r dim ( A U x) + dim ( A U 2) - dim ( A % + A U 2) = (» &)-[(P)+{P)-[t)l " ( r ) = r r r r dim A (UjOUg) and s i n c e A ( U ^ ^ ) S ( A U1)n( A U 2) the r e s u l t f o l l o w s . Lemma $ . 1 0 Let dim U = n = r + 1 . Let T J ^ and U 2 be (r + 1 ) -dimensional subspaces of U and l e t and V 2 be ( r - l ) - d i m e n s i o n a l sub&paces of U. Then r r ( i ) dim ( A t^n AU 2) -•- = 0 i f dim ( U ^ ^ ) ^  r - l , = 1 i f dim (U^Ug) = r , =r+l i f U1 = U 2. r ( i i ) dim f A U x0 W ^ ) J = 0 i f V 1 <Jf U x, = 2 i f V x C U r ( i i i ) dim[ V(V 1 )n1/(V 2 ) J = 0 i f dim(V- LnV 2) £ r - 3 , = 1 i f dim(V 1nV 2) = r - 2 , = n-r+1 15 P r o o f : ( i ) Th i s f o l l o w s at once from lemma 3 « 9 . ( i i ) Suppose V-ptU-p Then i f z e f ^ ) and z # 0, r U(z)<^U 1. Therefore z £ A t ^ . Suppose Y^jj^ Then /1/(V]L) c o n t a i n s 1/0^)11^ and V(V]_)ITJ 1 i s a two dim e n s i o n a l subspace of A U-jH *]/(v"^)• r I f z ^ l / O ^ ) and z ^ V t V 1 ) j U ] L then U ( z ) ^ U 1 . Therefore z * A U r and so 1/0^)1% = I f ^ r i A ^ , ( i i i ) I f 0 * z £ V(V 1)0V(V 2) then I . C U ( z ) f o r i = 1,2 which i m p l i e s dim (V^nVg) ^  r-2. T h i s proves the f i r s t statement of ( i i i ) . I f 1/(1^) H V ( V g ) co n t a i n s two independent v e c t o r s z^ and z 2 then Y±d U ( z 1 ) H U ( z 2 ) , i . e . V x = V 2. F i n a l l y , i f dim (V^ n V 2 ) = r-2 then 0 # /1/(V 1 )1| V 2 = 1/(V 2) |. V x = T (V ] L ) n V ( V 2 ) and has dimension 1. Of course, dim f V ( V 1 ) = n-r+1. CHAPTER IV THE 1:1 C-TRANSFORMATIONS Let W and U be f i n i t e dimensional v e c t o r spaces over a r r f i e l d E. A l i n e a r t r a n s f o r m a t i o n T: A W~>A U such t h a t T(H r(W))£ C r ( U ) i s c a l l e d a c - t r a n s f o r m a t i o n . We note t h a t T [ C ( A W| Q C ( A U) and T preserves the dimensions of the r- r elements of L_( A W) • A l s o note t h a t i f dim W = r then dim AW = 1 and /\W- !0? = L r ( W ) . In t h i s case the image r r of Aw under T i s a one-dimensional subspace i n £ ( A U ) , say r r < z > . We have t h e r e f o r e T( A W) = Au(z). In what f o l l o w s we assume dim W ^ r+1. r r Theorem 4.1 Let T: A ^ ^ A ^ ^ f . §. 1 J1 c - t r a n s f ormat i o n such t h a t T(0l (W)) S OT (U). Then there i s a subspace W'£U such t h a t r r r r dim W1 = dim W. and T( A W) = A W . r P r o o f : I f dim W = r+1 then Awe OTr(W) and the hypo t h e s i s of the theorem p r o v i d e s us w i t h the subspace W'. We proceed by i n d u c t i o n on dim W. Let dim W = n+1> r+1 and suppose f o r any proper subspace HCW there e x i s t s a subspace r r H'CU such t h a t T( A H) = A H 1. L e t , . . . ,x ^ be a b a s i s of W and l e t W± = < x x , . . . , x i _ 1 , x i + 1 , . . . , x n + 1 > , i = 1,2,...,r+1. r r Let W ^ C U be chosen such t h a t T( A W±) = A w i » . By lemma 3 . 9 , and s i n c e T i s 1:1, 16 17 T( A (w±a wd)) = T(( A w ±)n ( A w..)) = T( A w i)nT( A wd) = A w-^n A w..r = A (Wj'n w^ ) . Therefore dim (W. • H W .' ) = dim ( f f . H f f J = n-1. Therefore ^W1', Wg',..•,Wr1+1j i s a s e t of n-dimensional p a i r w i s e adjacent subspaces of U. We show t h i s set i s independent and of Type I I . Suppose i t i s dependent. Then by lemma 2.6, r r r Aw.'c AW^ +...+ A Wi<|>1 f o r some i > l . But t h i s i m p l i e s r r r A W^GAw i + ...+ A W^ _^  which i s f a l s e s i n c e X ^ A . . . A X ^ A X ^ A r r A " , A X r + l ^ o r x i / x ' * * A X ; r i f i = r + l ) i s i n A Wi but not i n Aw^-t-r + ...