The theory of rational integral functions of several sets of variables and associated linear transformations

THE THEORY OF RATIONAL INTEGRAL FUNCTIONS OF SEVERAL SETS OF VARIABLES AND ASSOCIATED LINEAR

TRANSFORMATIONS Andrew Hugh Wallace

A Thesis Submitted for the Degree of PhD at the

University of St Andrews

1949

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SJS E T rs OF VARIABLES AND ASSOCIATED LINEAR TRANSFORMATIONS.

T h e s is s u b m itte d by Andrew H. W a lla c e , M.A.,

f o r t h e d e g re e of

D octor o f P h ilo so p h y in t h e U n i v e r s i t y of S t . Andrews.

U U n i v e r s i t y C o lle g e , DUNDEE.

A p r i l , 1949.

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INTRODUCTION

The theme of t h i s p a p e r i s th e u n i f i c a t i o n o f two t h e o r i e s w hich a r o s e and were d e v e lo p e d in d e p e n d e n tl y of one a n o t h e r i n t h e l a t t e r p a r t o f t h e I 9 t h c e n tu r y an d t h e b e g in n in g of t h e 2 0 t h , namely t h e t h e o r y o f s e r i e s e x p a n s io n o f r a t i o n a l

i n t e g r a l f u n c t i o n s of s e v e r a l s e t s o f v a r i a b l e s , homogeneous i n t h e v a r i a b l e s of ea c h s e t , t h a t i s t h e s e r i e s e x p a n s io n o f a l g e b r a i c fo rm s i n s e v e r a l s e t s o f v a r i a b l e s , and t h e t h e o r y o f in d u c e d l i n e a r t r a n s f o r m a t i o n s , or i n v a r i a n t m a t r i c e s .

I have d i v i d e d t h e work i n t o f i v e c h a p t e r s o f which t h e f i r s t and t h i r d a r e p u r e l y h i s t o r i c a i l ; C h a p te r I i s an a c c o u n t of v a r i o u s m eth o d s, d e v i s e d b e f o r e t h e i n t r o d u c t i o n of t h e i d e a s of s t a n d a r d o r d e r and s t a n d a r d t a b l e a u x , o f fo rm in g s e r i e s e x p a n s io n s o f a l g e b r a i c fo rm s , w h ile C h a p te r I I I i s m a in ly o c c u p ie d by a n a c c o u n t o f S c h u r 's work on i n v a r i a n t m a t r i c e s . C h a p te r s I I , IV and V e s t a b l i s h th e l i n k be tv/e en t h e two

t h e o r i e s a n d , a t t h e e x p e n se of one o r two p o i n t s o f r e p e t i t i o n of d e f i n i t i o n s , a r e s e l f - c o n t a i n e d and may be r e a d c o n se cu ­

t i v e l y , more or l e s s w i t h o u t r e f e r e n c e t o t h e o t h e r two c h a p t e r s .

m a t r i c e s a n d I n v a r i a n t s " I owe my i n t r o d u c t i o n t o s y m b o lic m e th o d s, t o t h e G o r d a n -C a p e lli s e r i e s and t o t h e i d e a o f

s t a n d a r d f o r m s , t o m en tio n o n ly a few t o p i c s , and t o I T o f e s s o r D.E. L i t tl e w o o d * s " T h e o ry o f Group C h a r a c t e r s " 1 owe my i n t r o ­ d u c t i o n to t h e t h e o r y of i n v a r i a n t m a t r i c e s . A ls o , a l t h o u g h 1 have b o l d l y q u o te d i n t h e t e x t Young’s p a p e r s on s u b s t i t u ­ t i o n a l a n a l y s i s , my knowledge o f t h i s s u b j e c t i s e n t i r e l y due t o a r e a d i n g of Dr. R u t h e r f o r d ’s book " S u b s t i t u t i o n a l

i U i a l y s i s " . T'wo f u r t h e r books w hich have e n a b le d me t o s e e t h e p ro b le m o f i n v a r i a n t m a t r i c e s an d t i i e i r t r a n s f o r m a b l e s o t s from a somewhat d i f f e r e n t s t a n d p o i n t a r e % eyl’ s "The C l a s s i c a l Groups'*, an d S c h o u te n ’ s "D er R ic c i- K a lk û l" •

To a v o id c lu m s in e s s o f e x p r e s s i o n , a s w e l l a s p o s s i b l e c o n f u s i o n , 1 s h a l l n ot alw ays employ, i n t h e h i s t o r i c a l

s e c t i o n s , t h e n o t a t i o n s u s e d by t h e a u t h o r s q u o te d , b u t v / i l l r e p l a c e them by t h e f o l l o w i n g w herev er n e c e s s a r y .

S e t s o f v a r i a b l e s w i l l be d e n o te d by s i n g l e l e t t e r s , t h e l e t t e r s a , v , w o r u ( ^ ) , u ( ^ ^ . . . d e n o tin g row v e c t o r s ea ch o f m e le m e n ts and t h e l e t t e r s x , y , z o r column v e c t o r s o f m e le m e n ts . S e t s o f sy m b o lic c o e f f i c i e n t s a r e a l s o t o be d e n o te d by s i n g l e l e t t e r s .

An i n n e r p r o d u c t o f a row v e c t o r u and a column v e c t o r x ,

nam ely ^

I - 1

1 1 1 .

b e a c o e f f i c i e n t v e c t o r r a t h e r thsLC. a v a r i a b l e v e c t o r . The d e t e r m i n a n t s

4 " <

^ ( 0

X f - ' — 1 ^ 1 i r J 1 1 1

a ^ v \ c L

< x f 1 1

u f i ■ - - <

/ /

i ï t ' ^ K

1

w r < - ' '

o f r rows and columns a r e w r i t t e n , f o r r < m a s

( x“> x 'V - - - ( uf" U-'^L - - ■ w - '- 'i . y . . .

r e s p e c t i v e l y , and f o r r - m as

( x'V - - -

x f " ! ) J ' ’- - -

u‘~'j

r e s p e c t i v e l y .

The compound i n n e r p r o d u c t o r b i d e t e r m i n a n t

(k.'" x ‘' j

i s d e f i n e d a s

„ » u ~

where t h e summation i s t a k e n over a l l d i s t i n c t s e t s o f r nuiabear i , j . . . k c h o se n from t h e s e t 1 , such t h a t ...<^k.

The th e o re m on t h e d e t e r m i n a n t o f a m a t r i x p r o d u c t g i v e s - ---- ---Uty.) X* \ I

I J

LX^(W o r

o r

= ( w'" w."'. - . u “~j(x-“'

The o p e r a t i o n o f p o l a r i z a t i o n i s d e f i n e d a s

i=t

and 1 s h a l l ta k e t h e o p e r a t i o n s of s u c c e s s i v e p o l a r i z a t i o n t o

oe siiû p ly

(4> ‘-

( 4 / -

H H L —

w i t h o u t any e x t r a n u m e ric a l f a c t o r b e in g i n t r o d u c e d . y may o r may n o t be e q u a l to x.

The G a p e l l i o p e r a t o r of compound p o l a r i z a t i o n i s d e f i n e d as

*

w ith th e sumination e x te n d e d o v e r a l l s e t s o f r d i s t i n c t numbers i , su ch t h a t Ù c.} c - - ^ k ^ s e l e c t e d from 1 , 2 . . .m. But t h e a l t e r n a t i v e de t e ra i n a n t a l e x p r e s s i o n ,

a v a i l a b l e f o r o r d i n a r y compound i n n e r x ^ o d u c ts , i s o n ly v a l i d h e r e i f none of t h e x*s c o i n c i d e s w ith any o f t h e y 's * The

c o r r e s p o n d in g r e s u l t when x^^) = y (^ ) i s g iv e n below ( ^ 5)* O th e r n o t a t i o n a l d e v i c e s a r e i n t r o d u c e d l a t e r when n e c e s s a r y .

V .

Chap.

§

a

CONTENTS

I - S m iE S DEVELOPMENT OF ALGEBRAIC FORMS

1

Pap;e

Sym bolic Methods and R e d u c tio n Theorems o f I n v a r i a n t Theory . . . .

2 3 4 5 6 7 8 1 4 9

The C leb sch -G o rd an S e r i e s

C le b s c h * s Theorem . . . . .

The R e d u c tio n Theorems of M artens and G a p e l l i 13 The G a p e l l i O p e r a to r s . . . 14 The f i r s t M e r t e n s - G a p e ll i R e d u c tio n Theorem 18 The sec o n d M e r te n s - G a p e lli R e d u c tio n Theorem 19 C e r t a i n B erj.es E x p an sio n s an d t h e A p p l i c a t i o n

o f t h e R e d u c tio n Theorems . . 22

Chap. I I - STANDARD FORMS

9 - Young T a b le a u x and Double Forms é 10 - P o l a r i z e d Double Forms

a 11 - The C a lc u lu s o f T ab leau x .

