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UBC Theses and Dissertations

MacDonald characters of Weyl groups of rank ≤4 1973

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MACDONALD CHARACTERS OF WEYL GROUPS OF. RANK <4 by AGNES ANDREASSIAN B . S. ,M. S . , M . A . , A m e r i c a n U n i v e r s i t y o f B e i r u t , 1968 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e d e p a r t m e n t o f MATHEMATICS V7e a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may he granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MoiMruL^AAi- cs The University of British Columbia Vancouver 8, Canada Date R^dt 5 3 , 19 73 S u p e r v i s o r : D r . B . C h a n g ABSTRACT I n t h i s t h e s i s we o b t a i n a l l t h e o r d i n a r y i r r e d u c i b l e c h a r a c t e r s o f Weyl g r o u p s o f r a n k <_4 by u s i n g M a c D o n a l d ' s m e t h o d . T h i s method e n a b l e s us t o f i n d a l m o s t a l l t h e c h a r a c t e r s , and t h e r e m a i n i n g ones may be o b t a i n e d by c o m b i n i n g M a c D o n a l d c h a r a c t e r s and c h a r a c t e r s o f e x t e r i o r p r o d u c t s o f t h e r e f l e c t i o n r e p r e s e n t a t i o n . i i i TABLE OF CONTENTS INTRODUCTION A2 A 3 A4 D . B 2 and C 2 and C^ B 4 and C 4 BIBLIOGRAPHY 1 4 5 7 11 18 20 26 41 61 i v LIST OF TABLES Table I: Character Table for (A 2) . . . 4 Table II .-Character Table for (A 3) . . . 6 Table I I I : Character Table for (A 4) . . . 1 0 Table IV: Character Table for (D4) . . . 1 6 Table V: Character Table for (B 2)=(C 2) . . . 1 9 Table VI: Character Table for (B 3)=(C 3) . . 2 4 Table VII: Character Table for (B 4)=(C 4) . . 3 8 Table VIII: Character Table for (F.) . . 5 8 V ACKNOWLEDGMENT I would l i k e to express my gratitude to Dr. B.Chang for suggesting the topic of t h i s thesis and for a l l the help he so r e a d i l y offered during i t s preparation.I would also l i k e to thank Dr.R.Ree for reading the thesis. 1 INTRODUCTION I n a p a p e r e n t i t l e d "Some I r r e d u c i b l e R e p r e s e n t a t i o n s o f Weyl G r o u p s " [5] , I . G . M a c D o n a l d d e s c r i b e s t h e f o l l o w i n g c o n s t r u c t i o n w h i c h g i v e s m a n y , b u t i n g e n e r a l n o t a l l o f t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f a W e y l g r o u p . L e t V be a f i n i t e - d i m e n s i o n a l v e c t o r space o v e r t h e r a t i o n a l f i e l d , a n d suppose V has a p o s i t i v e - d e f i n i t e i n n e r p r o d u c t . L e t V * be t h e d u a l o f V . L e t R be a r o o t s y s t e m o f a Weyl g r o u p (R) and l e t S be a s u b s y s t e m o f R . F o r a g i v e n o r d e r i n g on R , l e t n(S) d e n o t e t h e p r o d u c t o f a l l t h e p o s i t i v e r o o t s o f S . T h e n II (S) i s . a homogeneous r a t i o n a l - v a l u e d p o l y n o m i a l f u n c t i o n on V . T h e space o f a l l r a t i o n a l - v a l u e d p o l y n o m i a l f u n c t i o n s on V i s t h e s y m m e t r i c a l g e b r a E = S y m ( V * ) . L e t t h e W e y l g r o u p (R) a c t on E , a n d l e t P(S) d e n o t e t h e s u b s p a c e o f E spanned by t h e p o l y n o m i a l f u n c t i o n s g(n ( S ) ) f o r a l l g i n ( R ) . M a c D o n a l d p r o v e s t h a t P (S ) i s an a b s o l u t e l y i r r e d u c i b l e ( R ) - m o d u l e . The p u r p o s e o f t h i s t h e s i s i s t o use t h i s c o n s t r u c t i o n t o compute t h e i r r e d u c i b l e c h a r a c t e r s o f t h e W e y l g r o u p s o f r a n k <4.We w i s h t o f i n d o u t w h i c h c h a r a c t e r s c a n be so o b t a i n e d and w h e t h e r t h e m i s s i n g c h a r a c t e r s a r e r e l a t e d t o t h o s e o b t a i n e d . T h e c h a r a c t e r s t h a t a r e o b t a i n e d by t h i s method w i l l be c a l l e d •MacDonald c h a r a c t e r s . U s i n g t h i s method we f i n d a l l t h e i r r e d u c i b l e c h a r a c t e r s i n t h e c a s e o f t h e g r o u p s (A^) .We f i n d t h a t n e i t h e r t h e subsys tems o f a l o n e n o r t h e subsys tems o f C^ a l o n e p r o v i d e a l l t h e i r r e d u c i b l e c h a r a c t e r s o f t h e g r o u p ( B ^ ) = ( C ^ ) . H o w e v e r , u s i n g t h e s u b s y s t e m s o f b o t h B. and C . we o b t a i n a l l the. ' - -• i r r e d u c i b l e c h a r a c t e r s . I n t h e c a se o f ( D . ) , t h e method g i v e s 2 e l e v e n o f t h e t h i r t e e n i r r e d u c i b l e c h a r a c t e r s . T h e e x t e r i o r p r o d u c t s o f t h e r e f l e c t i o n r e p r e s e n t a t i o n p r o v i d e t h e m i s s i n g c h a r a c t e r s . I n t h e c a s e o f (F^) we f i n d s e v e n t e e n c h a r a c t e r s . T h e r e m a i n i n g e i g h t c a n be w r i t t e n as p r o d u c t s o f two M a c D o n a l d c h a r a c t e r s . We d e s c r i b e b e l o w t h e n o t a t i o n u sed and t h e method o f o b t a i n i n g a l l t h e subsys tems o f a g i v e n r o o t s y s t e m o f a W e y l g r o u p . We use D y n k i n d i a g r a m s t o r e p r e s e n t r o o t sys tems and t h e s e t o f p o s i t i v e r o o t s i n t e r m s o f o r t h o n o r m a l b a s i s as g i v e n by N . B o u r b a k i [1] . A s i n [2] , i n l i s t i n g t h e subsys tems o f and ,we d i s t i n g u i s h be tween a d i a g r a m w i t h l o n g r o o t s and one w i t h s h o r t e