+ A w j _ _ i . Suppose ( W^ ' ,... » W r ^ ] i s of Type I . Then si n c e i t i s independent, dim ( + Aw^+1) = (n~1) + (r+1) ( ^ ( 1 1 r " I " ) = < i i m ^ A w ) . T h i s i s impossible- s i n c e r r r A w i ' + " •+ A wr+x i s contained i n the image of A W. Therefore [W 1 •,... ,W ^  \ i s independent Type I I . I f we l e t W' be the (n+1)-dimensional subspace of U c o n t a i n i n g W-^', W ',..., r r W ^  then i t i s c l e a r t h a t T( A W) = A w ' a*1*3- d i m w ' = d i m w-r r Theorem 4.2 Let T: ^ W~>/\U be a 1:1 c - t r a n s p o r t a t i o n such t h a t T (0T r(W))£ 07 (U). Then f o r some aeF and f o r some l i n e a r  t r a n s f o r m a t i o n A:W—>U, T = a C r ( A ) . 18 Proof: Let dim W = n+1 ^  r+1 and l e t fx-p... * x n + ] _ } ^ e a o a s i s f o r W. Let Wi = <x^,... > xi_i» xi +i» • • • »xn+l ^  l e t W^'CUbe chosen r r such t h a t T( A W. ) = A W. 1. We l e t W. . denote the subspace x x 1 * * r <x. X. y of U, 1 £ i,<.. ,<i ^ n+1 and then note t h a t l-p . .., J..i x r w . = n w, L'' l / l j=l.2i ...,rn-i J Let ,--,iK By a repeated a p p l i c a t i o n of lemma 3-3 and u s i n g the f a c t t h a t r r T i s 1:1, we conclude t h a t T ( A l . ) = Al- 1 4 • I t X-* • • • X IT • • • X f o l l o w s t h a t dim ( A W; 1 * ) = 1-1 * * * r We now show t h a t ( W^ ' ,... *Wn^ ] i s an independent Type I I set of p a i r w i s e adjacent subspaces of U. Let I ' C U be the r r subspace such t h a t T( A w) = A W. Then C W f o r i = 1 , 2 , . . . , n + 1 and dim W' = n+1. Theref ore [ W.^' ,... ,Wn+-^| i s a Type I I set of p a i r w i s e adjacent subspaces of U. I t remains to show i t i s independent, or e q u i v a l e n t l y , Suppose not and t e •* C . Let 0 4 y. 4 £ i=i 1** r r Aw.' . . Then s i n c e i x. A A X . 1 ^. i,<.. ,<i ^ n+1 \ * i i . . . x _ I i i ... x x r * 1 r r 1 r i s a b a s i s f o r A W, [7± ± 1 ± * ± < . m.<± i s a b a s i s 1 r i 1 r -* r f o r A W . C l e a r l y Z e W. » - a n d t h e r e f o r e Z A J , . = 0 x i ' " x v x r m m i f f o r i ^ , . . . , i r such t h a t 1 ^  i - ^ .. »<i ^ n+1. Therefore Z A J = 0 r f o r a l l y€ A W . But t h i s i s p o s s i b l e o n l y i f dim W ^ r which 19 i s i m p o s s i b l e s i n c e by assumption dim W r+1. Therefore n+i n w = o. By lemma 2.4 there e x i s t v e c t o r s yi>***»y n +i such t h a t WA' - < y 1 , . . . , 7 1 . 1 . y i + 1 . . . . . y n + i > . Therefore W.'^.^ = < 7* ,•••,y ± > and ^ • l r T(x. A...AX. ) = a. .... . y H A . . . * y . x l ^-r x l x r x l 1 r where 0 # a. . t P, 1 < i n< .. .<i_ 6 n+1. i x . . . i r x r /HI Let a = TT a, * . I f we l e t where r then T ( x . A ...AX. ) = a'. . y'. A ...Ay^ x l 1 r V ^ r ^ - i x, a'. = a when 1< i n < .. .<i £ r+1. x, .. .x, 1 r 1 r Now, suppose a. . = a f o r 1 ^  i-,< .. .<i ^ t - 1 f o r 1 r some t such t h a t t - 1 =^  r+1. Then we show t h a t a. . . = a. . . whenever l ^ i 1 < . . . < i t - 1 and ^ l * * * ^ r — 1 r—x 1 ^ ^<...<Dr t - 1 . I t i s s u f f i c i e n t to show they are equal whenever { i 1 , . . . , i r _ 1 j and { o-p ..., J r - 1 "] have r-2 i n t e g e r s i n common. L e t i € I ± 1 , . . . , i r _ 1 \ - f k x , . . . , k r _ 2 } and J e f d x , . . . , D r - 1 ] -£ k ,.. . > k r_ 2 j • Then { k-p ... , k r _ 2 , i , j ] i s a s e t of r d i s t i n c t i n t e g e r s l e s s than t . Let k be an i n t e g e r , 0 < k < t 20 such t h a t k [ k-^ , •.. , k r _ 2 , i , j \ . Such a k e x i s t s s i n c e t > r + l . Let X = x K _ A . . . A X K 2 A ^ x i + x j ^ ^ k ^ t ^ ' T h e n T ^ x ) e C r(U). We note t h a t T ( x k A .. .AX k A x ± A X k ) = a y k A .. .Ay k A y ^ y ^ . 1 r - 2 1 r - 2 s i n c e i f h^<...