^ 12 - The E x p r e s s io n o f a A l u l t i l i n e a r Form i n Terms o f P o l a r i z e d Double S ta n d a r d Forms

§ 13 - G e n e ra l E x p an sio n i n Terms o f P o l a r i z e d Double S ta n d a r d Forms

ë 14 - U n i l a t e r a l S t a n d a r d i z a t i o n

26 26 55 56 41 42 48

Chap. I l l

^ 15 ë 16 § 17 ^ 18

TRANSFORL'IABEE SETS AND INVARIANT MATRICES T ra n s fo rm a b le S e t s .

I n v a r i a n t M a tr ic e s . . . . S ch u r* s D i s s e r t a t i o n

TTirther Advances by S chur

55

55

57

59

L

Page

Chap. IV - INVÆIAM) MATRICES AND THB GORDAN-CAPELbl

SERIES . . . 75

§ 1 9 - A i 'r e s h S t a r t . . . . . 7 6 ^ 20 - T ra n s fo rm a b le S e t s f o r N o n -S in g u la r

I n v a r ia n t M a tr ic e s . . . 79

§ 21 - P r e l i m i n a r y R e d u c tio n o f an i l r b i t r a r y

I n v a r i a n t M a trix . . . . . 8 1

§ 22 - R e d u c tio n o f a Homogeneous I n v a r i a n t M a tr ix 85 à 2 5 - The C o n s t r u c t i o n of t h e I r r e d u c i b l e

I n v a r i a n t M a trix T /J; . . . 84 § 24 - T ra n sfo rm a b le S e t s f o r t h e I r r e d u c i b l e

I n v a r i a n t M a trix T%) and t h e R e d u c tio n

Theorem . . . . _{• •} 88

§ 2 5 - Symmetric F u n c tio n s of t h e L a t e n t R o o ts o f A 9 0

§ 26 - 'Traces o f I n v a r i a n t M a tr ic e s _{• •} 9 2 2 7 - The T ra ce of th e M a trix T m ; . _{. • 93}

ë 28 - The T ra ce Automorphism _{• .} 97

a 2 9 - ‘The C e n t r a l Core . . . 98

Chap. V - INVARIANT MATRICES AND THE THEORÏ OE EORMS . 100 ^ 3 0 - C a l c u l a t i o n of t h e G o r d a n - C a p e lli

C o e f f i c i e n t s

3 1 - A S p e c i a l G o r d a n -C a p e lli E x p a n sio n ë 3 2 - T ra n sfo rm a b le S e t s an d I n v a r i a n t s 6 33 - P ro o f o f Theorem V .

y 3 4 - P r o o f o f Theorem I I I

100

1 0 7

111

1 1 3 1 1 5

Ï 8 35 - The F i r s t Fundam ental Theorem o f I n v a r i a n t s 120 If

T S 36 - The S t r u c t u r e o f a P o l a r i z e d Double Form . 122

RETROSPECT . . . 124

CHAPTER I - SEPIES DEVELOPMENT OF ALGEBRAIC IDE MS ^ 1 . S^rmbolical Methods and R e d u c tio n Theorems

of I n v a r i a n t Theory

Towards t h e end of l a s t c e n tu r y two m ajo r s t e p s were

t a k e n i n t h e u n i f i c a t i o n of i n v a r i a n t t h e o r y , w h ic h , up t o t h a t t i m e , h a d c o n s i s t e d m a in ly of e x p l i c i t c a l c u l a t i o n s o f s e t s o f i n v a r i a n t s f o r p a r t i c u l a r fo rm s or s e t s of fo rm s . T hese w ere, nam ely, t h e i n t r o d u c t i o n o f t h e method of e q u i­ v a l e n t sym bols and t h e r e d u c t i o n of i n v a r i a n t t h e o r e t i c

p ro b le m s by e3q:>anding t h e r e l e v a n t form s i n te r m s o f s t a n d a r d fo rm s . B e f o r e any s y s t e m a t i c sy m b o lis e d c a l u l u s was i n t r o ­ d u c e d , s y m b o lic a l methods c e r t a i n l y had been em ployed. I n h i s work on i n v a i 'i a n t t h e o r y S y l v e s t e r liad made u s e o f sy m b o ls, w hich he c a l l e d um brae, ( e . g . S y l v e s t e r QJ and [2] J, The s,ym bolical method i n t r o d u c e d by ^ r o n h o ld ( [Ij p . 106) and C le b s h ( [ i j p . 117 and [2] ) was much more t h a n j u s t an

a b b r e v i a t e d n o t a t i o n , and i t endowed i n v a r i a n t t h e o r y n o t o n ly w i t h a b r e v i t y of e x p r e s s i o n , b u t a ls o w i t h a f a c i l i t y o f

m a n i p u l a t i o n # i i c h had been h i t h e r t o unknown.

The G le b sh -A ro n h o ld method o f s y m b o l iz a t io n was to %vrite t h e c o e f f i c i e n t s o f an a l g e b r a i c form as p r o d u c t s o f sy m b o lic f a c t o r s i n such a way t h a t t h e symmetry p r o p e r t i e s o f t h e co­ e f f i c i e n t s w ere p r e s e r v e d ; - E .g . f o r t h e m -a ry t r i l i n e a r form

f = \ I j k ^ i ^ j c o e f f i c i e n t a ^ j ^ may be w r i t t e n a s

0 J k

r

2

.

2 = ^111^1^1 + ^112^1^2 + - - + 2 a i2 1 ^ 1

^ J k

^2^1 *’■ ~ tias t h e symmetry p r o p e r t y = aj and so t h e c o e f f i c i e n t s must be sy m b o liz e d as a ^ j ^ = ^ i ^ j ^ k "

(b = a^b • i'o e x p r e s s s y m b o l i c a l l y p r o d u c ts o f c o e f f i c i e n t s X y

o f a fo rm , i t i s n e c e s s a r y to i n t r o d u c e e q u i v a l e n t sy m b o liz a ­ t i o n s o f t h e same fo rm . In t h e f i r s t example g iv e n h e r e t h e

c o e f f i c i e n t a^^j^ c o u ld be r e p r e s e n t e d e q u a l l y w e ll as

o r a ^ t j C ^ o r SL^^b^^c^^ e t c , ; th e n t h e p r o d u c t ^ o u ld be w r i t t e n u n am b ig u o u sly as E v i d e n t l y , to

a v o id a m b ig u ity , t h e sy m b o lic e x p r e s s io n of a p r o d u c t o f two o r more c o e f f i c i e n t s o f a form must be l i n e a r i n e a c h o f t h e s e t s o f sym bols i n t r o d u c e d . This c o n d i t i o n b e in g s a t i s f i e d t h e sym bols may be m a n ip u la te d j u s t l i k e o r d i n a r y num bers, w ith t h e a d d i t i o n a l o p e r a t i o n o f interciriange o f e q u i v a l e n t •y m b o ls, vdiic-i s im p ly means t h a t , t o ta k e t h e above t r i l i n e a r form a s an exam ple, i n an e x p r e s s io n i n v o l v i n g t h e f a c t o r s a f , b j , c^ t h e s e may be changed t o a ^ , b j , c^ p r o v i d e d t h a t t h e l i n e a r i t y c o n d i t i o n i s not v i o l a t e d .

I n t h e above q u o te d p a p e r ( G leb sc h j2] ) C le b s c h c o n s i d e r s a number o f fo rm s i n a s i n g l e u n a ry s e t o f v a r i a b l e s ,

e x p r e s s e d s y m b o l i c a l l y

-(afjc, +- ^

---where t h e d i f f e r e n t l e t t e r s a , b, c — d e n o te t h e d i f f e r e n t e q u i v a l e n t s y m b o l iz a t io n s ^ C le b s c h , by t h e way, does n o t use t h e ab j r e v i a t e d n o t a t i o n f o r t h e summations b u t l e a v e s them

3

w r i t t e n out i n f u l l ) and i ru n s from 1 t o k, s a y . The problem o f f i n d i n g th e i n v a r i a n t s of th e forms cfj^^ i s th u s re d u c e d t o th e d i s c u s s i o n of i n v a r i a n t s of t h e

sym bolic l i n e a r forms u s in g

t h i s p r i n c i p l e C leb sch pro v es t h e f i r s t fundam ental theorem of i n v a r i a n t th e o r y f o r t h e case of a number of forms in one m -ary s e t o f v a r i a b l e s , nam ely, t h a t any i n v a r i a n t or c o v a r i a n t o f th e forms may be e x p r e s s e d r a t i o n a l l y and i n t e g r a l l y i n term s of t h e sym bolic in n e r p ro d u c ts

and th e sym bolic d e te r m in a n ts whose columns a r e any m of th e column v e c t o r s

From t h i s s ta g e th e p r o g r e s s of i n v a r i a n t th e o r y was a lo n g two main l i n e s , f i r s t l y th e e x te n s io n of t h e f i r s t

fundam ental theorem t o system s in v o lv in g more th a n one s e t of v a r i a b l e s , and s e c o n d ly t h e r e d u c t i o n s of system s in v o lv in g a number o f s e t s of v a r i a b l e s t o sy stem s i n v o l v in g few er s e t s . I t i s t h i s second l i n e o f developm ent which I want t o s k e tc h now. The f i r s t s t e p c o n s i s t e d in th e in d e p e n d e n t d is c o v e r y by Gordan and C le b sc h o f t h e th eo rem t h a t a form i n two b in a r y s e t s of v a r i a b l e s x = and y = ^ m a y be e x p r e s s e d

^ 2. The Gk>rdaii-Clehsch S e r i e s .