r r o o t s by u s i n g S t o d e n o t e a D y n k i n d i a g r a m w i t h l o n g r o o t s and S one w i t h s h o r t e r r o o t s . W e do t h e o p p o s i t e f o r subsys tems o f C ^ . The c o n j u g a c y c l a s s e s and t h e o r d e r s o f t h e c e n t r a l i z e r s f o r (A^) , (B^) o r (C^) f o r i = 2 , 3 , 4 , a n d (D^) a r e s t a i g h t f o r w a r d t o f i n d . F o r (F^) we use t h e c o n j u g a c y c l a s s e s o b t a i n e d by R . W . C a r t e r [ 2 ] , a n d i n a l l c a s e s we use C a r t e r ' s n o t a t i o n f o r c o n j u g a c y c l a s s e s . F o r t h e sake o f c o n v e n i e n c e , w e a l s o use p e r m u t a t i o n n o t a t i o n and s i g n c h a n g e s . I n t h e t a b l e s , t h e e n t r i e s under h ^ d e n o t e t h e numbers o f e l e m e n t s i n t h e c o n j u g a c y c l a s s e s . d e n o t e s a c h a r a c t e r . F o r any r o o t a ,w d e n o t e s t h e r e f l e c t i o n d e f i n e d by a.We use t h e n o t a t i o n V ( a , , . . . , a ) t o d e n o t e t h e Vandermonde 1 ' n d e t e r m i n a n t o f o r d e r n , . n-1 a n G i v e n a r o o t s y s t e m R,we use D y n k i n 1 s method [3] t o o b t a i n a l l t h e subsys tems ,We s t a r t by a d j o i n i n g t o t h e r o o t s y s t e m R t h e m i n i m a l r o o t o f ( R ) . T h e d i a g r a m so o b t a i n e d i s c a l l e d t h e e x t e n d e d D y n k i n d i a g r a m f o r R . B y d e l e t i n g r o o t s f r o m t h e e x t e n d e d D y n k i n d i a g r a m we o b t a i n s u b s y s t e m s . R e p e a t i n g t h e p r o c e s s w i t h t h e subsys tems o b t a i n e d we e v e n t u a l l y f i n d a l l t h e subsys tems o f t h e g i v e n r o o t s y s t e m . l . A 2 . Subsystems of ; The extended Dynkin diagram for i s -(x 1-x 3) x 1 - x 2 x 2 _ x 3 From t h i s diagram we obtain the following subsystems (1) A 0: X l X2 X2 X3 (2) A, X l X2 We note that II(A2) =-V (x 1 ,x 2 ,x^) where V(x-^,x 2,x 3) i s the Vandermonde determinant of order 3. Table I: Character Table for (A 2) A, A l Conjugacy Class Representative C h a r a c t e r i s t i c Polynomial h. I \ \ $ (1) x 2-2x+l 1 1 1 2 A l (12) x 2 -1 3 1 -1 0 A, (123) x 2 + .x+1 2 1 1 -1 5 2.A3. Subsys tems o f A., The e x t e n d e d D y n k i n d i a g r a m f o r A ^ i s r (XT_-X 4) From t h i s d i a g r a m we o b t a i n t h e f o l l o w i n g s u b s y s t e m s : (1) A^ : o o o x l x2 X2 X3 X3~ X4 (2) 2A X (3) A 2 : (4) A1 P ( A ^ ) : We have X l X 2 . X 3 _ X 4 X l ~ x 2 X2 X3 X l x2 II (A^ ) =V (x^ , x 2 , x 3 , x ^ ) , t h e Vandermonde d e t e r m i n a n t o f o r d e r 4. P(2A^): I n t h i s c a s e , i t i s c o n v e n i e n t t o w r i t e t h e p o l y n o m i a l s i n t e rms o f t h e r o o t s (where a i ~ x i - x j _ + i ) r a t h e r t h a n t h e x ^ ' s . W e have II (2A^)-a-^OL^ r and so P(2A^) i s spanned b y : b 1 = a 1 a 3 , ^ 2 ~ a 2 ( a n + a 2 + a 3 ^ ' b 3=(a-^+a 2) (a 2+a 3 ) =b-^+b2 . P ( A 2 ) : The g r o u p ( A 3 ) a c t i n g on H ( A 2 ) g i v e s r i s e t o t h e f o l l o w i n g p o l y n o m i a l s : b-^V (x 1 ,x 2 ,x3) , b 2=V(x 2,x 3,x 4) , b 3=V(x 1,x 3,x 4), b 4=V(x 1,x 2,x 4). Then 1 1 1 1 1 1 1 1 x l x2 x3 x4 x 2 X2 x3 x 2 x4 shows b 1 - b 2 + b 3 - b 4 = o and {b 1,b 2,b 3} may be taken as a basis for P(A 2). Table II:Character Table for (A-,) A 3 2A X A, A l Conjugacy Class Representative C h a r a c t e r i s t i c Polynomial h. l *o Xl x 2 x 4 $ (1) x 3-3x 2+3x-l 1 1 1 2 3 3 A l (12) x 3- x 2- x+1 6 1 -1 0 -1 1 A, (123) x 3 -1 8 1 1 -1 0 0 A 3 (1234) x 3 + x 2+ x+1 6 1 -1 0 1 -1 2A X (12)(34) x 3+ x 2- x-1 3 1 1 2 -1 -1 3 .A, Subsystems of A^: The extended Dynkin diagram for A^ i s ( x r x 5 ) From t h i s diagram we obtain the following subsystems (i) A. : n _ 0 o (2) A 3: x l ~ x 2 x2 x 3 X3~ X4 X4 X 5 (3) A 2+A r X l X2 X2 X3 x3~ x4 (4) 2A±: X l X2 X2 X3 X4~ X5 (5) A, X l X2 X3 X4 ( 6 ) A, X l x2 X2~ X3 X l X2 P(A^): We have n ( A 4 ) = V ( X 1 , X 2 , X 3 , X 4 , x 5 ) . P ( A 3 ) : The ( A 4 ) - o r b i t of II ( A 3 ) consists of +b± (i=l,...,5) where b 1=V(x 1,x 2,x 3,x 4) , b 2=V(x 2,x 3,x 4,x 5) , b 3 = V ( x , x 3 , x 4 , x^) , b 4 = V ( x l ' X 2 ' X 4 , X 5 * ' b^ =V (x-^,X 2 f x 3 f x^) . As i n t h e c a s e o f P ( A 2 ) o f (A^) ,we have and { b ^ , b 2 , b 3 } forms a b a s i s f o r P ( A 3 ) P ( A 2 + A 1 ) : We have n (A2+A1)=- ( x 4 - x 5 ) V ( x 1 , x 2 , x 3 ) Hence P ( A 2 + A ^ ) i s spanned b y : b l = ( x 4 - X 5 V ( x x , x 2 r x 3 ) V ( x 3 - X 5 V ( x 1 , x 2 b 3 = ( x 2 - X 5 V ( x l f x 3 r X 4 ) b4 = X 5 V ( x 2 , x 3 ( x r X4 ) V ( x 2 , x 3 . x 5 ) V ( x 2 - X4 V ( x 1 , x 3 r X 5 ) ( x 3 - X4 V ( x 1 , x 2 r X 5 ) V ( x r X3 > V ( x 2 , x 4 r X 5 ) b9= ( x 2 - x3 ) V ( x l f x 4 ' X 5 ) ho x2 ) V ( x 3 , x 4 r X 5 ) We have b 1-b 2 +b 3-b 4=o w h i c h c a n be s een f rom t h e i d e n t i t y : X 1 " X 5 x2 X 5 X3 x 5 x 4 - x 1 • 1 1 1 X l x2 x3 x4 x 2 X l x| X3 x4 S i m i l a r l y , w e have 10 2 5 6 1_ 1_ 1_ T_ 1— 1' 2' 3' 5' 6 are e a s i l y seen to be l i n e a r l y independent, and hence form a basis for P(A 2+A^). P(2A^): Writing the polynomials i n terms of the roots a where a.=x.-x.,,,we have IT (2A1) '=a1a3 • I t i s easy to see that P(2A^) i s spanned by the following l i n e a r l y independent elements: a l a 3 ' a l a 4 ' a 2 a 4 , a2 ^ a l + a 2 + a 3 ^ ' a3 ( a 2 + a 3 + a 4 ) • P ( A 2 ) : A basis for P(A 2) i s given by the polynomials: -IT (A2)=V ( x 1 , x 2 , x 3 ) , V(x 1,x 2,x 4) , • V ( x 1 , x 2 , x 5 ) , V ( x 1 , x 3 , x 4 ) , V ( x i r x 3 ,x 5)', V ( x 1 , x 4 , x 5 ) . 