<hr and ( h - ^ , . . . , ^ j = [ k-^,... ,k g* 1*^ ] and i f TT i s the permutation k-^,... , k r _ 2»i, k h l ' - " ' n r - 2 ' h r - l ' h r then T ( x k A . . . A x k A x j * x k ) = ( s g n T r ) T ( x H _ A . . . A X ^ ) = (sgn TT ) a y h A • • .Ay h = a y k A . . . A y k * J ± * y k « S i m i l a r l y , T(x k_A . . , A x k A x j A x k ) = a.y k A • • 'Ay k A J^AJ-^. K X ^ A . . . A X k ^ A x i A x t ) = a i i > > e l ^ i t y k i - . . . A y k r _ A y i A y t T ( x k A . . . A x k ^ A x . A x t ) - a ^ . . . ^ y k A . . . A y ^ A y ^ L E T Y - V - W y i + V - W y r T h e n T(X) = y k A . . . A y k 2 A ( y ± + y n ) A y k + y v A . . . A y v A y A y+. • *1 ^ r - 2 v Since T(X) e C r ( U ) , < y k i , . . . , y k r _ 2 , y i + y d , y k > n < y k i , . . . , y k r _ 2 , y, y t > has dimension r - l . S ince y £ < y i , y p and ^  y 1,... > y n + 1 j i s an independent set we must have y = b(y.+y.), b e F . This i m p l i e s Let C denote the common value of the a. ^ 1 l * " 1 r - l t 21 Let y ' i - 7± i * t y ' t " a y t Then T(x. A . . . A X . ) = a'. . y'. A .../\y'j and a*. = a f o r l U , < . . . < i i t . We can assume t h e r e f o r e 1-, ... i 1 r 1 r t h a t y-p... , y n + - j _ were chosen at the o u t s e t so t h a t T(x. A . . . A X . ) = a.y. A...#sy. , 1 1 x r x l x r U i 1 < . . . < i r ^ n + 1 . Let A:W—>TJ such t h a t A x ± = j±. Then T = a C! (A) on the b a s i s set \ x. A . . . A X . 1 l^i-,< ...<i i n + l r < in ^ 1 1 r ) r and thus T = a.C r(A) on a l l of /\ W. r r Theorem 4.3 Let T: A U — > A U be a 1:1 c - t r a n s f o r m a t i o n and l e t dim U 2 r . Then f o r some a e F and f o r some l i n e a r  t r a n s f o r m a t i o n A:U—»U, T = a.C r(A). Pro o f : By theorem 4.2 i t i s s u f f i c i e n t t o prove t h a t T ( 0 e r ( U ) ) C 0 f r ( U ) . Suppose dim U < 2 r . I f TJ-, i s an ( r + l ) - d i m e n s i o n a l r r r subspace of U then T( A TJ^ e £ ( A U) and dim [ T( A U]_) J = r+1. r Therefore, by lemma 3-8 p a r t ( 2 ) , T( A ^ e O^CU). Suppose dim U > 2r and T( Of r(U) ) ^ 07p(U) . Let be an (r+1)-dimensional subspace of IT such t h a t T( A U^) \ Ol (J3). r Then by lemma 3.7 T( A U x) - V(W))W 16 1/^(U), dim W = r - l . L e t Ug be an ( r - l ) - d i m e n s i o n a l subspace of U^. Then dim f T( U ( U 2 ) ) ] = d i m [ < V ( U 2 ) ] = <iim U-r+1 > r+1. 22 Therefore T [ 1/(U 2) ] = 1/XU') f o r some ( r - 1 ) - d i m e n s i o n a l r subspace U 1 of U. By lemma 3.10 dim [ A U^n 1/(U 2) J = 2. Therefore dim [ T( A U ( U 2 ) ) J = dim [ V ( w ) l % H V ( U ' ) ] = 2. Therefore dim [ U ( W ) n 1 f ( U 1 ) ] ^ 2 and so by lemma 3.10 W = U« . This i n t u r n i m p l i e s t h a t 1/(W)) W 1 C V ( U ' ) and dim [ # V(W)lW 1 O I ^ U ' ) ] = dim ( <U(W)|W 1) = r + l > 2 , a c o n t r a d i c t i o n . Thus T( 0 T r ( U ) ) Q 0 l r ( U ) . r r Theorem 4-.4- Let T: Au—>AU be a 1:1 c - t r a n s f o r m a t i o n and l e t dim U = 2r . L e t 1£(U)- {W) subspace of UJ . U' i s an ( r - l ) - d i m e n s i o n a l Then e i t h e r or T i o y u n Q O I A ( U ) Tl%Lu)l c 14(U) TCaiA(u)] c %(v) - a r u i T t VA(U)] £ Oyu) P r o o f : I f V e ^ ( U ) then dim V = r+1. Therefore, by lemma 3-7 T( cyu) u T£(u)) & O y u ) u %iU) . I f U-j^  are two subspaces of U such t h a t dim = r - 1 , dim U-^  = r+1 then 2 = dim [ A ^ n T O p ] = d i m [ T( A u x ) n T [ V ^ ) ] ] . r T herefore, by lemma 3.10, e i t h e r T( A u i ) e 0 f r ( U ) , i n which case TL 1/(V,)] c %(U) or T(Au.) e 0£(U) , i n which case T(1 / ( V , ) ) e C \ ( U ) . The c o n c l u s i o n of the theorem now f o l l o w s from the o b s e r v a t i o n 23 t h a t . i f and U1-^ are any two ( r + l ) - d i m e n s i o n a l subspaces of U then there e x i s t s a sequence of subspaces U 1» vl»U2'. V2» , , ,' I Ik-l' Vk-l» uk = U ' l s u c h t h a t dim U. = r+1, dim V. = r - 1 , V. C IT. and V. CU. r X X X X X X ~ r X Theorem 4.5 Let dim U = n and l e t (x-^,... ,x n}be a b a s i s of U. n n-n. Let i e A U and e A U ^ two v e c t o r s such t h a t i - r , and where t>' = o- whenever n ...» r t - . - u ? e C ^ ( U ) i f a n d o n l ^ i f C n - * ( U ) . Proof: I t i s s u f f i c i e n t t o show ~£. £ U.