G ordan*8 argument i s as f o l l o w s (Gordan [ i j ) -

The g iv e n b in a r y form i n t h e two in d e p e n d e n t s e t s x and y i s e x p r e s s e d s y m b o l ic a l ly as

Then t h e c o v a r i a n t s

éo-j V r ^ ( 1 )

a r e e x p r e s s i b l e a s l i n e a r c o m b in a tio n s of p o l a r s of t h e form s

( 2 )

m u l t i p l i e d by powers of t h e d e te r m in a n t ( x y ) ; t h i s i s

o b v i o u s l y t r u e f o r k « q, f o r (a b )^ a -^ '^ "^ o y i s ( a p a r t from a n u m e ric a l f a c t o r ) a p o l a r of ( ab)^ , and so i t must be p ro v e d t r u e f o r a g e n e r a l v a l u e , s a y j , o f k by i n d u c t i o n on

q - k . Suppose, t h e n , t h a t i t i s t r u e f o r a l l v a l u e s o f k such t h a t J ^ k ^ and c o n s i d e r th e v a lu e o f

...(3 )

I f t h i s p o l a r i s w r i t t e n o u t i n f u l l i t may be s e e n t o c o n s i s t o f t h e comifion f a c t o r (a b )^ m u l t i p l i e d i n t o a sum o f te rm s e a c h o f w h ich i s a p r o d u c t o f (p - j ) a's and ( q - j ) b*s w i t h h + 1 o f t h e s u f f i x e s e q u a l to y and th e r e s t e q u a l to x , t h e s e t o f s u f f i x e s b e in g a r r a n g e d i n a l l p o s s i b l e v/ays to g i v e t h e

5 .

Any o t h e r te r m , T, may be changed i n t o G by a number of i n t e r ­ changes o f s u f f i x e s , p a s s i n g th r o u g h t h e i n t e r m e d i a t e s t a g e s

-f'^ T / ' T ' - - 7 7 ' s a y . Then

'

r

= T - r ' ^ T - T " + T - T " : — -

-Each o f t h e d i f f e r e n c e s I / and / — (r c o n t a i n s th e f a c t o r s ( a b ) ''^ | (xy) a s a r e s u l t o f t h e a l t e r n a t i o n o f s u f f i x e s . Hence each o f t h e s e d i f f e r e n c e s i s e q u a l t o (xy) tim e s one o f t h e co v a r i a n t s (1 ) f o r some v a l u e s o f h and 1 and t h e v a l u e j + 1 o f k and s o , by h y p o t h e s i s , may be e x p r e s s e d l i n e a r l y i n te r m s of p o l a r s of t h e form s (2 ) tim e s pov/ers o f ( x y ) . Hence t h e form (3 ) i s e x p r e s s i b l e a s a ( n o n - z e r o )

n u m e ric a l m u l t i p l e o f G added to p o l a r s of t h e fo rm s (2 ) e a c h tim e s a pov/er o f ( x y ) , and s i n c e th e form (3 ) i s i t s e l f a p o l a r o f one of t h e form s (2 ) w h ile G i s t h e co v a r i a n t o f t h e

ty p e (1 ) f o r k = j , th e i n d u c t i o n i s c o m p le te d .

from c o n s i d e r a t i o n s o f t h e d e g r e e s i n t h e x- and i n t h e y - v a r i a b l e s , t h e s e r i e s e x p r e s s i o n f o r G must be of t h e form

-where t h e a r e n u m e ric a l c o e f f i c i e n t s . Gordan o b t a i n e d r e c u r r e n c e r e l a t i o n s f o r t h e s e c o e f f i c i e n t s , and h en c e e v a lu a ­ t e d them. I n p a r t i c u l a r , f o r j - 0—

= 2 " — ( È l i ü r

J tz ïîl—

Where

( r j ” n ( t - r ) !

i s a s f o l l o w s

-L e t t ^ ^ be t h e g iv e n fo rm , e x p r e s s e d s y m b o l i c a l l y . I'hen i f j T . i s t h e C ay ley o p e r a t o r

i t may be e a s i l y v e r i f i e d t h a t

f I ^ ) J 1 j- ( 5 )

I h i s i s a s p e c i a l case o f th e G a p e l l i i d e n t i t y u s e d below 6 and 7). C le o sc h t h e n a s s e r t s , and p r o v e s by i n d u c t i o n on k t h a t f may be e x p r e s s e d a s a l i n e a r c o m b in a tio n o f t h e fo rm s

( y h T 2- - - - k )... ( 6 )

where k does n o t ex c eed q. The r e s u l t i s t r u e f o r k = 1 , b e in g e x p r e s s e d , i n f a c t , by e q u a t i o n (5)* Assuming t h e r e s u l t t r u e f o r k = j , t h e n i f h o ld s f o r k = j + 1*— f o r , a p p ly in g e q u a tio n (5 ) t o t h e form

7 .

8Lnd t h e n m u l t i p l y i n g th r o u g h o u t by , i t f o l l o w s t h a t

an d so t h e form s (<) w i t h k « j and ^ ^ I y j a r e

e x p r e s s i b l e l i n e a r l y i n te rm s of t h e fo rm s (<î) w i t h k = j + 1

and ^ » 0, ij jj-t-l * ai^d t h i s c o m p le te s ttie i n d u c t i o n . I n p a r t i c u l a r t h e r e s u l t h o ld s f o r k » q. f u t i n t h i s

c a s e t h e form s ^ i i a r e

in d e p e n d e n t o f y an d so th e r e l a t i o n

r

H o ld s, vv iore th e ^ a r e f r e e o f y and a r e . g i v e n b y

i'iow,

H i = M M

and 0 0, by i n d u c t i o n ,

‘ <■"

Then, p o l a r i s i n g

-(

hence

= 2 ( 3 )

r

N ext, i i i s a form o f t o t a l le?çree S su ch t h a t

J% '=^ O , i t may be v e r i f i e d by d i r e c t d i f f e r e n t i a t i o n t l u

Hence i s z e ro i f r ^ 5 , and c o n t a i n s (xy) as a f a c t o r i f r ^ s , w h ile

J 2 ^ { — rir-lh - bfS)U'*-^)P.

The foims ^ a l l s a t i s f y th e c o n d i t i o n

-TL.

(^ ^ jV =

O

s i n c e S h commutes w ith • and <p^ i s in d e p e n d e n t of y . And so i f JTIT i s a llo w ed t o a c t on th e r i g h t hand s i d e of (8) t h e r e s u l t i s

£ + term s

I r ' % ' / • n Q

I f y i s put equal t o x th e s o le s u r v i v i n g term i s

containing th e fa cto i

( x y ) .

< ...( S )

B ut, p u t t i n g y = X in e q u a tio n (7)

-T r h ^ '. ... E q u a tin g th e e x p r e s s io n s (9) and (1 0 ),

. / _ - , _

which a g r e e s w ith (Jordan’ s r e s u l t (Eqn. ( 4 ) ) .

A second s e r i e s developm ent, t h i s tim e f o r a form u j ! ' ^

where x and y a r e two m -ary s e t s of v a r i a b l e s was a l s o r i v e n by (Jordan ( fkj , S i and Z) namely

I I ; ... ( 1 1 )

For b in a r y v a r i a b l e s t h i s s e r i e s re d u c e s t o t h e s e r i e s (4) ; w h ile f o r t h e case of t e r n a r y v a r i a b l e s , p u t t i n g

-9 .

( r u r n b u l i [L] p. 255 and S tu d y [l] pp. 55-PÇ/

Ü 3* O l e ü s c li 's Tiaeorem

The O ie js c h -G o rd a n s e r i e s e x p a n sio n o f a form i n two b i n a r y s e t s o f v a r i a b l e s i s a s p e c i a l c a s e o f t h e th e o r e m o f C le b sc h which was d i s c o v e r e d v e r y s h o r t l y a f t e r t h e s e r i e s

a p p e a r e d . I n t h e e x p r e s s i o n of t h i s th e o re m , compound co­ o r d i n a t e s a r e i n t r o d u c e d . I f x, y , z, a r e r s e t s o f

»VI^ /

m. v a r i a b l e s ( r ^ ) t h e n t h e s e t o f yrjZXr)i ‘^"•^owed d e t e r m i n a n t s

... =*-/

---I i ( k

%

----i s c a l l e d a v e c t o r of compound c o o r d ----i n a t e s o f r- tk. c l a s s . I f a form f depends on an a r b i t r a r y number o f compound v e c t o r s o f ea ch c l a s s , th e n C le b s c h * s theorem s t a t e s t h a t f can be e x p r e s s e d a s a l i n e a r c o m b in a tio n o f p o l a r s of f o r m s , each o f w hich depends on j u s t one compound v e c t o r of each c l a s s .