10 Table I I I : Character Table for (A.) <S> A4 A3 A l A+A 2 1 2AX A 2 Conjugacy Class Representative C h a r a c t e r i s t i c Polynomial h. l *0 Xl x 2 *3 x 4 *5 (1) x'*-4x3 + 6x 2-4x+l 1 1 1 4 4 5 5 6 A l (12) x"-2x 3 +2x-l 10 1 -1 -2 2 -1 1 0 A 2 (123) x1*- x 3 - x+1 20 1 1 1 1 -1 -1 0 A3 (1234) x* -1 30 1 -1 0 0 1 -1 0 A4 (12345) x"*+ x 3 + x 2+ x+1 24 1 1 -1 -1 0 0 1 (12)(34) x" -2x 2 +1 15 1 1 0 0 1 1 -2 A 2 + A l (123) (45) xlf+ x 3 - x-1 20 1 -1 1 -1 -1 1 0 4.D 4 Subsystems o f D The extended D y n k i n d i a g r a m f o r D^ i s Q * 3 * 4 x r x 2 o- o x 0+x. x 2 - x 3 3 4 - ( x 1 + x 2 ) From t h i s d i a g r a m we f i n d t h e f o l l o w i n g subsystems: (1) D (3) A (7) A c > X3 X 4 4 : i *• v */ X 1 " X 2 X 2 " X 3 x 3 + x 4 4A 1: 0 0 0 X 1 " X 2 x l + x 2 x 3 - x 4 A 3 : o X 1 ~ X 2 X 2 " x 3 x 3 ~ x 4 A l : /-> o X l " X 2 X 2 " X 3 x 3 + x 4 A": • j '"" '" \J \j x 3 + x 4 x 2 - x 3 x 3 ~ x 4 3A X: o o o X 1 ~ X 2 X 3 ~ x4 x 3 + x 4 A 2 : o X 1 ~ X 2 X 2 " x 3 2A±: 0 0 X 1 ~ X 2 X 3 + X 4 2A£: 0 0 X 1 ~ X 2 x l + X 2 x 3 + x 4 12 (10) 2 A " : x o o x x - x 2 x 3 - x 4 (11) A , : 1 o X 1 ~ X 2 P ( D ^ ) : We have n ( D 4 ) = V ( x ? L , x | , x | , x | ) . • P ( 4 A 1 ) : A b a s i s f o r P (4A X ) c o n s i s t s o f n ( 4 A 1 ) = ( x 2 1 - x | ) ( x 2 - x 2 ) , ( x ' - x 2 ) ( x ' - x * ) . P ( A 3 ) : The ( D 4 ) - o r b i t o f ^ A ^ c o n s i s t s o f + b.^ ( i = l , , . . , 4 ) where b 1 = V ( x 1 , x 2 , x 3 , x 4 ) , b 2 = V ( x - ^ f x 2 , — x 3 , — x 4 ) , b 3 = V ( x 1 , - x 2 , x 3 , - x 4 ) , b 4 = V ( x l ' - X 2 ' ~ X 3 ' X 4 * ' A s i m p l e c a l c u l a t i o n o f d e t e r m i n a n t s shows t h a t b 1 + b 2 + b 3 - b 4 = o and b ^ , b 2 , b 3 a r e l i n e a r l y i n d e p e n d e n t . P ( A l ) : We have II ( A ^ ) = V ( x 1 , x 2 , x 3 , - x 4 ) , and so P(-& 3) i s spanned b y : b^—V (x-j^, x 2 , x 3 , —x4 ) , 4 b 2 = V (x-^ , x 2 , —x3 , x 4 ) , b 3 ~ V (x-j^, — x 2 , x 3 , x 4 ) , b 4 = V ( - x 1 , x 2 , x 3 , x 4 ) . As a b o v e , a s i m p l e c a l c u l a t i o n shows t h a t b 1 +b 2 +b 3 +b 4=0 and {b-^,b 2 , ,b 3 } forms a b a s i s f o r P ( A 3 ) . P (A 3') : We f i n d n ( A 3 ' ) = - V ( x 2 , x | , x | ) . T h i s c a s e i s a n a l o g o u s t o t h a t o f P ( A 2 ) o f ( A 3 ) , a n d we o b t a i n a b a s i s c o n s i s t i n g o f : V ( x 2 , x 2 , x 2 ) , V ( x * , x 2 , x * ) , V ( x J , x * , x | ) . p (3A^) : We have II (3A1) = ( x 1 - x 2 ) ( x 3 - x 4 ) . The e l e m e n t (12) [-1 1 - 1 l ] o f '(D 4) a c t i n g on n(3A 1) g i v e s ( x x + x 2 ) ( x 3 - x 4 ) . Then ( x 1 ~ x 2 ) ( x 3 ~ x 4 ) + ( x l + x 2 ) ( x 3 ~ x 4 ) = 2 x l ( x 3 ~ x 4 } i s i n P ( 3 A 1 ) . S i n c e P(3A 1) i s i r r e d u c i b l e , w e may t a k e t h e ( D 4 ) - o r b i t o f x 1 ( x | - x 4 ) i n s t e a d o f t h a t o f II(3A1) t o span P(3A 1 ) . W e c a n now e a s i l y w r i t e down a b a s i s as f o l l o w s : x. x n X, X. X. X. x , 2 _ 2 ) 2 2 . X 3 ~ X 4 ; ' x l " x 3 J ' 2 2 x 3 ~ x 4 ; ' X 1 ~ X 2 ) ' 2 _ 2 > x2 4 ; ' x l x 2 ; ' X 4 ^ X2 X P " P ( A 2 ) : We have H ( A 2 ) = - v ( x 1 , x 2 , x 3 ) We n o t e t h a t N ( A 2 ) i s i n P ( 3 A ^ ) a b o v e , f o r Hence P ( A 2 ) C P ( 3 A ^ ) . B u t b o t h s p a c e s a r e i r r e d u c i b l e , h e n c e P ( A 2 ) = P ( 3 A 1 ) . P ( 2 A 1 ) : We have n ( 2 A X ) = ( x 1 - x 2 ) ( x 3 + x 4 ) . W r i t i n g t h i s i n t e r m s o f t h e r o o t s o f D ^ , w h e r e a - j _ = x i - x i + i f o r i = l , 2 , 3 and a ^ x ^ + x ^ w e c a n e a s i l y f i n d a b a s i s f o r P ( 2 A 1 ) , s u c h as a l a 4 ' a 3 ^ a l + 2 a 2 + a 3 + a 4 ) i a 2 ( a 1 + a 2 + a 4 ) . P ( 2 A | ) : A b a s i s f o r P ( 2 A | ) i s g i v e n by 2 2 x l ~ x 2 ' 2 2 x l x 3 ' 2 2 X l X 4 ' P ( 2 A £ ) : We have n (2A^) = ( x 1 - x 2 ) ( x 3 - x 4 ) • A g a i n , i t i s more c o n v e n i e n t t o use t h e r a t h e r , t h a n t h e x We may t a k e a l a 3 ' a 4 ( a 2 . + 2 a 2 + a 3 + a 4 ^ ' a 2 ( a 1 + a 2 + a 3 ) as a b a s i s f o r P ( 2 A ^ ) . I n c a l c u l a t i n g t h e c h a r a c t e r s , t h e f o l l o w i n g f a c t e n a b l e s us t o s i m p l i f y our c o m p u t a t i o n . The group (D^) c o n t a i n s a c e n t r a l i n v o l u t i o n z ( t h e element c o r r e s p o n d i n g t o 4A^).Then f o r any c h a r a c t e r x and element g o f t h e group,we have X(zg)=ex(g) where X d ) F u r t h e r m o r e , e=l o r -1 a c c o r d i n g as II (S) i s o f even o r odd d e g ree. Denote by [x] an element i n t h e c o n j u g a c y c l a s s d e s i g n a t e d by X i n t h e t a b l e . T h e n we have t h e f o l l o w i n g c o n j u g a c y r e l a t i o n s : z = [ 4 A i ] , [2A|1 ~ z [ 2 A j ] , [ A j ~ Z [ 3 A j , LD4(al>] ~s[Val>] ' [A 2] ~z[D 4] , [ 2 A " ) . z [ 2 A g , [ 2 A j ~ Z [2AJ , [AJ] ~ Z [A'] , [ A 3 ] ~ Z [ A 3 ] ' [A-] -z [A'] . Using the subsystems of we obtain a l l but two of the i r r e d u c i b l e characters of (D.). Table IV: Character Table for (D„). $ D4 4 A 1 2 A1 2A^ 2A^ Conjugacy Class Representatives C h a r a c t e r i s t i c Polynomial h. X Xl x 2 x 3 x 4 X5 $ (1) [1111] x 4-4x 3+6x 2 -4x+l 1 1 1 2 3 3 3 (1) [-1-111] x^ -2x 2 +1 6 1 1 2 -1 3 -1 4 A 1 (1) [-1-1-1-1] x' t + 4x 3 + 6x 2 +4x+l 1 1 1 2 3 3 3 A l (12) [1111] x"-2x 3 +2x-l 12 1 -1 0 1 