^  (V) i m p l i e s C h _ A ( U ) . Let * = . ZL. p . XfA- • • A * . € C . ( l / j Let 1 p. ,^ 1 be skew-symmetric i n the s u f f i x e s . Then 1 p. . ] s a t i s f y the qua d r a t i c p - r e l a t i o n s [ 2 D , * A where { i , > . . Ln_(3 and £ j ^ - - , j n 3 a r e se^s o £ d i s t i n c t i n t e g e r s chosen from i •,2t • -, n ] . 24 Let P 1 . . f o r 1^ i 1 < . . . < i ^ n be d e f i n e d as i n i n • • • i „ x n—r 1 n-r the statement of the theorem and l e t { P'. . ] be skew-l i • • • x 1 n-r symmetric i n the s u f f i x e s . I t s u f f i c e s t o show t h a t H p; £ p' • • = ° f o r subsets { i ^ , . . . _ ^ 5 and {3 0»•••»d n - r 1 o f d i s t i n c t i n t e g e r s taken from f 1,...,n \ . We note f i r s t t h a t i f jx i s a permutation o f ( l , . . . n - r - 1 j and ( T a permutation o f ^ 0 , 1 , . . . , n - r j t h e n ZL (-° p • • ? • • » ZL ( _ , ) & ...t i pi . i i . i Therefore i t s u f f i c e s t o prove (1) f o r any a r b i t r a r i l y chosen o r d e r i n g of the s u f f i x e s { i - ^ , . . . , i n _ r _ i 1 and £ Jo'*'*'^n-r ^ * Let I % » • • •»l n_ r.i ? and \ J 0 » " ' » 3 n _ r \ b e ordered such t h a t i = j ,, s = l , 2 , . . . , t where t i s the number of i n t e g e r s in common i n the two s e t s and * 1 c i 2 < '' * K H ' \+l < it+2 < * * * < ^ - r - l ' Then (1) becomes ^ 5 2 1 (-1) p' . . pf s i n c e P- • = o when 7, = o, > .. t-/ • Since i - ^ , . . . »ln_r_i><3-fc> • • • ' ^ r i - r a r e d i s t i n c t , 0-=2r+t-n. 25 Let q = 2r+t-n and l e t £ b x < b 2 < • • • < b q ] D e t n e i n t e g e r s i n { l , . . . , n \ - i i i ^ . ^ V ^ i ^ t * - - - ^ ^ ^ • I f q = 0 then the set i s empty. Then, f o r A ^ t , where £^= i 1 and - = ± 1. We w i l l show t h a t £ A-V^ i s independent of 7v. To t h i s end i t i s s u f f i c i e n t to show t h a t £. 4A £, f _ = 1 f o r each A • For ^=TL ,J'=ZK^\ an<J - f o r , , / > ' . . . =• E.' U p 1 J o " " J ? - . J T H Y ' ' J « - A . )x< * ' h\" ^ t . t r + | • c r t_ A_ ( where £. =- ( - i / 1 - * - 7 1 . Let {1± < i 2 < • • • ^ i n _ r . * • *' ^ -n-r-l ^ 0 2 < *** "^n-r- o'dl» * * * '^A-l'^+2' * * • ' J n - r ^  U » < < ... < i ^ ^,... ^ q ^ t + l ' * * • , : L n - r - l 1 [d£< ^2 ^ ' * * *~ ^r-1 I ,bq,jt,..•»dA_2» 0 A +2»"*'J n. Then for = A , J( and for je=-\ + i t = ej • u 26 L e t k = - - !• Then there are k i n t e g e r s g r e a t e r than j ^ and l e s s than Suppose k^ of these i n t e g e r s are i n { i£,... » i ^ _ r - 1 } and k 2 of them are i n { o£,... , ^ - r - l ? * Then k - ^ of them are i n £j£,... »dp_]_ ^  and k-k 2 are i n i -L2' * * * ' r - l * I f CTJ" j ... , are the permutations t h a t order the s e t s ( 1 ) l±{,. ' * ^ n - r - l ' ^ 3 > (2) " » 1n-r-l'^+l^ > ( 3 ) " ' ^ n - r - l ' ^ A ' * ' ^ n - r - l ' ^ A + 1 * > (5) rd£,. • •' J r - l ^ A + 1 ? ; (6) (d£,. '•»°r-l ' J A . 3 , (7) *i£,. " • ^ r - l ' ^ + l ? > (8) {i£,. then (sgn^r-L)(sgn o~2) = k l = ( -D 1 (sgn rjr5)(sgn U"4) = k 2 (sgn cr^)(sgn <r 6) = - ( - l ) k - k l (sgn CTyXsgncj-g) = - (-l) k" k2 F i n a l l y , s i n c e = (sgn O^Xsgn q-^ ) £ ' M ) = (sgn cr2)(sgn q~6) = (sgnu-4)(sgn^r8) V ^ + ) = (sgntr 5)(sgncr 7) , 27 Therefore ^ -^^ i s independent of and so we set £ = <f^i^ Then = 0 . Therefore fP.' . } s a t i s f i e s the qu a d r a t i c P - r e l a t i o n s 1 l * * , : L n - r i m p l y i n g e C h . * ( U ) • Theorem 4.6 Let dim U = 2 r . Then t h e r e e x i s t s a 1:1 r c - t r a n s f o r m a t i o n T: A U —> A U such t h a t T L<XAw)l C %(U) r r Proof: Let ^ x x,... ,x 2 r^be a b a s i s of U. Let T: A U —-> /\ TJ be the l i n e a r t r a n s f o r m a t i o n such t h a t T ( x . A . . . A X . ) x l x r x . A . . . A X . where 1 ^  i-,< .. .