( C le b s c h [4] and [5j ) *

o in c e t h e r e e x i s t n o n -z e ro compound v e c t o r s o f c l a s s r o n ly f o r r^*vw, C le b sc h * s theorem i m p l i e s t h a t , i f t h e b a s i c v e c t o r of e a c h c l a s s i s s u i t a b l y c h o s e n , th e n a n a r b i t r a r y form may oe e x p r e s s e d as a sum of p o l a r s o f fo rm s ea ch of

w i l l be e x p l a i n e d below , t h e d i s c u s s i o n o f c o n c o m ita n ts of a r b i t r a r y s e t s o f fo rm s may be made to depend on t h e d i s ­ c u s s io n of c o n c o m ita n ts of fo rm s i n n o t more t h a n m s e t s of v a r i a b l e s .

The p r o o f o f 0 1 eb sc h * s th eo rem ( g iv e n i n C le b s c h , [4^ b e in g o n ly a s k e t c h of t h e r e s u l t s ) s t a r t s o f f fro m t h e s e c o n d s e r i e s developm ent d i s c o v e r e d by Gordan^ t.e. (^11). The symbols a _X and b _y stan.d f o r i n n e r p r o d u c t s , but t h e r e i s n o th in g i n G o rd a n 's w orking w hich r e q u i r e s t h a t th e y s h o u ld be s im p le i n n e r p r o d u c t s , each fo rm e d from two s i n g l e v e c t o r s ; th e y may s t a n d f o r compound i n n e r p r o d u c ts o f r-tL c l a s s

-Ck)rdan*3 r e s u l t shows t h a t ^ , w i t h t h i s i n t e r p r e t a t i o n o f t h e sym bols, may be e x p r e s s e d i n te rm s o f p o l a r s o f form s l i k e

where now t h e p o l a r i z a t i o n s a r e compound, i . e . i f i s

1 1 .

r — k- , k b e in g a p o s i t i v e i n t e g e r .

C o n s id e r now a form F homogeneous i n t h e e le m e n ts o f e a c h o f a number o f compound v e c t o r s of each c l a s s . I f F i s

e x p r e s s e d s y m b o l i c a l l y , i t ^M.11 appear a s tlie p r o d u c t o f

f a c t o r s l i k e t h e compound i n n e r p r o d u c ts a _X and b _J of t h e l a s t p a r a g r a p h . i/ioreover, i f F c o n t a i n s two d i s t i n c t compound v e c t o r s x and y o f r - t k c l a s s i t must c o n t a i n a f a c t o r l i k e a ^ b^ t o wiiich th e Gordan r e s u l t may be a p p l i e d as i n t h e l a s t p a r a g r a p h .

I'he w eig h t o f a compound v e c t o r o f c l a s s , c o n s t r u c t e d from ut-ary v e c t o r s , may be d e f i n e d as r.( - r ) , and th e

w e ig h t of a form as » where u> i s t h e w e ig h t of a v e c t o r o c c u r r i n g i n 1 , )p i s t h e d e g re e t o w hich i t a p p e a r s , and t h e summation i s t a k e n o v e r a l l compound v e c t o r s a p p e a r i n g i n F . I n p a r t i c u l a r , th e w e ig h t of b^ i s (p + q) r . - r ) and t h i s i s a l s o th e w e ig h t of each o f t h e fo rm s a ^ ^

i n t h e Gordan e x p a n s io n , sind of any compound p o l a r o f t h e s e fo rm s i f o r w eig h t i s o b v io u s ly i n v a r i a n t u n d e r

su ch p o l a r i z a t i o n s . The c o n t r i b u t i o n to t h e w e ig h t o f

( a ^ b ^- a^b^)^ rftte t o a s i n g l e f a c t o r a ^ b ^ - a ^ b ^ i s Z r ( K v - r ) ; b u t ;ln t h e a p p l i c a t i o n o f t h e S y l v e s t e r i d e n t i t y t h i s f a c t o r i s r e p l a c e d by a sum o f p r o d u c ts b ^ , w h ich , X^ and y^ b e in g of c l a s s e s and r - L r e s p e c t i v e l y , i s of v/eight

( r f k . ) ( y—k.) + k , ] ( ^

fiais i s l e s s t h a n % v-), t h e w e ig h t of a^ b ^ . And so t h e e f f e c t of t h e S y l v e s t e r i d e n t i t y i s t o e x p r e s s

a b - a b a s a sum of form s of low er w e i g h t , and by r e p e a t e d X y y X

a p p l i c a t i o n o f t h i s p r o c e s s , a^ b^ i s r e p l a c e d by an a g g r e g a t

X y

o f form s o f low er w e ig h t. I f , t h e n , F c o n t a i n s more t h a n one v e c t o r o f t h e same c l a s s , i t may be e x p r e s s e d a s a sum o f

p o l a r s of form s o f low er w eig h t t h a n F ; and s i n c e t h e p r o c e s s o f d e c r e a s i n g w e ig h t must t e r m i n a t e i n a f i n i t e number o f

s t e p s , i t f o l l o w s t h a t F w i l l u l t i m a t e l y be e x p r e s s e d as an a g g r e g a te of p o l a r s of form s each o f w hich c o n t a i n s j u s t one v e c t o r of e a c h c l a s s , the p o l a r o p e r a t i o n s b e i n g , o f c o u r s e ,

compound p o l a r i z a t i o n s , i n which e le ra e n ts o f compound v e c t o r s a r e s u b s t i t u t e d and perm uted w i t h o u t r e g a r d to t h e i r c o m p o site n a t u r e .

I n t h e l i g h t o f l a t e r knowledge t h e r e s u l t o f C le b s c h * s th eo re m i s se e n to be t h e e x p r e s s i o n of an a r b i t r a r y fo rm i n te r m s o f compound p o l a r s of s t a n d a r d f o r ia s , and s u c h compound p o l a r s can alw ays be e x p r e s s e d i n term s of a g g r e g a t e s of

o r d i n a r y p o l a r s of s t a n d a r d form s ( c f , i h r n b u l l [j] Chap. XXIII

§

i ) .

3 .

a 4 . 'Che d e d u c ti o n Theorems o f Mertô.ns an d G a p e l l i

Both iviertens and C a p e l l i , i n t h e works j u s t q u o te d , made a d e t a i l e d stu d y of p o l a r o p e r a t i o n s , a n d , u s i n g t h e r e s u l t s o f t h e i r i n v e s t i g a t i o n s , d e r iv e d t h e f o l l o v a n g fo rm u la e

-I f f i s a form d epending on n v e c to r ^ e a c h o f m v a r i a b l e s , where ^ th e n

f

=

... (12)

where A r e p r e s e n t s an ag£p?egate of p o l a r o p e r a t i o n s , and ÿ i s a form d e p e n d in g on j u s t m v e c t o r s .

I f f i s a form d e p e n d in g on m v e c t o r s , e a c h o f m v a r i a b l e s , t h e n

f =... ... (13)

where t h e r e p r e s e n t a g g r e g a te s of p o l a r o p e r a t i o n s , and each of t h e form s 4^' depends on only m - 1 v e c t o r s , w h ile J) i s t h e d e te r m in a n t whose columns a r e ttie v e c t o r s on w h ich f dep e n d s .

The l a t t e r r e s u l t i s a d i r e c t g e n e r a l i z a t i o n o f t h e G leb sc n -G o rd an s e r i e s (4 ) f o r double b in a r y f o r m s , and th e two r e s u l t s ta k e n t o g e t h e r a r e a s t r o n g e r form o f C le b s c h * s th e o re m , t h e r e d u c t i o n now b e in g t o form s i n o n ly m - 1

ij 5. The *^apelli Opérateurs

I f X, y t a r e n iit-ary column v e c t o r s t h e compound i n n e r p r o d u c t

H = (1 4 )

i s c a l l e d a C a p e l l i o p e r a t o r o f n - t h o r d e r . I f t h e p o l a r o p e r a t o r ) i s w r i t t e n a s t h e n i t may be p ro v e d t h a t

Ax K - Z .

---:

I

I

Dxt '

(15)

w h e re , i n ex p a n d in g t h e d e t e r m i n a n t , t h e p o l a r o p e r a t o r s must be k e p t i n t h e same o r d e r from l e f t to r i g h t a s t h e columns from which t h e y a r e p i c k e d ^ C a p e l l i [Ï) ; c f . T u r n b u ll [ i j p . 116 J .