1 1 A3 ' (12) [-11-11] -1 24 1 -1 0 -1 1 -1 3 A l ) A" J (12) [-1-1-1-1] x"+2x3 -2x-l 12 1 -1 0 1 1 1 3 J A 2 (123) [ l l l l ] x"- x 3 - x+1 32 1 1 -1 0 0 0 D4 (123) [-1-1-1-1] xh+ x 3 + x+1 32 1 1 -1 0 0 0 2 A 1 (12) (34) [ l l l l ] xh -2x 2 +1 6 1 1 2 -1 -1 3 2A 1 (12) (34)[-l-lll] xh -2x 2 +1 6 1 1 2 3 -1 -1 D 4 (a x) (12) (34)[-ll-ll] <t 2 x +2x +1 12 1 1 2 -1 -1 -1 A3 (1234) [ l l l l ] <• x -1 24 1 -1 0 -1 -1 1 A3 (1234)[-l-lll] X -1 24 1 -1 0 1 -1 -1 T a b l e I V : C h a r a c t e r T a b l e f o r (D.) C o n t i n u e d . A 3 A 3 A 3 A l 3A, A 2 X 6 X 7 X 8 x 9 X 1 0 x l l X 1 2 3 3 3 4 4 6 8 -1 -1 3 0 0 -2 0 4 A 1 3 3 3 - 4 -4 6 -8 A l -1 -1 -1 2 -2 0 0 a " ' 3 1 1 -1 0 0 0 0 3 A l ) -1 -1 -1 -2 2 0 Aj J A 2 0 0 0 1 1 0 -1 D 4 0 0 0 -1 - 1 0 1 2A£ 3 -1 -1 0 0 -2 0 2 A X -1 3 -1 0 0 -2 0 -1 - 1 -1 0 0 2 0 A 3 -1 1 1 0 0 0 0 A 3 1 -1 1 0 0 0 0 5. B 2 and C 2 Subsystems of B 2: The extended Dynkin diagram for B 2 i s -JO x l * ~ x 2 x2 ( x ± + x 2 ) From t h i s diagram we obtain the following subsystems ( 1 ) B 2 : ,, (2) 2A1 (3) A x: (4) A 1 X1~ X2 X2 x r x 2 - ( x 1 + x 2 ) X l " X 2 x 2 P (B 2) : We have H (B 2)=x 1x 2(x£-x|) P (2A1) : P(2A 1) i s spanned by n(2A 1)=-(x 2-x 2) . Subsystems of C 2: The extended Dynkin diagram for C 2 i s •3 -2x 1 x l ~ x 2 2 x2 We obtain the following subsystems: (2) 2A X: (3) A 1 (4) A x: X1~"X2 2x 2 9 9 -2x^ o X l ~ x 2 2x 2 Combining the P(S) for the subsystems S of B 0 and of C 0, we obtain a l l the characters of (B 2)=(C 2). Table V: Character Table for (B 0)=(C o). B2 2A 1 A l A l $ C2 2A 1 A l A l Conjugacy Class Representative C h a r a c t e r i s t i c Polynomial h. l xo X l x 2 X 3 X4 B2 c 2 (1) [11] x 2-2x+l 1 1 1 1 1 2 A l (1) [-11] x 2 -1 2 1 -1 1 -1 0 2A 1 (1) [-1-1] x 2+2x+l 1 . 1 1 1 1 -2 A l A l (12) [11] x 2 -1 2 1 -1 -1 1 0 B2 C2 (12) [-11] x 2 +1 2 1 1 -1 -1 0 Note:The f i r s t l i n e of the table gives the subsystems of B_ and the second the subsystems of C r 20 6. and C^. Subsystems of B.,: The extended Dynkin diagram for B^ i s ? - ( x 1 + x 2 ) X1~ X2 X2~ X3 We obtain the following subsystems: (1) B 0: (2) A 3: (3) 2A 1+A 1 (4) A. (5) B, (6) A1+A1: (7) 2An (8) An (9) A- o- X l " X 2 x 1 - x 2 X l x2 x2 X3 X l x2 X l X2 x l x2 x2 X3 x2 x 3 - ( x 1 + x 2 ) 0 o x l ~ x 2 x2~ x3 o.. - -— 7 * — ® X, X 3 o -(x 1+x 2) -o - ( x 1 + x 2 ) x. X. P ( B 3 ) : We have n ( B 3 ) = ~ X l X 2 X 3 V ( x i f X 2 ' x 3 ) * P ( A 3 ) : I t i s e a s y t o see t h a t P ( ^ 3 ) i s spanned by IT ( A 3 ) = V ( x 2 , x 2 ,x 2) P ( 2 A 1 + A 1 ) : We have n ( 2 A 1 + A 1 ) = - x 3 ( x ^ - x 2 ) . Hence P(2A^+A^) i s spanned b y : x l ( x 2 ~ X 3 ) ' X 2 ^ i - 5 ^ ' i 2 _ 2 \ X 3 1 x 2 ' w h i c h a r e l i n e a r l y i n d e p e n d e n t . P ( A 2 ) : We have n ( A 2 ) = - V ( x l r x 2 , x 3 ) and V ( x l f x 2 , x 3 ) - w x ^ ( V ( x 1 , x 2 , x 3 ) ) = V ( x 1 / x 0 , x ^ ) - V ( x 1 , x 2 , - x 3 ) = 2 • l ' " 2 ' * 3 1 1 0 x l x 2 x 3 x ! x 2 0 = 2 x 3 ( x 2 - x 2 ) =-2n (2A 1 +A 1 ) T h i s shows t h a t P ( A 2 ) = P ( 2 A 1 + A 1 ) P ( B 2 ) : We f i n d n ( B 2 ) = X 2 X 3 ( x 2 ~ x 3 ) . Thus P ( B 2 ) i s spanned by t h e f o l l o w i n g l i n e a r l y i n d e p e n d e n t e l e m e n t s : X 2 X 3 ( x 2 ~ x 3 ) ' X 1 X 2 {X1~X2} P(A 1+A 1): The ( B 3 ) - o r b i t of IlfA^A^) consists of x ^ X j - x ^ ) and +x.(x.+x, ) where i , j , k are d i s t i n c t elements of {1,2,3) — 1 j K Therefore X 1 X 2 ' X 1 X 3 ' X 2 X 3 form a basis for P(A 1+A 1). P (2A1) : We have n ( 2 A 1 ) = - ( x 2 - x 2 ) , and a basis for P(2A^) i s given by: 2 2 x l _ x 2 ' 2 2 x l x3* Subsystems of C 3 The extended Dynkin diagram for C 3 i s 3= 2 x l x l x2 X2 x 3 2 x 3 From t h i s diagram we obtain the following subsystems (1) 3 o (2) C2+A-L (3) A 2 —o 2~X3 u x l ~ x 2 X2~ X3 2 x 3 •2x1 x2~ x3 2 x 3 X l " x 2 X2 X3 x2 X3 2 x 3 <5>2V -2x-^ 2x^ <6> A l + A l : (7) A i : (8) A 2: 2x„ X l X2 We next consider subsystems of the above.From the extended Dynkin diagram of C^+A^, 9 w > Q / -n •2x1 ~^x2 x2~ x3 2 x 3 we obtain the subsystem (9) 3A^: & 0 "~2x̂  ^^3 P (C 2+A 1) : We have II ( C 2 + A i ^ = - 8 x i X 2 X 3 ^ X2~ X3^ * Hence a basis for P (Ĉ +A-̂ ) i s given by; X-j^X^X^ (x^—:X2) e X 1 X 2 X 3 ( x l ~ X 3 ) ' P(3A 1): P (3A 1) i s spanned by IT (3A 1) =8x 1x 2x 3. We e note that X 3 A N ( 3 x^fO^tained from subsystems of C^, cannot be obtained using subsystems of B^,whereas x 2 a n <^ X5 obtained from subsystems of B^,cannot be obtained from C^. The group (B^) contains a n o n - t r i v i a l central element, the element corresponding to 2A^+A^.As i n the case of (D^), denoting t h i s central element by z and an element i n the conjugacy c l a s s X by [x] ,we observe the following conjugacy r e l a t i o n s : [A-J] ~ Z [ 2 A j , [ A j ~ Z [ A I + A J , [Aj ~ Z [ B J . Table VI: Character Table for (B-) = (C.J $ B3 A 3 <D C3 3 A 1 Conjugacy Class Representative C h a r a c t e r i s t i c Polynomial h. X X 0 X l x 2 X3 B3 C 3 $ $ (1) [111! x 3 -3x 2+3x-l 1 1 1 1 1 A l A l (1) [ l l - l ] x 3- x 2- x+1 3 1 -1 1 -1 2AX 2A 1 (1) [1-1-1] x 3+ x 2- x-1 3 1 1 1 1 2A1+A] 3 A 1 (1) [-1-1-1] x 3+3x 2+3x+l 1 1 -1 1 -1 A l A l (12) [111] x 3- x 2- x+1 6 1 -1 -1 1 B2 C2 (12) [-111] x 3- x 2+ x-1 6 1 1 -1 -1 A l + A l A l + A l (12) [ l l - l ] x 3+ x 2- x-1 6 1 1 -1 -1 A3 C 2 + A l (12) [ l - l - l ] x 3+ x 2+ x+1 6 1 -1 -1 1 A 2 A 2 (123) [111] x 3 -1 8 1 1 1 1 B3 C3 (123) [-111] +1 8 1 -1 1 -1 25 Table VI: Continued 2A 1 A l A l A2 2A 1 +A 1 B2 • V A 1 C 2 + A l A l A l A2 C2 A l + A l x 4 X5 X6 X7 X8 x 9 (1) [111] 2 2 3 3 3 3 (1) [11-1] -2 2 1 1 -1 -1 (1) [1-1-1] 2 2 -1 -1 -1 -1 (1) [-1-1-1] -2 2 -3 -3 3 3 (12) [111] ' o • 0 1 -1 -1 1 (12) [-111] 0 0 1 -1 1 -1 (12) [11-1] 0 0 -1 1 -1 1 (12) [1-1-1] 0 0 -1 1 1 -1 (123) [111] -1 -1 0 0 0 0 (123) [-111] 1 -1 0 0 0 0 Note:The f i r s t l i n e of the table gives the subsystems of B and the second the subsystems of C-.. 