-^i^ 2 r , 1 ^  j - , ^ .. .< j 2 r and { i-p ... ,i 5 O i j-p • • • , d r 1 = 0' Then by theorem 4.5jT i s a c-t r a n s f o r m a t i o n . I f we l e t TJ-^  = < X p . . . , x r + ^ ) and U 2 = < x r + 2 , . .. , x 2 r > then T [ (X ( t ^ ) ] = 1/(U 2) im p l y i n g by theorem 4.4 t h a t T [ 07 r(U) 3 Q V r ( U ) . r r Theorem 4.7 Let T: AU—> Au be a 1:1 c - t r a n s f ormat i o n and r r l e t dim U = 2r . Let J : A U —> A u £e aSZ f i * e d 1:1 c-t r a n s f o r m a t i o n f o r which J ( Of r(U)) Q 1/ r(U). Then one of T or TJ i s equal t o a.C (A) f o r some a e F and f o r some l i n e a r t r a n s f o r m a t i o n A:U—>U. P r o o f : C l e a r l y , e i t h e r T( 0T r(U) ) C CT r(U) or TJ(rJT r(U) ) CZ £T r(U). The theorem now f o l l o w s from theorem 4.3. 28 In the remainder of t h i s chapter we show t h a t there e x i s t Grassmann spaces which have c- t r a n s f o r m a t i o n s t h a t are not one-to-one. Lemma 4.8 Let r >1 be an i n t e g e r . L e t F be a f i e l d such t h a t i f a . g F , i = l , . . . , r + l and ZL a i = 0 then a]_= a2 =* • • = a r + l = 0 * Let U be a 2r-dimensional v e c t o r space over P. Then there e x i s t s a ( 2 r " 1 ) -dimensional subspace V C A u such t h a t V Ci C p ( U ) = fS. P r o o f : L e t f x - ^ , . . . , X 2 R } be a b a s i s of TJ. Por each set { i ^ < . . . < i r ] C f l . . . , 2 r - l \ we d e f i n e a permutation TJ . . of ( l , . . . , 2 r ] and a v e c t o r 2. s . e A U as f o l l o w s . 1 l * - * 1 r ^ - l " * 3 ^ L et { ^  < o 2 < • • • < 0 r 1 = (l»---»2r } - £ ... ,i p 1 , (1, 2, .. . , ' • . j 2.x \ _, i l ' i 2 ' " * , i r , ^ l , * * , ^ r y and l r Z. . = x. A • • . A X . + (sgn TT • , ) { x. A . . . A X . ) . ^ • • • ^ r x l r 1 l * * * 1 r d l dT ThenVz. . 1 ^  i , <.. .<i £ 2 r - l X i s an independent set of c x^ • • » v e c t o r s i n A U c o n t a i n i n g y ~ ) elements. Suppose Now we w r i t e A . and (2.) i m p l i e s J \ H J V . . P . , = o 9 29 where f P . . 1 [ i - , ,... , i _ } CZ f l , . . . ,2r } \ i s skew-symmetric i-^... \ ± T -> i n the s u f f i x e s . Let \J0< 31< ••• <drl CZ { l , 2 , . . . , 2 r - l 1 and [ i x < ... < i I > 1 ] = [ l , . . . , 2 r 5 - f d 0 , . . . , j r ] . Then P . . . . = a, . . . . D o * ' * a A - l J A + l * * * a r <V * • ^ ^ - I ^ A + I * ' ' ^ r Suppose < < +i Then • V V , = '^^  S" ' PT''- % ^ V< • ' l - O 7 1 ' * * " ' ^ ^ .... , . . ) ^  • • . = 14 1 . - \ — / V s * / >n_> „ / 1 a . . . Zn \ Therefore and n. * (->) p. p. 21 t-o* "A = o = O . Therefore 21 a * . . . = O - 0 and s o a - i i i i = 0 whenever 1 ^ ' * , < ; J A - 1 < °'^+l<* * * ^ r 1 ^  [ l , 2 , . . . , 2 r - l ] j i m p l y i n , t h a t 0 6 Cj.CU), a c o n t r a d i c t i o n . •S 30 2 We note t h e r e f o r e t h a t i f r = 2 then A U c o n t a i n s a 3-dimensional subspace W w i t h no non-zero pure v e c t o r s . Let V 2 "be any 3-dimensional pure subspace of A U. Then any 2 2 T: /\XS —> /\U w i t h k e r n e l W and image Y i s a c - t r a n s f o r m a t i o n . 2 Of course such t r a n s f o r m a t i o n s e x i s t s i n c e dim A U = 6 . CHAPTER V THE MAIN RESULTS In t h i s s e c t i o n we show t h a t any c - t r a n s f o r m a t i o n r r T: A U —> A U i s 1:1 i f the f i e l d P i s a l g e b r a i c a l l y c l o s e d . We need f i r s t t hree p r e l i m i n a r y lemmas. Lemma 5*1 Let U be an (n+l) - dimensional v e c t o r space over a f i e l d P and l e t f x - p ...,x n | j \be a b a s i s of U. Let U i = < x 1 5 . . . , x i _ 1 , x i + 1 , . . . , x n + 1 > f o r i = l,2, . . . , s + l where 1 ^ s = r ^  n . Let U' = < x ~ ^ > • l f r r Z e A U , + .,.