The numbers 1, 2 , n - 1 added on to t h e d i a g o n a l e le m e n ts o f H i n i t s d e t e r m inant a l fo rm , so c a u s in g t h e d i f f e r e n c e betw een t h i s e x p r e s s i o n and t h e o r d i n a r y d e t e r ­ m in a n t a l form f o r a compound i n n e r p r o d u c t , a r e i n s e r t e d t o

com pensate f o r th e f a c t t h a t th e p o l a r o p e r a t o r s a p p e a r i n g i n t h e d e t e r m i n a n t H a c t n o t o n ly on t h e v a r i a b l e s a p p e a r i n g i n some g iv e n fo rm , f , b u t a l s o on the v a r i a b l e s i n t r o d u c e d by t : ie p o l a r o p e r a t o r s i n l a t e r columns o f H. I n o r d e r to p ro v e t h e e q u iv a le n c e o f t h e e x p r e s s i o n s (14) and (1 5 ) f o r H, I

i n t r o d u c e o p e r a t o r s of t h e type

1 5 .

p o l a r i s a t i o n a s r e g a r d s t h e v a r i a b l e s a p p e a r i n g i n some g i v e n form f , b u t t r e a t i n g a s c o n s t a n t s any v a r i a b l e s i n t r o d u c e d i n t o f by o t h e r p o l a r i z a t i o n s . T his m eans, f o r ex am p le, t h a t

l)

i n s t e a d o f t h e c o r r e s p o n d in g r e s u l t u s i n g o r d i n a r y p o l a r i z a ­ t i o n s , namely

-IVien, by t h e theorem on t n e e x p r e s s i o n o f an o r d i n a r y compound i n n e r p r o d u c t a s a d e te r m .in a n t,

d: . D i E(J

DL, AV

4: C o n s id e r th e r - r o w e d d e te r m in a n t

Dly

---; i

(1 6)

i n t h e

Diz byi

- o p e r a t o r s . T his d e te r m in a n t may be expanded a s

where th e sequence x ^ y ' . . . ^ z ' i s o b t a i n e d by p e r m u tin g th e sequence x ^ y . . . , z and t h e suTomation i s o v e r a l l p o s s i b l e p e r m u t a t i o n s , even ones g e t t i n g a p o s i t i v e s i g n and odd ones a n e g a t i v e s i g n . I n t h i s d e te r m in a n t l e t t h e f i r s t column be r e p l a c e d by the column o f o r d i n a r y p o l a r i z a t i o n s

0 ^2^* l ‘hus i n t h e e x p a n s io n o f t h i s m o d i f i e d d e te rriiin a n t t h e d i f f e r e n t i a t i o n s w i t h r e s p e c t to x a r e

I

0,

6

:

a 'zz

= ...

yz

where c ^ y --- a r e iironecK er d e l t a s . S in c e i n d e t e r ­ m in a n t a l suioiriation o ver t h e p e r m u ta tio n s o f x y . . . z an in te r c h 'in '^ e o f x'' and ^ o r x^ and 'z/ , e t c . , s im p ly changes s i g n , t h e e x p r e s s i o n on t h e r i g h t hand s i d e o f t h e l a s t e q u a ti o n may be w r i t t e n a s

a'z'

Hence

1

o ; ,—

f e y - -₁

-" f e y

(

—

f e z

j

f e z - - ' - f e z

DU D U

I

f e .

-f e r

---1

' f e l

- fey₁

f e z

i

f e z - ' - f e z

O % y

--O %%

Z-2, a

6», fez

i n w ords, i n a D - d e t e r m i n a n t of o r d e r r t h e P - o p e r a t o r s i n t h e f i r s t column may be r e p l a c e d by o r d i n a r y p o l a r s , i . e . by D- o p e r a t o r s , p r o v id e d t h a t th e number r - 1 i s added t o t h e P - o p e r a t o r w i l l e q u a l s u f f i x e s .

Hut i t i s c l e a r from t h e p r o o f g iv e n h e r e t h a t th e same r e p la c e m e n t co u ld be c a r r i e d o u t i n any d e t e r m i n a n t whose f i r s t coiurm c o n s i s t s o f P - o p e r a t o r s even i f t h e o t h e r

cL

....

a .

_4.

i

/#

j

4 /

₁

1

Pty

)

i

1(

1

D

f'K'-V-

P

T

₁

i

i

i

Pyl"---

k t i

4;

L 4%' "

^

p;

1 7 .

( 1 7 1

The e q u a l i t y of t h e e x p r e s s i o n s (14) and (15) f o r H may now be p r o v e d by i n d u c t i o n , assum ing t h i s e q u a l i t y t o h o l d f o r a

compound i n n e r p r o d u c t of o r d e r n - 1 and t h e c o r r e s p o n d i n g ( n - 1) rowed D - d e t e r m i n a n t . 'Inen e x p a n d in g t h e D - d e t e r m i n a n t (1C) i n te r m s of i t s f i r s t column and c o f a c t o r s , and r e p l a c i n g t h e s e c o f a c to r s by th e a p p r o p r i a t e D - d e t e r m i n a n t ii i s s e e n to be e q u a l to th e l e f t hand s i d e of ( 1 7 ) , an d t h e r e s u l t f o l l o w s a t once.

I t may be see n from t h e e x p r e s s i o n (14) f o r H a s a com­ pound i n n e r jproduct t h a t i f t h e n H v a n i s h e s i d e n t i c a l l y , w h ile f o r k ^

H ^ t ) S l ~

w here H i s t h e C ay ley o p e r a t o r

i 'h ,

---^

2

.

-i f

à

6. The F i r s t M e rte n s -C a p e lli R e d u c tio n Theorem

Let f be a form depending on n m -ary column v e c t o r s X, y • • • t , and l e t H be th e C a p e lli o p e r a t o r

I f H e x p re sse d as a d e te r m in a n t i s f u l l y expanded, t h e l e a d i n g term w i l l be

----t h e e f f e c ----t of w hich on f i s sim ply ----t o m u l ----t i p l y i ----t by a n o n -z ero c o n s t a n t , p ro v id e d t h a t f a c t u a l l y c o n t a i n s t . I n t h e o t h e r te r m s of th e e x p a n s io n o f H, t h e f a c t o r s c o n t a i n i n g p o l a r

o p e r a t o r s w ith e q u a l s u f f i x e s , i . e . I

, may be r e p la c e d by n u m e ric a l f a c t o r s , and t h e r e s u l t i n g o p e r a t o r s w i l l always have as l a s t f a c t o r t o t h e r i g h t a p o l a r o p e r a t o r s e l e c t e d from above t h e p r i n c i p a l

d i a g o n a l of H. The e f f e c t of such a f a c t o r i s t o i n c r e a s e t h e d e g r e e o f f i n one of th e s e t s x, y . . . z , t , w h ile d e c r e a s i n g i t s d e g r e e in a s e t o c c u r r in g l a t e r i n t h i s

s e q u e n c e . A f or m d e r iv e d in t h i s way from f w i l l be c a l l e d * a form o f low er r a n k th a n f . Hence Hf i s e q u a l t o a n o n -z e ro m u l t i p l e of f added t o an a g g re g a te of p o la r s of forms of

1 9 .

s e t s of v a r i a b l e s o n ly .

r e p e a t e d a p p l i c a t i o n of t h i s r e d u c t i o n , f may be ejcp ressed as an a g g r e g a te o f p o l a r s o f form s i n v o l v i n g n - 2 s e t s o f v a r i a b l e s o n ly , th e n n - 3 s e t s o f v a r i a b l e s o n ly , and f i n a l l y m s e t s of v a r i a b l e s o n ly . f i l l s i s t h e f i r s t r e s u l t s t a t e d above, namely e q u a tio n (16) o f â 4 .

7, f h e becond M e r t e n s - C a p e l l i x(eduction theorem

L e t f be a form depending on m m-ary column v e c t o r s X, y . . . t , and l e t H be t h e c o r r e s p o n d in g C a p e l l i o p e r a t o r ,

e q u a l i n t h i s c a se to (x y . . . t ) l 2 .