7. B 4 and C 4. Subsystems o f B 4 : The extended D y n k i n diagram f o r B 4 i s 9 - ( x 1 + x 2 ) X l X 2 x 2 X 3 X 3 X 4 From t h i s d i a g r a m we o b t a i n t h e f o l l o w i n g subsystems (1) B„: „ „ X 1 ~ X 2 (2) A 3+A 1: ^ x 2 X 3 X 3 X 4 X l x 2 X 2 ~ X 3 ( x 1 + x 2 ) (3) D, 9- ( x 1 + x 2 ) -o x l ~ x 2 X 2 ~ X 3 X 3 X 4 (4)B„+2A, : 2. 1 O X 1 ~ X 2 - ( x 1 + x 2 ) X 3 ~ X 4 (5) B 3 : - C r x 2 x 3 X 3 ~ X 4 x (6) B 2+A 1: az X l X 2 X 3 ~ X 4 x (7) A 2+A 1 (8) A. X 1 ~ X 2 - X 2 X 3 © x, X l X 2 x 2 X 3 - ( x 1 + x 2 ) (9) A' X l x 2 x 2 x 3 X 3 x 4 (10) 2A 1+A 1: Q X 1 ~ X 2 - ( x 1 + x 2 ) 27 (11) 3A±: Q X 1 " X 2 " ( X 1 + X 2 ) X 3 " X 4 (12) A 2 : (13) 2A : (14) 2A| X 2 " X 3 X 3 X 4 x r x 2 ~(x 1+ x2 ) x l X 2 X 3 ~ X 4 (15) A 1 + A 1 : D X2~ X3 X4 (16) B 2 : Q ^ X3~ X4 X4 (17) A± (18) A x : X l " X 2 X4 From t h e e x t e n d e d D y n k i n d i a g r a m f o r B 2 +2A- L , O O Cn x 1 - x 2 - ( x 1 + x 2 ) x 3 ~ x 4 x 4 - ( x 3 + x 4 ) we o b t a i n t h e s u b s y s t e m : (19) 4A 1 x l ~ x 2 ~ ( x 1 + x 2 ) x 3 - x 4 - ( x 3 + x 4 ) P ( B 4 ) : P ( B 4 ) i s spanned by H ( B 4 ) = x 1 x 2 x 3 x 4 V ( x 2 f x 2 , x 3 , x 2 ) P ( A 3 + A ^ ) : We have n (A 3+A 1)=x 4V(x 2_,x 2,,x 2) , and so PCA^+A^ i s spanned by: x j V ( x 2 ' ^ 3 ' ^ 4 ^ ' x 2 V ( x 2 , x * , x 2 ) , x^V (X-^ , x 2 f x^) , x^V (x-̂  f x 2 , x^) . It i s easy to see that these are l i n e a r l y independent. P(D 4): P(D^) i s spanned by one element: n(D 4)=V(x 2,x 2,x 2,x 2). P(B 2+2A 1): We fi n d IT (B 2+2A 1)=-x 3x 4 (x£-x 2) ( x 2 - x 4 ) . It can be seen that the following elements are l i n e a r l y independent and span P (B2+2A^); x.^x4 (x^—x 2) (x 3~x 4) , x.^x2 (x^ —x 2) ( x 3 _ x 4 ) f ^2 X4 ^ xl~~ X3 ̂  ̂ x2~~X4 ̂  ' x^x 4(x^—x 4) (x^ -x 3) , X 2 X 3 ^ X l — x 4 ^ ^ x2~ x3 ̂  * P (B 3) : We f i n d i r ( B 3 ) = x 2 x 3 x 4 v ( x 2 / X 2 , x 2 ) , hence P(B 3) i s spanned by: X2 X3 X4^^ X2' X3' X4^' X 1 X 3 X 4 ^ ^ x l ' X 3 ' X 4 ^ ' X2. X2 X4^ ^ X l ' X2 ' X4 ^ ' x-j^x2^ 3V ( x i , x2/• x^) • We o b s e r v e t h a t t h e s e a r e l i n e a r l y i n d e p e n d e n t . P ( B 2 + A 1 ) : We have n ( B 2 + A 1 ) = x 3 x 4 ( x | - x 4 ) ( x 1 ~ x 2 ) . S i n c e ( I - w x ) ( x 3 x 4 ( x 3 ~ x 4 ) ( x 1 - x 2 ) = 2 x 1 x 3 x 4 ( x 3 - x 4 ) , P(B2+A^) i s spanned by t h e ( B 4 ) - o r b i t o f x ^ x ^ x ^(x^-xp We c a n now e a s i l y w r i t e down a b a s i s as f o l l o w s : X 1 X 2 X 3 <*i - * 2 X 1 X 2 X 3 "«! X 1 X 2 X 4 -x| x i x 2 X 4 -4 X 1 X 3 X 4 "*3 X 1 X 3 X 4 - 4 X 2 X 3 X 4 (x| "*3 X 2 X 3 X 4 (x» P ( A 2 + A 1 ) ; We have 11 ( A 2 + A 1 ) = x 4 V ( x 1 , x 2 , x 3 ) I n t h i s c a s e i t i s e a s i e r t o c o n s i d e r t h e ( B 4 ) - o r b i t o f ( I - w x ) ( n ( A 2 + A 1 ) ) = 2 x 1 x 4 ( x 2 - x 2 ) r a t h e r t h a n t h a t o f II (A^+A^) .Then a b a s i s f o r P ( A 2 + A 1 ) i s g i v e n b y : X 1 X 2 ( x 3 " x 4 ' ' X 1 X 3 ( x 2 ~ X 4 ) ' X 1 X 4 ( x 2 ~ X 3 ) ' X 2 X 3 ( x l _ x 4 ) ' X 2 X 4 ( X 1 X 3 ) ' x^x^ (x.^—x 2)• P ( A 3 ) : We f i n d n ( A 3)=V(x 2 ,x 2,x 2) . The ( B 4 ) - o r b i t i n t h i s case i s the same as the ( A 3 ) - o r b i t 1 T ( A 2 ) with each x^ replaced by x? . Therefore,a basis for P ( A 3 ) i s given by: V(x 2,x 2,x 2)., Vfx* ,x*,x2) , VCx^x 2 ,x 2) . P ( A 3 ) : We have n ( A 3 ) = V ( X 1 , X 2 , X 3 , X 4 ) . Let Q=(I-wv ) (n(A') ) , X 3 R=(I-wv ) (Q) , X 4 S=(I+v/ ) (R) . X 2 Writing these i n determinant form we see that S=8II ( B 2 + 2 A 1 ) . Therefore P ( A ^ ) = P ( B 2 + 2 A 1 ) . P ( 2 A 1 + A 1 ) : It can be seen that the following elements are l i n e a r l y independent and that they span P ( 2 A 1 + A 1 ) : - n ( 2 A 1 + A 1 ) = x 4 ( x 2 - x 2 ) , • x 4 (x-^— x 3) f X l ^2~*l) ' X l ^ X 2 * ~ X 4 ^ ' 31 x 2 ( x l x 3 * ' X 2 ^ X l ~ X 4 ^ ' (x.^—x2) / x ^ (x-^""X^) . P ( 3 A 1 ) : We have n ( 3 A X ) = - ( x ^ - x 2 ) ( x 3 - x 4 ) , and .2 „ 2 ( I - w x ) ( n ( 3 A 1 ) ) = 2 x 4 ( x £ - x ^ ) =7-211 ( 2 A 1 + A 1 ) . T h e r e f o r e P (3A 1 )=P (2A 1 +A ] L ) . P ( A 2 ) : We have and n ( A 2 ) = - V ( x 2 , x 3 , x 4 ) , (I-w ) ( w X i - X 3 ( n ( A 2 } ) ) = 2 x 4 ( x l _ x 2 ) =-211 (2A 1 +A 1 ) Hence P ( A 2 ) =P (2A 1 +A ] L ) . P (2A-^) : A b a s i s f o r P (2A^) i s g i v e n by -n ( 2A 1)=xJ-x 2, 2 2 x l ~ x 3 ' 2 2 x 1-x 4. P ( 2 A | ) : L e t c 1 = I I ( 2 A | ) = ( x 1 - x 2 ) ( x 3 ~ x 4 ) , . C 2 = W X 2 ( C 1 ) = ( X 1 + X 2 ) ( X 3 ~ X 4 ) ' C 3 = W X 4 ( C 1 ) = ( X 1 ~ X 2 ) < x 3 + x4 }' We have C 4 = W x 2 ( c 3 ) = ( X l + X 2 ) ( X 3 + X 4 } C 1 + C 2 + C 3 + C 4 = 4 X 1 X 3 ' and we o b t a i n t h e f o l l o w i n g b a s i s f o r P(2A£): x l x 2 ' X 1 X 3 ' X l X 4 f X 2 X 3 ' X 2 X 4 ' X3 X4* P ( A 1 + A 1 ) : We have n ( A 1 + A 1 ) =x4 ( x 2 - x 3 ) , showing t h a t P(A-j+A^)=P(2A£) P ( B 2 ) : We f i n d n ( B 2 ) = x 3 x 4 ( x 2 - x 4 ) , and P ( B 2 ) i s spanned by: X 1 X 2 ( x l ~ X 2 i 2_ 2 X l X 3 l X l X 3 X 1 X 4 ^ X1~ X4 X 2 X 3 ( X 2 ~ X 3 X 2 X 4 ( X 2 _ X 4 ( 2 _ 2 x 3 X 4 l X 3 X 4 I t i s c l