+ A U ~\ then there e x i s t 1 s+1 r r oi £ A u1+...+ A u g _ 1 „ r-s+1 p e A u-r - s Y and % e A U' such t h a t 2. = CX -+- ^ A X , * • xs_( •+ ( X/\ys +• V A X ? < . ( ) A X , A • A X S _ ( . I f s = | we have = p + § „ X ) + Y A x z . I£ S=A wehave ^ = a + A . A X S _ , + (§x5 + Y x s J A X ( A . • *XS_, where S and Y e AU 1 = F and (3 e < x s ^ , . .. , X n + , > . P r o o f : Let W = £ z | z e A u i s o f t n e form (3) ^ . Then W i s a r subspace of A ^ a n d furthermore W c o n t a i n s the b a s i s v e c t o r s 31 32 of A u i > i = l , . . . , s + l i m p l y i n g A U l + < * * + ^ Us+1 C W. Lemma 5.2 Let U be an (n+1) - dimensional 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 P. Let r and s be i n t e g e r s  such, t h a t H s ^ r t n . Let { U-^ ,... >U g + 1 } be a n independent Type I I set of n-dimensional p a i r w i s e adjacent subspaces of IT. Let _ f l C /\ TJ-j_+. • • + A u s + i ^ §. subspace such t h a t dim _Q = (£~g j . Then there e x i s t s a set [ vi>***» v s ] 2 l s independent p a i r w i s e adjacent n-dimensional subspaces of TJ such t h a t r r J Q _ n ( A 7!+...+ A v s) * 0. Proof: Since { U-p ...»U S + 1 ] i s independent Type I I we can f i n d v e c t o r s xi»'-'» x n+i i n u such t h a t TJ^ = < x-^,... , xi_^ 1 x i + l 1 * * *' x n + l ^  ' ^ = "^ * *""' s+^" * "^e^ Z l ' * * * , Z k w n e r e k = ( r - s ) b e a b a s i s o £ * I f r = s then k = 1. I n t h i s case we have by lemma 5«1, z-, = . r r r = *i + X'AX-JA .. . A X g _ 1 e A U 1+.. .+ A U g - i + A U' where x' = (3 + 5x s + V x s + ) , p U , ; Y e F and where TJ' i s any n-dimensional subspace of U c o n t a i n i n g x' ,x^,... » x s_2 and such t h a t fU^,...,U ^,U' \ i s independent. Suppose s < r . By lemma 5-1 there e x i s t cX,- , ^ . , Tt- , St- J i = i, z, ..., k such t h a t where oii G A u 1 + . . . + A T J g - 1 r-s+1 (SL € A u' r — s X. and € A u' 3 3 and XT' « < x s + 2 > - - - > x n + l > • it Suppose { 5^,..., S K ] i s dependent and £ 3<; £<; = 0 where not a l l a. = 0. Then "21 a i 2-L = 1 i — i f>A • A-A -v ^ A X S A X , A - - A , where Therefore o 4 L ^ , ^ ( £ A U , + - t A U S . + A LL„ . I f ( V - L , . . . , JT^  } i s dependent we show i n a s i m i l a r way t h a t SL n ( A U, •+ ' + AUj., + A U S ) * O • The only remaining case to c o n s i d e r i s t h a t i n which both I " t f ^ , . . . , ] and 16"^,..., are independent. I n t h i s case both s e t s are bases of A U' si n c e dim /\ TJ' = (pig)* Therefore there e x i s t a. . e F, i = 1,... ,k and j = 1,... ,k such t h a t Y- = JL ; ^, where det (a..) 0. Since F i s a l g e b r a i c a l l y c l o s e d there e x i s t s A f 0 and b.^e F, i = l , . . . , k not a l l zero such t h a t A b ; = E b . a r , Then k where 34 Now k K Therefore ZL ^ «. A U , + " + A U S _ ( -t A U " where U n = < 3 C 1 , . . . , x s - 1 , A x s + x s + 1 , x s + 2 , . . . , x n + 1 > . Since f a k J k i s independent , Z L b ( ^ O . i = i Lemma 5.3 Let U he an n-dimensional 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 P and l e t and U 2 be ( r - 1 ) -dimensional adjacent subspaces of U. I f _Q_ c "MtU)-taV~(Uz) and dim S~L = n-r then X L r\ C A ( U ) 0 . P r o o f : I f U-^  = U 2 the statement i s t r i v i a l so suppose dim (D^nUg) = r"~2' L e t x i ' * , , ' x r - 2 y , Z b e c l l o s e 1 1 s u c n t h a t Ul = < xi» • • • ' x r - 2 , y > a n d U 2 = < xl» • * * » x r - 2 , Z > * Let X = x L A . . . A X R _ 2 and l e t & ± = X/»J*\X±+XKZ * <v±, i = l,..,n-r be a b a s i s of J7_ , ( u t , vc e U , <.' = <., ..,n-/0 . I f ( x x , . . . ,x r_ 2,y,Z,U 1,... , l i n _ r ] i s a dependent set then we can suppose ZL. * i -+ - Z L pfu£ = o j and not a l l ^ t. = o . Then O * 2 ^ 3 ; =: X A * A ( Z L ^ - - f ^ ) * C N ( U ) . Suppose ( x 1 ? . . . j X ^ g j y ^ Z j ' u ^ , . . . ,11^ } i s independent. Since t h e r e are n v e c t o r s i n the set we can w r i t e v. = 2 * t j X j + ft y + + 21 S£. u. ; , ft , Yj , 8tj e F . I f det a . . ) . = | - o then f o r some e F , i r W t and not a l l zero we have U • O and therefore,,.