Hf a (x y — t ) _ d f = k f + a g g r e g a t e of p o l a r s o f fo rm s o f lo w e r r a n k t n a n f , where k i s a n u m e ric a l c o n s t a n t , non­ z e ro i f f a c t u a l l y c o n t a i n s v a r i a b l e s o f t h e s e t t ; t h i s f o l l o w s by t h e argument u se d i n s 6. R e a r r a n g in g t h i s

e q u a t i o n

-f = (xy— t)J % -f 4" a g g r e g a te o -f p o l a r s o -f -fo rm s o -f lo w e r

r a n k t h a n f (18)

i f f ^ i s one o f t h e forms o f low er r a n k , t h e n s i m i l a r l y

f ^ = (x y . . . t ) - O - f ^ +■ a g g r e g a te o f p o l a r s of form s o f low er ra n k th a n f ^ (and so o f lo w e r ra n k t h a n f ) . i t may be e a s i l y v e r i f i e d t h a t a C a p e l l i o p e r a t o r coimiiutes w ith any o f t h e p o l a r o p e r a t o r s a p p e a r i n g i n i t s de term i n a n t a l e :{ p r e s s io n , -and so i f i s any p r o d u c t o f su c h p o l a r

A t , = + a g îs re g a te o f p o l a r s of form s

o f lower r a n k t h a n f ^ (1 9)

A ll t h e p o l a r i z a t i o n s on t h e r i g h t hand s i d e o f (1 8 ) w i l l commute w itn H, and so a p p l y i n g (1 9) t o a l l t h e form s i n t r o ­ duced i n ( 1 8 ) , an d c o l l e c t i n g t o g e t h e r a l l th e form s from w hich (x y t ) may be e x t r a c t e d a s a common f a c t o r , i t f o l l o w s t h a t

^ ^ -- t f j

where f ^ i s o f lo w er t o t a l d e g r e e t h a n f , a s may be s e e n from c o n s i d e r a t i o n s o f h o m o g en eity , w h ile i s an a g g r e g a t e of p o l a r s o f fo rm s of lo w e r r a n k th a n t h o s e on th e r i g h t hand s i d e o f ( 1 8 ) . R e p e a te d r e d u c t i o n o f t h i s t y p e g i v e s

f ^

where i s an a g g r e g a t e o f p o l a r o p e r a t i o n s , A i s in d e ­ p e n d e n t o f t h e s e t of v a r i a b l e s t a n d D z (x y . . . t ) .

fh e form f ^ may be t a c k l e d i n t h e same way as f h a s b ee n , and so s t e p by s t e p t h e e x p r e s s i o n (13) of f * s a s e r i e s o f

powers of D i s b u i l t up

Comparison o f t h i s r e s u l t w ith C le b s c h * s th eo re m s u g g e s t s t h a t th e s e r i e s g i v e n by M ertôns and C a p e l l i d o es n o t t e l l t h e whole s t o r y , f o r a s i t s t a n d s t h e r e i s no e x p l i c i t m e n tio n o f

compound v a r i a b l e s of any c l a s s e x c e p t t h e But t h e method o f C a p e l l i o p e r a t o r s may be c a r r i e d s t i l l f u r t h e r .

I f f i s now a form i n t h e n s e t s o f v a r i a b l e s x , y , . . . t , w ith

n < Jn^

t i f B - ' 1

2 1 .

T h is e q u a t i o n i s d e r i v e d i n th e same way a s b e f o r e , by con­ s i d e r i n g t h e de term i n a n t a l form of H. I f f i s e x p r e s s e d s y m b o l i c a l l y , th e n t h e l e f t hand s i d e o f (2 0 ) i s a sum o f form s e a c h o f w hich c o n t a i n s as a f a c t o r a compound i n n e r p r o d u c t o f n-tK o r d e r , ( x y tlA) ^ # h e re A i s some n-colum ned m a t r i x o f sy m b o lic c o e f f i c i e n t v e c t o r s . The argum ent u s e d t o

o b t a i n th e se c o n d M e r t t o s - C a p e l l i r e d u c t i o n th eo re m g i v e s i n t h i s c a s e

Where e a c h o f t h e form s i s an a g g r e g a t e o f p o l a r s o f form s d e p e n d in g on n - 1 s e t s of v a r i a b l e s , each i s a p r o d u c t o f i compound i n n e r p r o d u c ts o f K-CL o r d e r , an d t h e sum iaation, f o r each i , i s t a k e n over a l l s e t s o f i compound i n n e r p r o d u c t s each fo rm e d from t h e v e c t o r s x , y , • • . t and some s e t o f a c o e f f i c i e n t v e c t o r s .

Combining now a l l t h e s e r e s u l t s , f o r n. g r e a t e r t h a n , e q u a l to and l e s s t h a n , an e x p a n s io n i s o b t a i n e d o f an a r b i t r a r y fo rm , s y m b o l ic a l ly e x p r e s s e d , i n te r m s o f p o l a r s o f t h e form s i n t h e m v e c t o r s x^y, . . . t^

( x y — • z t j - z / JJl ^

i

( 2 1 )

w here each o f t h e p r o d u c ts JJ i s t a k e n o v e r a number o f cont- pound i n n e r p r o d u c ts o f t h e o r d e r i n d i c a t e d , a l l fo rm e d fro m t h e same s e t o f v a r i a b l e v e c t o r s , b u t p o s s i b l y fro m d i f f e r e n t

a r e now p r o d u c ts o f o r d i n a r y p o l a r o p e r a t i o n s , and n o t comr- pound p o l a r s a s i n ^ 5» and th e components of t h e compound v e c t o r o f r-tL c l a s s i n v o l v e d i n th e e x p a n s io n i n te rm s o f t h e fo rm s (2 1 ) a r e t h e minors o f o rd e r r of t h e m a t r i x whose

columns a r e t h e f i r s t r o f t h e s e t o f m v e c t o r s x , y ^ . , t .

é 80 C e r t a i n C e r i e s E x p an sio n s and t h e A p p l i c a t i o n o f t h e R e d u c tio n Theorems

As t h e r e s u l t a t t h e end of th e l a s t s e c t i o n shows, th e a l g e b r a i c i d e a s in v o lv e d a r e m o re th a n can be c o m f o r ta b ly

d e a l t w i t h i n t h i s n o t a t i o n . But b e f o r e p r o c e e d in g t o d i s c u s s t h e a p p l i c a t i o n o f s u b s t i t u t i o n a l a n a l y s i s t o th e p ro b lem o f s e r i e s e x p a n s io n , and th e i n t r o d u c t i o n o f a more

a b b r e v i a t e d n o t a t i o n , I s h o u l d l i k e t o re m a rk on c e r t a i n s p e c i a l s e r i e s e x p a n s io n s , and on th e a p p l i c a t i o n o f t h e g e n e r a l r e s u l t s t o i n v a r i a n t t h e o r y .

G e n e r a l i z a t i o n s o f t h e G lebsch-G ordan s e r i e s (4 ) were o b t a i n e d in d e p e n d e n tly by Godt [ij and W aelsh [ij - t h e s e a r e e x p a n s io n s i n s e r i e s o f fo rm s i n more t h a n two b i n a r y s e t s o f v a r i a b l e s . D eru y ts [l] and P e t r [l] b o t h c o n s i d e r e d s e r i e s e x p a n s io n s of form s i n n s e t s of m v a r i a b l e s e a c h , b u t t h e p ro b lem o f f i n d i n g t h e c o e f f i c i e n t was l e f t i n r a t h e r a

d i f f i c u l t um aanageable s t a t e . - An a c c o u n t o f a l l t h e s e s e r i e s i s g iv e n i n th e t h e s i s o f I r o s t [l] .

2 3 .

CO ) i s of s p e c i a l i n t e r e s t on acco u n t o f t h e mode o f d e r i v a ­ t i o n , I’lie n s e t s of v a r i a b l e s a re t r a n s f o r m e d c o g r e d i e n t l y by a m a t r i x whose columns a r e th e n - a r y v e c t o r s

----sa;/, fh e c o e f f i c i e n t s o f t h e t r a n s f o r m a t i o n a r e t h e n s t r i p p e d o f f by o p e r a t i n g on th e c o e f f i c i e n t s of t h e t r a n s ­ fo rm ed f w ith co m p o site p o l a r o p e r a t o r s

-fh e r e s u l t i n g f o r m s , o b t a i n e d f o r d i f f e r e n t c h o i c e s o f t h e numbers k a r e t h e "Normal fo rm s " , and f may be

e x p r e s s e d l i n e a r l y i n te rm s o f t le i r p o l a r s , T h is r e l a t i o n between c o m p o site d i f f e r e n t i a l o p e r a t o r s and t h e form s i n te r m s o f wtiich s e r i e s e x p a n s io n s may be o b t a i n e d may be s e e n i n t h e work of D e ru y ts q u o te d , and in d e e d i t l i e s a t t h e

h e a r t o f t h e r e p e a t e d u s e o f C a p e lli o p e r a t o r s i n th e d e r i v a ­ t i o n of ( 1 2 ) , (13) and t h e expansion i n te r m s o f t h e fo rm s

( 2 1 ) , I s h a l l show l a t e r (^ 30) how t h e r e l a t i o n s h i p may be e x p r e s s e d i n a sim p le and s t r a i g h t f o r w a r d way by em ploying t h e i d e a o f i n v a r i a n t m a t r i c e s and t n e i r t r a n s f o r m a b l e s e t s .