e a r t h a t t h e s e a r e l i n e a r l y independent, P (4A 1) : We have n(4A 1) = (xJ-x|) ( x 3 - x 2 ) , V and the ( B 4 ) - o r b i t of IT(4A1) consists of ~<±=1,2 , 3) ,where b 2 = ( x 2 " x 3 ) ( x i ~ X 4 ) ' b 3 = ( x j - x 5 ) ( x | - x j ) . As i n P(2A X) of A 3, b 3=b 1 +b 2 and {b^,b2} forms a basis for P (4A^) Subsystems of : The extended Dynkin diagram for i s 2 x l x l x2 X2~ X3 X3" X4 2 x4 From t h i s diagram we obtain the following subsystems: (1) C, '4 x l x2 x2 X3 (2) 2C 2: ^ '"' ~ 2 x l X1~ X2 (3) C 3+A i : e (5) C 2+A 1: Q X1~ X2 (6) C2+A1 (7) A 3: ^ 2 x l X2 X3 (4)A 1+2A 1: o © a —=*\ :3~X4 s 2x o -i 4 *s 2x 4 s —<j - N " x4 2x 4 2 -JC **™2JC-Ĵ  2X^ ~ x4 2x 4 O 6S> 2x-̂  x 3~x 4 2x 4 x l ~ x 2 x2 X3 x3 x4 34 (8) C x2~ x3 (9) A 2+A 1: (10) 2A. X1~ X2 X l X2 (11) A ^ A ^ 0 X2~ X3 (12) 2A1: (13) A, (14) C. (15) A, (16) A, X3~ X4 x l x2 2x. X3 X4 X2~ X3 X3 X4 2x, 2x, 2x, 2x, -2x^ 2x 4 u x2~ x3 X3~ X4 < o BI Next,we consider subsystems of the above.From the extended Dynkin diagram for 2C 2 1 o *C E -2x- x l x2 2x. -2x. x3~ x4 2x, we obtain (17) 4A, : -2x, 2x, 0 -2x. '1 ""2 3 From the extended Dynkin diagram for C^+A^ 2x, SJC^ ^ 2 ~~ ̂ 3 ^*3^4 ^ ̂* 4 we obtain (18) C 2+2A 3: . Q o < 25C-^ ^^2 ^3 "̂4 2x Fina l l y , f r o m the extended Dynkin diagram for C„+A, Jt—A. „, •2x^ ~ 2 x 3 X 3 - x 4 2 x we obtain (19) 3A i : -2x^ - 2 x 3 2 x4 P (2C 2) : We have n (2C 2)=-16x 1x 2x 3x 4(x|-x|) (x 2-x 2) As i n P(4A 1) of (B 4), :1 X2 X3 X4 X-. « ~ . ( X -T ~ X ~ ) ( X - — x. ) / ^1^2^3^4 (x 2~x^) (x^—x 4) i s a basis for P(2C 2). P (C 3+A 1) : We f i n d II (C 3+A 1) =16x 1x 2x 3x 4V (x 2 ,x 3 ,x4) . As i n P(A 3) of (B 4) ,a basis for P(C3+A-^) i s given by: X 1 X 2 X 3 X 4 V ( X 1 ' X 2 ' X 3 ) ' X1 X2 X3 X4^^ X2' X3' X4^' X 1 X 2 X 3 X 4 V ( X 1 ' X 3 ' X 4 K P (A 1+2A 1) : We have 36 and we f i n d Thus JI (A 1+2A 1)=-4x 1x 4 (x 2-x 3) , (I+wx ) ( n(A 1+2A 1))=-8x 1x 2x 4 X 1 X 2 X 3 ' X 1 X 2 X 4 ' X 1 X 3 X 4 ' ' X 2 X 3 X 4 form a basis for P(A^+2A^) P(4A 1): P(4A ) i s spanned by II (4A 1) =16x 1x 2x 3x 4 P (C 2+2A 1) : We f i n d n(C 2+2A 1)=16x 1x 2x 3x 4(x|-x|) A basis for P (C2+2A^) i s given by: X 1 X 2 X 3 X 4 ^ l " * ! * ' X l x 2 x 3 x 4 ( x 2 - x 2 ) , X 1 X 2 X 3 X 4 ( x l ~ x l * ' P(3A 1): We have II (3A 1)=8x 1x 3x 4 and P(3A 1)=P(A 1+2A 1). We note that X 3 , X 5 / Xg^ X 9,and x 1 3 obtained from subsystems of C 4,cannot be obtained using subsystems of B 4,while x 2' X4» Xg, X-jran& X-^Q robtained from subsystems of B 4 ,cannot be obtained from C.. A g a i n , i n c a l c u l a t i n g t h e c h a r a c t e r s we use t h e f o l l o w i n g r e l a t i o n s : z= [4Aj , [ A j - z [2A 1+A ]] , [2AJ ~ z [ 2 A j , [ A j ~Z [3A.J , [ B J ~z [B2+2AJ , [ A x + A j - Z ^ + A J , [ A 3 ] . z [ A 3 ] , [ A 2 ] ~ Z f D J ' ' [ B ^ - z f A ^ A j , [2A-] ~ z [ 2 A j ] , [ B 2 + A j - z [ B 2 + A j , [ D 4 ( a i ) ] -^VM , [ A 3 ] . Z [ A 3 j , [ B 4 ] ~ Z [ B 4 1 • 3.8 Table VII: Character Table for (B.) = (C/,). $ B4 D4 C4 4 A 1 Conjugacy Class Representative C h a r a c t e r i s t i c h. i X l X2 X3 C4 Polynomial * $ (i) [ 1 1 1 1 ] x 4-4x 3+6x 2-4x+l 1 1 1 1 1 A l (1) [-1111] x 4-2x 3 +2x-l 4 1 -1 1 -1 2A 1 (1) [-1-111] x 4 -2x 2 +1 6 1 1 1 1 2A 1+A 1 3AX (1) [-1-1-11] x 4+2x 3 -2x-l 4 1 -1 1 -1 4 A 1 4 A 1 (1) [-1-1-1-1] x' f+4x3 + 6x2+4x+l 1 1 1 1 1 A l A l (12) [ m i l x"-2x 3 +2x-l 12 1 -1 -1 1 B2 C2 (12) [-1111] x"-2x 3+2x 2-2x+l 12 1 1 -1 -1 A l + A l A l + A l (12) [11-11] x 4 -2x 2 +1 24 1 1 -1 -1 A 3 B 2 + 2 A i A 3 + A 1 J C 2 + A l C 2 +2A 1 (12) [-11-11] (12) [-11-1-1] x" -1 x'* + 2x 3 + 2x 2 + 2x+l 24 12 1 1 -1 1 -1 -1 1 -1 3 A 1 A 1 + 2 A 1 (12) [-1-1-1-1] x"+2x3 -2x-l 12 1 -1 -1 1 A 2 A 2 (123) [ m i ] x 4- x 3 - x+1 32 1 1 1 1 B 3 C3 ' (123) [-1111] x 4- x 3 + x-1 32 1 -1 1 -1 A 2 + A l A 2 + A l (123) [ l l l - l ] x 4+ x 3 - x-1 32 1 -1 1 -1 D4 C 3 + A l (123) [-1-1-1-1] x 4+ x 3 + x+1 32 1 1 1 1 2A| 2A£ (12) (34) [1111] x 4 -2x 2 +1 12 1 1 1 1 B 2 + A l C 2 + A l d2) (34) [-1111] x 4 -1 24 1 -1 1 -1 V a l > 2C 2 (12) (34) [-11-11] x 4 +2x2 +1 12 1 1 1 1 A3 A3 (1234) [ l l l l ] x 4 -1 48 1 -1 -1 1 B4 C4 (1234) [-1111] x 4 +1 48 1 1 -1 -1 Table VII: Continued 4 A 1 2A l A3 B3 A l A l 2C 2 3 + AL C3 A l A l 3A1 \ + 2 A l x 4 X5 X6 X7 X8 x 9 X1Q X l l K12 , X13 (1) [ l l l l ] 2 2 3 3 3 3 4 4 4 4 (1) [-1111] 2 -2 3 3 -3 -3 2 -2 2 -2 (1) [-1-111] 2 2 3 3 3 3 0 0 0 0 (1) [-1-1-11] 2 -2 3 3 -3 -3 -2 2 -2 2 (1) [-1-1-1-1] 2 2 3 3 3 3 -4 -4 -4 -4 (12) [ l l l l ] 0 0 1 -1 -1 1 -2 -2 2 2 (12) [-1111] 0 0 1 -1 1 -1 -2 2 2 -2 (12) [11-11] 0 0 1 -1 1 -1 0 0 0 0 (12) [-11-11] 0 0 1 -1 -1 1 0 0 0 0 (12) [-11-1-1] 0 0 1 -1 1 -1 2 -2 -2 2 (12) [-1-1-1-1] 0 0 1 -1 -1 1 2 2 -2 -2 (123) [ m i ] -1 -1 0 0 0 0 1 1 1 1 (123) [-1111] -1 1 0 0 0 0 1 -1 1 -1 (123) [ l l l - l ] -1 1 0 0 0 0 -1 1 -1 1 (123) [-1-1-1-1] -1 -1 0 0 0 0 -1 -1 -1 -1 (12) (34) [ l l l l ] 2 2 -1 -1 -1 -1 0 0 0 '0 (12) (34) [-1111] 2 -2 -1 -1 1 1 0 0 0 0 (12) (34) [-11-11] 2 2 -1 -1 -1 -1 0 0 0 0 (1234) [ l l l l ] 0 0 -1 1 1 -1 0 0 0 0 (1234) [-1111] 0 0 -1 1 -1 1 0 0 0 0 4 0 Table VII: Continued A 3 B 2 + 2 A 1 A 2 + A l A 1 + A 1 2 A . . B 2 B 2 + A l A 3 A | 2 A 1 + A 1 A 3 A 2 + A 1 2 A 1 A X + A : 2 A . . C 2 C 2 + A l C 2 + A l V X 1 4 X 1 5 X 1 6 X 1 7 X 1 8 X 1 9 (i) [ i m ] 6 6 6 6 8 8 (i) [ - 1 1 1 1 ] 0 0 0 0 - 4 4 (i) [ - l - i i i ] - 2 - 2 - 2 - 2 0 0 (i) [ - 1 - 1 - 1 1 ] 0 0 0 0 4 - 4 (i) [ - 1 - 1 - 1 - 1 ] 6 6 6 6 - 8 - 8 ( 1 2 ) [ 1 1 1 1 ] - 2 0 2 0 0 0 ( 1 