^ ri./L *.„. O + C d - a . = X A V A ( 21 c(;U- ) -+- X A ^ A C Z I o f i r ) - X A u A L LL <u. — ( n d L«) 2 J e C n(U) . I f det(S i.) ^ O , then s i n c e F i s a l g e b r a i c a l l y c l o s e d there e x i s t A * 0 and W^eF, i = l , . . . , n - r not a l l Z ero such t h a t AW. = W- . Then n-A. . n-A f\-A. v Z w-u- = ZLEw^i-.x, t l S H f J u + ( 2 W L Y t H v tuj and t h e r e f o r e p t ) y O * I L w £ 3 . = X A U + A'y)/\ L (.ZLWfp^y + A l l W4 u. ] Theorem 5*4 Let IT and W he f i n i t e dimensional v e c t o r spaces r r 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 F. Let T: AW be a c-1ransformation such t h a t T( Ol (W) )C0T T,(U). Then T i s 1:1. 36 P r o o f : I f d i m W ^ r + 1 t h e n c l e a r l y T i s 1:1 s i n c e Awe C( A w). L e t d i m W = n+1 > r + l . I f H i s a p r o p e r s u b s p a c e o f W r t h e n T r e s t r i c t e d t o A H s a t i s f i e s t h e c o n d i t i o n s o f t h e t l f b r e m . r T h e r e f o r e we c a n s u p p o s e T i s 1:1 on A H . T h e n , b y t h e o r e m 4.1, r r T ( A H) = A H ' where H ' C U h a s d i m e n s i o n e q u a l t o d i m H . P o r e a c h s , 1 6 s ^ r + 1 , l e t r r r 3) s = f V | V C A W a n d V = A W]L+...+ A ^ s where ^ W ^ , . . . , W g 3 i s a n i n d e p e n d e n t s e t o f p a i r w i s e a d j a c e n t n - d i m e n s i o n a l s u b s p a c e s o f W. ] . We now show t h a t i f T i s 1:1 o n e a c h e l e m e n t o f f o r l ^ i ^ k w h e r e k < r + l t h e n T i s 1:1 on D ^ + | i . r r _ i e t t A W 1 + . . . + AW k + ie j D k + 1 . L e t W | £ U be c h o s e n s u c h r r t h a t T( A W . ) = A w , ! > 1 = l , - . . , k + l . By lemma 3.9 we c a n 1 r r show e a s i l y t h a t T T A ( W . n W . ) l C ^ A(W!oWl) f r o m w h i c h i t 1 j x j f o l l o w s t h a t d i m (W.'nWl) ^ d i m (W.nW.) and W! , Wl a r e a d j a c e n t . Now, s u p p o s e [ W J ^ , . . . , W ^ + 1 ] i s d e p e n d e n t . T h e n r r r b y lemma 2 . 6 , A Wj_C A W ^ + . . . + A W ^ f o r some i , 2 * i « k + l . r r r T h e r e f o r e T ( A W±) Q T ( AW x+...+ AW j__1). S i n c e r r r k < r + l and i ^ k+1, A w ^ Aw ] L+...+ A w . ^ , and s i n c e T i s 1:1 o n e l e m e n t s o f JCL f o r j ^ k we must h a v e i = k+1 a n d T( Aw 1 + . . .+ A w k + 1 ) = T(AW 1+...+ Aw k ) . 37 Let XL C A W-L+.. .+ A ^ k + 1 b e t n e s u h s p a c e a n n i h i l a t e d by T. Then dim X L = ( ) said t h e r e f o r e by lemma 5.2 there e x i s t s V e l \ such t h a t X I n V # 0, a c o n t r a d i c t i o n . Therefore £ • • • » ^ k + l 1 1 S a n independent s e t . Now, T( A W1+...+ A W k + 1 ) = A wi +•••+ A W k + 1 and dim (AW'+...+ A W k + 1 ) * dim ( A W 1 + . . . + A W f c + 1 ) . I t f o l l o w s t h a t the dimensions are equal and T i s 1:1 on r r A W 1 + . . . + A W k + 1 . Therefore T i s 1:1 on D k + 1 f o r k ^ r . r c o n s i s t s of the s i n g l e element A W and so T i s 1:1 r on A W . Theorem 5*5 Let TJ be an n-dimensional v e c t o r space over an r r 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. Let T: ALT -» A U be a c- t r a n s f o r m a t i o n . Then T i s 1:1. r Proof: Suppose n < 2 r . I f A ^ e 07 r(U) then r r dim T( A u i ) =  r + 1 and s o ^7 lemma 3.8, T( A U ^ e # r ( U ) . Therefore by theorem 5.4,T i s 1:1. r Suppose n >2r and t h a t f o r some A > r r T( A % ) $ Of (U). Then T( A V = V(W)|W 1 f o r subspaces W and Wx of IT