Any i n v a r i a n t o r c o v a r i a n t o f t h e fo rm s o f t h e ty p e dis­

a p p e a r i n g i n t h e e x p a n s io n s of t y p e (1 3 ) of t h e g iv e n g ro u n d fo rm s i s t h u s a n i n v a r i a n t o r c o v a r i a n t o f t h e s e g ro u n d fo r m s . And c o n v e r s e l y any i n v a r i a n t o f t h e g iv e n g ro u n d f o r m s , b e in g a f u n c t i o n of t h e c o e f f i c i e n t s o f t h e s e f o r m s , i s a l s o a

f u n c t i o n o f t h e c o e f f i c i e n t s o f t h e fo rm s of t h e t y p e ÿr i n t h e e x p a n s io n s o f ty p e (13 )* and so i s an i n v a r i a n t o f t h e l a t t e r fo rm s . A co v a r i a n t o f t h e g ro u n d f o r m s , m o re o v e r, may i t s e l f be expanded i n a s e r i e s o f ty p e (1 3 ) i n te r m s o f form s w h ich , b e in g d e r i v e d from i t by p o l a r i z a t i o n s a n d t h e u s e o f t h e C a y le y o p e r a t o r , a r e a l s o c o v a r i a n t s . f h e co v a r i a n t i s t h u s e x p r e s s e d a s an a g g r e g a t e o f p o l a r s o f c o v a r i a n t s

m u l t i p l i e d by pow ers of t h e d e t e r m i n a n t D, b u t s i n c e t h e l a s t m e n tio n e d c o v a r i a n t s depend on m - 1 s e t s o f v a r i a b l e s o n l y ,

th e y may be t a k e n t o be co v a r i a n t s of t h e fo rm s o f t h e ty p e cJPj

i n t h e e x p a n s io n s of t h e g ro u n d f o r m s . Hence i n t h e problem o f d i s c u s s i n g a l l i n v a r i a n t s and c o v a r i a n t s o f a number o f g ro u n d form s i n a n a r b i t r a r y number o f s e t s o f v a r i a b l e s , i t * i s p o s s i b l e t o r e p l a c e t h e s e form s by a number o f form s i n

m - 1 s e t s o f v a r i a b l e s , and t h e n a l l i n v a r i a n t s o f t h e g iv e n form s a r e i n v a r i a n t s o f t h e new o n e s , w h ile a l l co v a r i a n t s may

be d e r i v e d by p o l a r i z a t i o n from co v a r i a n t s of t h e new s e t .

T h is r e d u c t i o n g r e a t l y s i m p l i f i e s t h e p r o o f s o f t h e fu n d a m e n ta l th e o re m s o f i n v a r i a n t s (Weyl [ l ] ) . Of c o u r s e t h e f i r s t may be p r o v e d e a s i l y w ith o u t t h i s p r e l i m i n a r y r e d u c t i o n ( T u r n b u ll

25*

CliÀFSim I I - STANDiU^D FORMS

§ 9 , Young T ab lea u x and Rouble Forms

The m a n i p u l a t i o n o f a l g e b r a i c form s i s v ery much f a c i l i ­ t a t e d by t h e i n t r o d u c t i o n o f s t a n d a r d fo rm s ^ T u r n b u ll [l] ,

c h a p t e r X X lIlJ and t h e a p p l i c a t i o n o f c e r t a i n r e s u l t s o f

S u b s t i t u t i o n a l A n a ly s is ( Young[l] ; i n p a r t i c u l a r t h e p a p e r i n P r o c . L. M. S. v o l . 33» R u t h e r f o r d [ i ] ) . For b o th o f t h e s e

s e t s o f r e s u l t s t h e i d e a o f t h e ïo ung t a b l e a u i s f u n d a m e n ta l. T ab leau x a r e form ed from n sym bols, vjhich may or may n o t be a l l d i s t i n c t . The symbols a r e a r r a n g e d i n ro w s, X, i n t h e f i r s t , ^ 2 i n t h e se c o n d , e t c . where ,

and [)i)= ( \ , \ ...) i s a p a r t i t i o n o f t h e number n. I f a s t a n d a r d o r d e r o f t h e symbols u sed i s d e f i n e d , th e n a t a b l e a u i s c a l l e d s t a n d a r d i f i t c o n t a i n s no r e p e a t s o f symbols i n any one coluiiin, and h a s t h e symbols o t h e r .vise i n s t a n d a r d o r d e r r e a d i n g down th e columns and alo n g t h e row s fro m l e f t to r i g h t . For example i f x , y , z a r e t h r e e d i s t i n c t sy m b o ls, a l p h a b e t i c a l o r d e r b e in g s t a n d a r d t h e n x x y and x x z a r e b o t h s t a n d a r d ;

y z y y

v/hereas x y x i s n o n - s t a n d a r d , b e c a u se x and y a r e o u t of y z

o r d e r i n t h e f i r s t row, and x y y i s n o n - s t a n d a r d b e c a u s e i t X z

c o n t a i n s x tw ic e i n one coluran.

L e t u ( ^ ) , u ( ^ ^ , u ( ^ ) . . . be a s e t o f row v e c t o r s , and x ( ^ ) , x ( ^ ) . . . a s e t o f column v e c t o r s , ea c h v e c t o r h a v in g m

-2 7 .

v e c t o r s ?are each a r r a n g e d i n t a b l e a u x of t h e same s h a p e , ( ^ )» s a y and r e s p e c t i v e l y . I’hen i f t h e sym bols o f t h e

t h e i- tL column o f ^ a r e u^*"^ , and t h o s e of

i-tk column o f X‘" x ‘% x ' f . . . , x f ^ , t h e d o u b le fo rm -^1/'” / i s d e f in e d a s

J J ( - - x ' " ^ )

(22)

t n e p r o d u c t b e in g e x te n d e d o v er a l l columns o f t h e t a b l e a u o f shape (X )• I f L/ and A a r e b o tn s t a n d a r d , t h e n

{ } i s a d o u b le s t a n d a r d form . I n T u r n b u ll [ij ,

chap. X X III, t h e s e form s a r e i n t r o d u c e d i n a n o t a t i o n w i t h o u t t h e c u r l y b r a c k e t s ; I have i n t r o d u c e d them to a v o id c o n f u s i o n when a number o f t h e form s a p p e a r c l o s e t o g e t h e r .

The fu n d a m e n ta l theorem on t h e s e d o u b le form s s t a t e s t h a t t h e d o u b le s t a n d a r d fo rm s , f o r d i f f e r e n t t a b l e a u x and

d i f f e r e n t a r ra n g e m e n ts o f t h e symbols w i t h i n t h e t a b l e a u x , a r e l i n e a r l y in d e p e n d e n t, w h ile n o n - s ta n d a r d dou b le fo rm s may be e x p r e s s e d l i n e a r l y i n te rm s of t h e s t a n d a r d o n e s . ( o f .

i 'u r n b u l l [ l ] , p . 357.)

The f i r s t p a r t o f t h i s theoreia may be p ro v e d by a s e r i e s o f i n d u c t i v e a rg u m e n ts, s t a r t i n g v /ith t h e BBSumption t h a t a l i n e a r r e l a t i o n does e x i s t between t h e s t a n d a r d d o u b le form s i n a g iv e n s e t o f x-sym bols and u -sy m b o ls. A t a b l e a u o f shape ( X ) w i l l be s a i d t o p r e c e d e a t a b l e a u of sh a p e (/*. ) i f

tn e f i r s t row of t h e t a b l e a u of sh ap e (>. ) which i s n o t o f t h e same l e n g t h a s t h e c o r re s p o n d in g row o f t h e t a b l e a u o f sh ap e

d e f i n e d i s a s i n g l e rowed t a b l e a u , an d t h e r e i s j u s t one

d o u b le s t a n d a r d form c o r r e s p o n d in g to t h i s s h a p e . I f a l l t h e u - v a r i a b l e s a r e p u t eq u a l t o one a n o th e r and a l l t h e x -sy m b o ls

e q u a l t o one a n o th e r t h e n a l l th e d o u b le form s v a n i s h e x c e p t t h i s one c o r r e s p o n d in g to t h e s i n g l e row ed t a b l e a u . And so i n t h e assumed l i n e a r r e l a t i o n t h i s p a r t i c u l a r form must have a ze ro c o e f f i c i e n t . ouppose t h a t th e c o e f f i c i e n t s o f a l l t h e fo rm s c o r r e s p o n d in g t o t a b l e a u x f o l l o w i n g t h e s ’aape ( X )

v a n i s h ; t h e th eo re m w i l l be p ro v e d i f i t can be shown t h a t t h e c o e f f i c i e n t s o f t h e form s c o r r e s p o n d in g to t h e sh a p e ( X ) a l s o v a n i s h .