2 ) [ - 1 1 1 1 ] 0 - 2 0 2 0 0 ( 1 2 ) [ 1 1 - 1 1 ] 0 2 0 - 2 0 0 ( 1 2 ) [ - 1 1 - 1 1 ] 2 0 - 2 0 0 0 ( 1 2 ) [ - 1 1 - 1 - 1 ] 0 - 2 0 2 0 0 ( 1 2 ) [ - 1 - 1 - 1 - 1 ] - 2 0 2 0 ' 0 0 ( 1 2 3 ) [ l l l l ] 0 0 0 0 - 1 - 1 ( 1 2 3 ) [ - 1 1 1 1 ] 0 0 0 0 1 - 1 ( 1 2 3 ) [ l l l - l ] 0 0 0 0 - 1 1 ( 1 2 3 ) [ - 1 - 1 - 1 - 1 ] 0 0 0 0 1 1 ( 1 2 ) ( 3 4 ) [ 1 1 1 1 ] 2 - 2 2 - 2 0 0 ( 1 2 ) ( 3 4 ) [ - 1 1 1 1 ] 0 0 0 0 0 0 ( 1 2 ) ( 3 4 ) [ - 1 1 - 1 1 - 2 2 - 2 2 0 0 ( 1 2 3 4 ) [ l l l l ] 0 0 0 0 0 0 ( 1 2 3 4 ) [ - 1 1 1 1 ] 0 0 0 0 0 0 Note: The f i r s t l i n e i n the table gives the subsystems of and the second the subsystems of C.. Subsystems of : The extended Dynkin diagram for F. i s O- O o 13 —O x l ~ x 2 X2~ X3 X3~ X4 x4 ~ h ( X 1 + X 2 + X 3 + X 4 From t h i s diagram we obtain the following subsystems: (1) F 4 : Q o ^ x 2 - x 3 x 3-x 4 x 4 -3 S(x 1+x 2+x 3+x 4) (2) B 4 : o o o > -J* x l x2 X2 x3 X3 X4 X4 (3) A 3+A 1: Q Q : o 0 x r x 2 x 2-x 3 x 3 - x 4 -J5(x 1 +x 2+x 3+x 4) (4) A 2+A 2: 0 Q # @ X 1 _ X 2 X 2 _ X 3 X4 -J2(x 1 +x 2+x 3+x 4) ( 5 ) C 3 + A l : o o x x - x 2 x 3 - x 4 x 4 -h(x1+x2+x3+x4) (6) B 3: Q o > x2 X3 x3~ x4 x4 (7) C 3: x3~ x4 x4 -%(x 1+x 2+x 3+x 4) (8) 2 A 1 + A l S 0 X l " x 2 • X3~ X4 ~ % ( x 1 + x 2 + x 3 + x 4 ) (9) A 3: X l x2 x2 X3 X3~ x4 (10) A 2+A 1: ^ X 2 - X 3 x3" x4 (x 1+x 2+x 3+x 4) (11) A 2+A i : 0 x2~ x3 x4 -%(x 1+x 2+x 3+x 4) x l x2 X3 X4 X4 (13) A 2 x2" x3 X3 X4 (14) A 2: x 4 -h (x 1+x 2+x 3+x 4) (15) 2A i : 0 (16) A^A-^ X l X2 ' X3 X4 x 3 - x 4 -%(x 1+x 2+x 3+x 4) <17) B 2 : :- (18) A 1 (19) A±. x 3 X4 X4 X l X2 ® x, 4 Repeating the process with the above systems of roots we f i n d four subsystems that are not congruent under (F 4) to any of the above,as seen below. The extended Dynkin diagram for B 4 i s o - (x 1+x 2) -o x l x2 x2 x3 x3 X4 x4 from which we obtain (20) D.: o - ( x 1 + x 2 ) The extended Dynkin diagram for C^+A^ i s x l ~ x 2 x3" x4 x4 ( x l + X 2 + X 3 + X 4 ^ X l + X 2 From t h i s diagram we obtain the following two new subsystems (21) B 2+2A i : Q Q " — > (22) 3AX x l ~ x 2 X l + X 2 X3 X4 X4 x l ~ x 2 X3~ X4 x l + x 2 F i n a l l y , from the extended Dynkin diagram for B2+2A-̂ o o o > x j ~ X2 xl"^~ x2 X 3 — x 4 X4 — ^X3"^X4^ we obtain (23) 4A 1: x l ~ x 2 x l + x 2 X3~ x4 ( x 3 + X 4 ) In the group (F^), we have the following coset decomposition (F 4) = (B 4)U(B 4)w rU(B 4)w x^w r where r=-h(x^+X2+x3+x4).Consequently the ( F 4 ) - o r b i t of a polynomial II (S) i s the union of the (B 4 ) - o r b i t s of n (S) ,wr (H (S) ) f and wv ŵ  (H (s) ) . x 4 In some cases we f i n d that ±w ( n(S)) and ±w w (n(S)) r x4 r are polynomials that are already i n the (B 4 ) - o r b i t of H(S). Then the ( F 4 ) - o r b i t of n(s) i s the same as i t s ( B 4 ) - o r b i t . V7e f i n d t h i s to be the case for the subsystems A2+A2,C3+A^, C3 , A 2 + A 1 , A2 , A2 , D4 ' 3 A 1 a n c ^ 4A^.In 9 1 1 t n e other cases i t turns out that e i t h e r wr (n(S) )=±n(S) or 44 w (n (s) )=±w v w (n (s)), r x 4 and so a b a s i s f o r P(S) can be found i n t h e u n i o n o f t h e ( B ^ ) - o r b i t s o f two o f n(s) ,w (ii(s)) ,wv w„(n(s)). x x 4 i P (F^) :: We have n(F„)=:. 1 V(y ,y ,y , y ^ ) V ( x ,x ,x ,x^) ¥ 0 9 6 " J- ^ -a « x ^ J * where y i = x 1 + x 2 , y 2 = x l ~ x 2 ' y 3 = x 3 + x 4 , y 4 = x 3 - x 4 . P (B 4) : We have II ( B 4 ) = x 1 x 2 x 3 x 4 V ( x 2 , x 2 , x 2 ,x|) We f i n d t h a t X 3 _ X 2 X 3 X 4 ^ ^ x l ' X 2 ' X 3 ' X 4 ̂  ' -16wr (n(B 4) )=16w wr (II (B 4) ) -• £(x 1-x 2) - ( x 3 + x 4 ) 2 ] [ ( x 1 + x 2 f - ( x 3 - x 4 ) 2] V ( x 2 , x 2 , x 2 , x 4 ) form a b a s i s f o r P ( B 4 ) . P (A 3+A 1) : We have IT (A 3+A 1)=-32 ( x 1 + x 2 + x 3 + x 4 ) V ( x 1 , x 2 , x 3 , x 4 ) We f i n d w r(n(A 3+A 1))=-n(A 3+A 1), w x ^ w r ( H ( A 3 + A x ) ) = x 4 V ( x | , x 2 , x 2 ) . As i n (B 4),we o b t a i n from x 4 V ( x 2 , x 2 , x 2 ) t h e f o l l o w i n g f o u r l i n e a r l y i n d e p e n d e n t p o l y n o m i a l s : We have b l = X l V * X 2 , X 3 , X 4 ) , b 2 = X 2 V ( x l ' x 3 ' X 4 ) , b 3 ^ x l , x 2 , X 4 ) , b 4=x 4V(x*,x 2,x* ) . 1 1 1 1 x l X2 X3 X4 -2(n(A 3+A 1))= x i 2 X2 X3 X l x l *2 X3 X4 =-b l + b 2 - b 3 + b 4 • We note that any other polynomial i n the (B 4 ) - o r b i t of H(A 3+A 1) can be obtained from t h i s by an appropriate change i n signs.We can,therefore,express each of these polynomials as a l i n e a r combination of b^,...,b 4 by making the corresponding change i n signs i n b^ , . . . b 4 . Thus (b-^ ,b 2 ,b 3 ,b 4 } i s a basis for P(A3+A^)- P(A 2+A 2): We f i n d n ( A 2 + A 2 ) = 4 X 4 [ ( x l + x 2 + x 3 ) 2 _ x 4 2 ] V(x 1,x 2,x 3) , and -w (II(A2+A2) )=w wr (n(A 2+A 2) )=IT(A2+A2) . X4 Therefore the ( F 4 ) - o r b i t of II (A 2+A 2) i s the same as i t s (B 4 ) - o r b i t . L e t Q= (l-w x ) (n (A 2+A 2) )., R=(I+wv ) (Q) . X 3 Then R=x 1x 4(x|-x 3) (x 2-x 2-x 3+x 4). The ( B 4 ) - o r b i t of R gives us the following l i n e a r l y independent polynomials spanning P(A 2+A 2): X 1 X 2 ( X3 "X4> (x 2+x 2-x 2-x 2), X l x 3 (x 2 - X4^ (x 2-x 2+x 2-x 2), X1 X4 ( X2 -x 2) ( x 2 - x 2 - x 2 + x 2 ) , X 2 X 3 (X1 V (-x 2+x 2+x 2-x 2), X2 X4 <*; -x|> ( - X i + x 2 ~ x 3 + x 4 ) ' X3 X4 (x 2 -xl) (-x 2-x 2+x 2+x 2). P (C^+A^ : We have n(C 3+A 1)=- x 3 x 4 ( x 2 - x 2 ) (x 2-x 2) [ ( x 1 - x 2 ) 2 - ( x 3 + x 4 ) 2 J [ ( x 1 - x 2 ) 2 - ( x 3 - x 4 ) 2 } We f i n d -wr (n(c 3+A 1) )=w w r(n(c 3+A 1)) 4 =w w (n(C-+A n)). x""*^3 2 ̂ 4 Therefore