where dim W = r - l . I f 1/*(U2) e 1/r(U) then dim T [ 1 T ( U 2 ) ] > r+1 i m p l y i n g by lemma 3.8 t h a t T [ V ( U 2 ) ] e ^ ( U ) and furthermore we get t h a t T [ 1 / ( U g ) l = VCU') f o r some ( r - l ) - d i m e n s i o n a l subspace U' of U. Now suppose l ^ C U - p 38 r Then dim [ A I^n 1/(TJ2) J = 2. Therefore dim [ V ( W ) | W 1 A V ( U ' ) ] * 2 which i m p l i e s dim [V(W) O V ( U ' ) J * 2. By lemma 3.10 then W = IT'. Let Ug and U 2 he d i s t i n c t adjacent subspaces of U-^  w i t h dimensions r - 1 . Then T [ 1/(U 2) + V ( U | ) ] = V(W). Let I L C V C U ^ ) + I f C U g ) be the subspace a n n i h i l a t e d by T. Then dim XL = n-r. Therefore by lemma 5.3 _0_ n C A ( U ) # 0^ a c o n t r a d i c t i o n . Therefore T( 0 T r ( U ) ) C 0T r(U) and T i s 1:1. F i n a l l y , suppose n = 2 r . Let "V^ (TJ) be d e f i n e d as i n theorem 4-.4-. Let and TJ2 be adjacent ( r + l ) - d i m e n s i o n a l r subspaces i n IT and suppose T( A U^) e. 0T r(U) w h i l e r — r h. r T( A U 2 ) 6 1£(U). Then T( A U 1) =AU' and T( A U 2) = V(V') where dim U* = r+1 and dim V = r - 1 . Let V-^ CL TJ-^  1TJ2 be an r ( r - l ) - d i m e n s i o n a l subspace. Then dim ( VO^) n A U i ) = 2> i = 1,2. Therefore <U[T( VO^)) a A U 1 ] ^ 2 and dim [ TCVCV-L)) 0 ] * 2. By lemma 3*10 then we must have e i t h e r T [ V ( V 1 ) ] = A U - or T [ 1 / ( V 1 ) ] = 1/(V). Since t h e r e are at l e a s t t h r e e d i s t i n c t ( r - 1 ) - d i m e n s i o n a l subspaces of U^O TJ2 t h e r e are two d i s t i n c t subspaces and V 2 such t h a t T r V O ^ ) J = T r U ( V 2 ) J . A l s o V± and V 2 are adjacent. The argument of the second paragraph now pr o v i d e s us w i t h a r c o n t r a d i c t i o n . T h i s i m p l i e s t h a t e i t h e r T( A U ^ ) e ^ r ( U ) , r — i = 1,2 or T( A tf±) e V r ( U ) , i = 1,2. I f 1^ and U 2 are 39 a r b i t r a r y ( r + l ) - d i m e n s i o n a l subspaces then there i s a sequence of subspaces YQ = U-^  yY^,... ,Vg = Ug such t h a t dim Y^ = r+1 and V i are adjacent, i = l , . . . , s . I t f o l l o w s then t h a t e i t h e r T( 0 T r(U)) S. 0? r(U) or T ( 0 T r(U)) £ 0^(11). In the f i r s t case T i s 1 : 1 by theorem 5 . 4 - . Suppose we have the second case. We w i l l show t h a t T 2 L 01 (U) ] C 0? r(U) i m p l y i n g T 2 , and t h e r e f o r e T, i s 1 : 1 . I f T 2 [ 0Tp(U) ] $ (U) then T [ ^ ( V 1 ) J e V r ( U ) f o r some 1/(V 1) e ^ ( U ) . I f I ^ C u such t h a t dim = r+1 and Y±C TJ1 then r T( A U X ) = T [ 1 / ^ ) J . Let V 2 be an ( r - l ) - d i m e n s i o n a l subspace adjacent to V^. I f i e - V ( V 2 ) then dim (U(?:)+V 1) £ r + 1 . Therefore U(0+V-,C U 1 f o r some subspace II* such t h a t r dim U' = r + 1 . Since V-jCU' and ? e A u l 2 <= T [ "VO^) ] . Therefore T [ V C V g ) 3 = T [ ^ O ^ ) ~] and as before t h i s p r o v i d e s us w i t h a c o n t r a d i c t i o n . We are now able to a s s e r t Theorem 5.6 Let TJ be an n-dimensional 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 . I f n 2r then every c - t r a n s f o r m a t i o n r r of A ^ ^ A ^ i ^ S compound. I f n = 2 r then there e x i s t s a r r f i x e d c - t r a n s f ormation J : A TJ —> A U such t h a t any c - t r a n s f o r m a t i o n i s e i t h e r a compound or i s the composite  of J and a compound. REFERENCES Chow, Wei-Liang: On the Geometry of A l g e b r a i c  Homogeneous Spaces. Annals of Math, v o l . 50 (1949), PP. 32-67. Hodge, W.V.D. and Pedoe, D: Methods of A l g e b r a i c  Geometry. V o l . I , Cambridge, 1947. Wedderburn, J.H.M.: Lectures on M a t r i c e s , v o l . X V I I , Amer. Math. Soc. C o l l o q . P u b l . , P r i n c e t o n , N.J., 1934. 

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