‘I'o p ro v e t h i s su p p o se t h a t a l l t h e double s t a n d a r d fo rm s now a p p e a r in g i n th e assumed l i n e a r r e l a t i o n ( i . e . fo rm s

c o r r e s p o n d in g t o shape ( X ) &nd e a r l i e r s h a p e s , by h y p o t h e s i s ) a r e expanded i n te rm s o f compound v a r i a b l e s . T his am ounts to w r i t i n g each su c h fo rm , sa y { } sls a sum of p r o d u c t s s u c h as ^

I S

5 I

where 8 i s a t a b l e a u o f sh ap e (yw, ) w i t h n u m e ric a l sym bols, which a r e a t t a c h e d to t h e sy m b o ls

I

o f L/ and A as s u f f i x e s ; e . g .

0) ..(x)i

i s t o be i n t e r p r e t e d as

I n t h e c a s e o f form s c o r r e s p o n d in g to t h e sh a p e ( X ) e a c h e x p a n s io n , s a y o f

{

I

, s t a r t s o f f w ith t h e te rm

29.

wnose ro\Y consiotG o f t h e nuiaber i r e p e a t e d Xi t i m e s ; b u t t h e s e t o f symbols a p p e a r in g i n ca n n o t a p p e a r i n t h e e x p a n s io n o f any form c o r r e s p o n d in g t o a s h a p e p r e c e d i n g ( X ) 5

f o r i f a t a b l e a u of su c h a shape were c o n s t r u c t e d from t h e symbols o f 4 i t WDuld c o n t a i n a t l e a s t one column w i t h a r e p e a t e d sym bol, and any d o u b le fo rm i n v o l v i n g t h i s t a b l e a u would v a n i s h . The assumed l i n e a r r e l a t i o n betw een t h e d o u b le s t a n d a r d form s i n t h e g iv e n u- and x - v a r i a b l e s must h o ld

s e p a r a t e l y f o r each d i s t i n c t s e t o f s u f f i x e s a p p e a r i n g i n t a b l e a u x l i k e 6 , and so i n p a r t i c u l a r i t m ust g iv e a l i n e a r r e l a t i o n betw een t h e p r o d u c ts

{uri 5j{s,ixp

I and Ay o f sh ap e ( X )» w i t h t h e same c o e f f i c i e n t s a s th e c o r r e s p o n d i n g fo rm s

{ Xj^^} o r i g i n a l l y h ad . I n f a c t a r e l a t i o n of t h e form

... , , 3,

m ust h o l d ; to com plete t h e i n d u c t i o n i t m ust be shown t h a t a l l t h e c^j v a n i s h .

A se c o n d i n d u c t i o n c o m p le te s th e p r o o f . Suppose t h a t t h e r e i s a l i n e a r r e l a t i o n between th e d o u b le form s

where th e sym bols i n t h e r i g h t hand t a b l e a u a r e x ( ^ ) , x ( ^ ) . . . x ( ^ ) , b u t t n a t no such l i n e a r r e l a t i o n e x i s t s betw een t h e

sai-e d e g re e a s i n t h e form s {3^ | Xj } $ and \7b.ose l e f t hand h a l v e s a r e t a b l e a u x i n w h i c h ,t h e h tL row c o n s i s t s e n t i r e l y o f

^ * H'or N = 1 t h e r e e x i s t s o n ly one d o u b le s t a n d a r d fo rm ,

e q u a l t o a power o f , and so t h e p r o p e r t y o f l i n e a r

in d e p e n d e n c e i s o b v io u s . zLxpand each o f t h e fo rm s { X l X j J

a s a p o ly n o ia i^ a l i n t n e e le m e n ts of x^^^^ and p ic k o u t from a l l t h e e x p a n s io n s t h e rjroduct o f t h e s e e le ia e n ts f o r w hich t h e sum

o f t h e s u f f i x e s i s a maximum ; suppose t h a t t h i s te rm o c c u r s o n ly i n t h e e x p a n s io n s of f Xl Xf> 3, j {5^IXy.j,

I'hen i t s c o f a c t o r i n each o f t h e s e exx^ansions i s o b t a i n e d by d e l e t i n g from t h e d o u b le form i n q u e s t i o n t h e symbol x^^ )

w h e r e v e r i t o c c u r s , and a l s o t h e sym bols i n t h e c o r r e s p o n d i n g p o s i t i o n s o f 3^ . ( t h i s would n o t n e c e s s a r i l y be t r u e f o r a p r o d u c t o f t h e x^^) w ith s m a l l e r s u f f i x sum

J

. The l i n e a r r e l a t i o n betw een t h e ^ 5, I Xj^J g i v e s r i s e t o a l i n e a r

r e l a t i o n betv/een t h e s e c o f a c t o r s , w hich a r e , h ow ever, th e

d o u b le form s i n x ^ , x ^ ^ ) . . . x ^ ^ ^ ^ v/hose l i n e a r in d e p e n d e n c e h a s b e e n assumed as t h e h y p o th e s is of t h e i n d u c t i o n . Hence t h e fo rm s { 5^ | X;*^} * and s i m i l a r l y t h e fo rm s '[ 5 ,} a r e l i n e a r l y in d e p e n d e n t.

T h is g i v e s f i r s t t h a t i n (25)

c

f o r a l l j ; and t h e n t h a t ~ G f o r a l l i and j .

co-3 1.

e f f i c i e n t s v a n i s h , and t h e f i r s t p a r t o f t h e th e o re m i s

p ro v e d .

To p ro v e t h e second p a r t of t h e th e o r e m , l e t IX }

be an ai‘bitrax*y double form w ith t a b l e a u x o f sh ap e (X ) con­ s t r u c t e d fro m a c e r t a i n s e t of u -sym bols and a c e r t a i n s e t o f X -sym bols. The symbols i n each column o f b o th t a b l e a u x may be made to ap p e ar i n s t a n d a r d o r d e r , r e a d i n g fro m to p t o

b ottom , b y , a t m ost, a change o f s i g n . Assuming t h a t t h i s

X (à)

i n t u r n u n t i l t h e f i r s t symbol i s r e a c h e d which i s o u t o f s t a n d a r d o r d e r ; t h a t i s , t h e f i r s t symbol x ^ ^ ) which i s im m e d ia te ly f o l l o w e d i n t h e same row by x^ , v /ith ù< J . The p a r t o f t h e p r o d u c t e x p a n s io n (22) o f ? which i n v o l v e s t h e o f f e n d i n g sym bols may be w r i t t e n a s

wnere i s t h e m a tr ix o f column v e c t o r s a p p e a r i n g above x^*^) i n t h e t a b l e a u X^^ , Xg i s t h e m a t r i x c o n s i s t i n g of x ^ a n d t h e v e c t o r s below i t , X^ i s t h e m a t r i x c o n s i s t i n g o f x^^ and t h e v e c t o r s above i t and i s t h e m a t r i x o f v e c t o r s below x^"^), w h ile 1% , U , Uj , 14 a r e m a t r i c e s whose rows a r e t h e

I f(^}

c o r r e s p o n d in g s e t s o f v e c t o r s s e l e c t e d from L/

The compound i n n e r p r o d u c t ( U I K ^ z ) i s t h e d e t e r ­ m in an t of t h e m a tr ix p r o d u c t

usin^;, t h e n o t a t i o n o f p fU ? titio n e d m a t r i c e s . o i m i l a r l y

( h

1

Xj x ^ )

-now th e determ inant

14 X,

u ,x .

V, X, M X, *

l/,X, KX, •

14 X, _{(4 X*} 14 X, <

4

x, (

4

X«.

(2 4 )

T h is i s t h e d e te r m in a n t of t h e m a t r i x prod.nct

u.

f X, X, X, ' -}

L '

X. X, x J

I'he v a lu e o f t h i s p ro d u c t i s u n a l t e r e d i f t h e c o n fo rm a b ly p a r t i t i o n e d r e c i p r o c a l m a t r i c e s and j J , i n

t h a t o r d e r , a r e i n s e r t e d betw een th e m a t r i x f a c t o r s

-"i/, •' p , X. X,- 1 _ u • I/, • L ‘ X, X, X J ^ •

• M • ^

L4.

~u, q

—

• 4 _■ ^0.

[ ■ ' «. «. ' S < » )

And so t h e d e t e r u i n a n t (24) i s a l s o e q u a l t o t h e d e t e r m i n a n t of t h e m a t r i x p r o d u c t ( 2 5 ) . xuvaluate t h e d e t e r m i n a n t (2 4 ) i n two ways, f i r s t by L a p la c ia n ex^; ans io n i n te rr a s o f m in o rs s e l e c t e d fro m t h e rows d e f i n e d by U, and , and t h e i r co- f a c t o r s , and second by w r i t i n g down th e d e t e r m i n a n t o f t h e p r o d u c t o f r e c t a n g u l a r m a t r i c e s (25) u s i n g t h e B in e t-C s u c h y

t ieorem. The f i r s t method of e x p a n s io n s im p ly g i v e s