we only need to consider the (B 4 ) - o r b i t of n(c 3+A 1) We f i n d that P ( C 3 + A 1 } i s spanned by: b l = x 3 X 4 ( x l ~ x 2 ) / 2 2 \ l x3 X 4 J [ ( x 1 - x 2 ) 2 - Cx 3 +x 4) 2J[( X 1 " X 2 ) 2 - (x 3-x 4) 2] , b 2 = X 3 X 4 ^ l " ^ , 1 2 \ tx 3 x 4; [ ( x 1 + x 2 ) 2 - ( x 3 + x 4 ) 2][ ( x 1 + x 2 ) 2 " ( x 3 - x 4 ) 2 ] , b 3 = x l x 2 ^ l - ^ / 2 2 \ [ ( x 1 + x 2 ) 2 - (x 3-x 4) 2] [ ' ( x r x 2 ) 2 ~ ( x 3 - x 4 ) 2 ] , b 4 = x l X 2 ( x l " " X 2 ) / 2 2 \ { x 3 X 4 J [ ( x 1 + x 2 ) 2 - (x 3+x 4) 2 J [ ( x V x 2 ) 2 - ( x 3 + x 4 ) 2 ] , b 5 = X l X 4 ( x 2 ~ x 3 } i 2 2\ ( x r x 4 ) [ ( x 1 + x 4 ) 2 - ' X2~ X3 ) 2][ - ( x 2 - x 3 ) 2 ] , b 6 = X l X 4 ( X 2 _ X 3 ) / 2 2 \ (x x-x 4) [ ( x 1 + x 4 ) 2 - x 2+x 3) 2][ ( x r x 4 ) 2 ~(x 2+x 3) 2] , b 7 = X 2 X 3 ( x 2 ~ X 3 } / 2 2 \ [ ( x r x 4 ) 2 - x 2+x 3) 2 J [ " ( x ^ ) 2 - ( x 2 - x 3 ) 2 ] , b 8 = X 2 X 3 ( x2~ x3* / 2 2 \ ( x r x 4 ) [ ( x 1 + x 4 ) 2 - ( x 2+x 3) 2][ ; ( x 1 + x 4 ) 2 - ( x 2 - x 3 ) 2 ] , b 9 = X 2 X 4 ^ X l - X 3 ^ ( 2 2 \ i x 2 - x 4 ; [(x 1~x 3)  2- ( x 2+x 4) ( X 1 ~ X 3 ) 2 - ( x 2 - x 4 ) 2 ] , b 1 0 = X 2 X 4 ( X I - X 3 ] <x|-x" ) [(x 1+x 3) 2- (x 2+x 4 ) 2] [ ( x 1 + x 3 ) 2 - ( x 2 - x 4 ) 2 ] , b l l = X l X 3 ( x i " x 3 } ) [ ( x 1 + x 3 ) 2 - (x 2-x 4 ). 2l [ ( x r x 3 ) 2 - ( x 2 - x 4 ) 2 J , b 1 2 = X l X 3 ( X 1 ~ X 3 ) (x|-x| ) [ ( x 1 + x 3 ) 2 - (x 2+x 4 ) 2J [(x 1-x 3) 2 - ( x 2 + x 4 ) 2 ] . If we substitute 4 7 y i=x 2+x 2 y 2=x 1-x 2 y 3 = x 3 + x 4 Y 4 = X 3 ~ X 4 in b^,b2,b3,fc>4 we obtain the simpler forms b.=-%y ny 0y.y / 1v(y 1 2,y 2 y 2) , b 4 r = - ^ i y 2 y 3 Y 4 V ( Y i ' Y 2 ' Y 3 ) From t h i s we can see that V b l " b 2 + b 3 Similarly,we f i n d W W b 1 2 s = b 9 " b 1 0 + b l l ' I t can be seen that {b^ ,b 2 ,b^ ,br-,bg ,b^ ,b^ ,b^^ ,b^^} forms a l i n e a r l y independent set. P (B 3) : We f i n d n (B 3)=-x 2x 3x 4V(x 2,x 2,x 2) , -wr (IT (B.J )=w w (II (B ) ) 1 = . — (x x-x 2+x 3+x 4) (x 1+x 2~x 3+x 4) (x 1+x 2+x 3-x 4)V(-x 1,x 2,x 3,x 4) Four l i n e a r l y independent polynomials are found from the (B 4) - o r b i t of JI (B ) ; b ^ = x 2 x 3 x ^ V ( x 2 , x 3 i X 4 ) , b 2 = x ^ x 3 x V(x^,x 3,x^), b 3 - x 1 x 2 x 4 V ( x 2 , x 2 , x 2 ) , b 4~x^x 2X 3V(x^,x 2,x 3). Let 48 Q=8wv (w -(n(B-))) = (x,+x„-x_-x.)(x.-x^+x-.-x.)(x.-x,-x.+x.)V(x,,x0,x,,x4) 4 4 4 = . ( ? x i . T ? X i x j t ? £ x i x j x k ) V ( x l ' x 2 ' x 3 ' x 4 ) i ^ j i<j<k This can be written i n terms of the following determinants B= 1 1 1 1 x l x2 x 3 X4 x 2 x l x 2  X2 *3 X4 x 6 x l x 6  X2 X3 1 1 1 i x l X2 X3 x4 x 3 x l X2 x 3 X3 *4 x 5 x l X2 x 5 X3 < 1 1 1 1 x 2 x l x 2  X2 x 2  X3 *4 X2 x 3 X3 x 3  X4 *2 x" X3 x* X4 = E X I t £ x i x j t £ X i X j x k ) V ( x 1 , x 2 , x 3 , x 4 ) , i=l xj = l Jij,k=l J ift i<j<k 4 4 = (&?x.+2r; x.x x k ) V ( x 1 , x 2 , x 3 , x 4 ) xj = l J 3j,k=l J i ^ j i<j<k :(Z]xixixk)v(xl'x2'x3'x4) * i i k = l i<j<k We have Let Q=A-2B+3C, Then R=(I+w ) (Q) , x2 S=(I+wv ) (R) , X3 T=(I+w )(S). x 4 49 T=< 1 1 1 1 1 1 x 2 * 3 x 2  X4 + 3x 3 *2 *; x 2 X4 x 6  X2 * 3 x 6 X4 * 3 x 4 X4 \ =-8 [x x (x 2+x 2+x 2)V(x 2,x 2 ,x 2)-3x 3V(x 2 ,x 2,x 2)] =-8x1 [(x 2+x 2+x 2+x 2)-4x 2] V(x 2 ,x 2 ,x 2) . We can now complete the basis for P(B^) by applying to T: b 5 = x x [(x 2+x 2+x 2+x 2 ) -4x 2] V(x 2,x 2,x 2) , b 6=x 2 [(x 2+x 2+x 2+x 2)-4x 2]v(x 2 ,x 2 ,x 2) , b ?=x 3 [(x 2+x 2+x 2+x 2)-4x 2]v(x 2,x 2,x 2) , b g=x 4 [(x 2+x 2+x 2+x 2)-4x 2]v(x 2,x 2,x 2) . w X . -x. i 3 P ( C 3 ) : In t h i s case i t i s simpler to write II (C 3) i n terms of the y^ as defined i n P(F 4).We have n ( c 3 ) = ~ gT Y 1Y 3y4V(y^,y 3,y 4) . We note that the elements w ,w , c fr, N x. x.+x.,w ,and w w of (F.) x l j r x4 can be expressed i n terms of w ,w ,w , , . . , N yL' y^Yj' -%(y 1+y 2+y 3+y 4), and w w , , , , , .,and conversely.Therefore,we can fi n d y 4 ( y i + y 2 + y 3 + y 4 ) the (F.)-orbit of II (C 0) by applying w ,w . ,w , , , , , w 4 3 y^ y i ± Y j (y 1+y 2 +y 3+y 4) and w w . . . y 4 ~h (y 1+y 2+y 3+y 4) to y 1 y 3 y 4 v ( y 2 , y 2 , y 2 ) . We observe that t h i s . i s the same as -w (II (B_) ) with y. replacing-the X £ Hence a basis for P(C 3) i s given by the polynomials i n the basis for P(B^) with y^ replacing the x^. P(2A 1+A 1): We have 50 n (2A 1+A 1)=-3s ( x 1 - x 2 ) ( x 3 - x 4 ) ( x 1 + x 2 + x 3 + x 4 ) We f i n d w r ( n ( 2 A 1 + A 1 ) ) = - n ( 2 A 1 + A 1 ) , w „ w_ (II ( 2 A 1+A 1) )=-x 4 ( x 2 - x 2 ) . x4 r As i n ( B 4 ) X 4 ( x 2 - x 2 ) , X 4 ( x i " T X 3 ) ' X l ( x 2 - x 2 ) , X l ( x 2 ~ X 4 * ' X 2 ( x 2 - x 2 ) l x l x 3 ; ' X 2 ( x 2 - x 2 ) , X 3 ( x 2 - x 2 ) , X 3 a r e l i n e a r l y i n d e p e n d e n t . l t c a n be seen t h a t H(2A^+A^) and o t h e r p o l y n o m i a l s i n i t s ( B 4 ) - o r b i t a r e a l l l i n e a r c o m b i n a t i o n s o f t h e s e e l e m e n t s . P ( A 3 ) : We f i n d n ( A 3 ) = V ( x 1 , x 2 , x 3 , x 4 ) , w r(II(A 3))=II<A 3) , w x ^ w r ( n ( A 3 ) ) = V ( x 2 , x 2 , x 2 ) . A s shown f o r P ( A 3 ) o f ( B 4 ) , t o f i n d t h e span o f t h e ( B 4 ) - o r b i t o f V ( x ^ , x 2 , x 3 , x 4 ) , w e may c o n s i d e r t h e o r b i t o f x .^x 4 (x^—x 2) (x^—x 4) • We have s een i n P ( B 2 + 2 A ^ ) o f (B 4 ) t h a t t h e f o l l o w i n g p o l y n o m i a l s a r e l i n e a r l y i n d e p e n d e n t : b 1 = x 3 x 4 ( x 2 - x 2 ) ( x 2 - x 2 ) , b 2 = x l x 2 ( x l - x 2 ) ( x 3 - x 4 ) ' 51 t> 3=x 1x 3 (x 2-x 2) (x^-x^) , b 4 = X 2 X 4 ( x i ~ x 3 ) ^ x2" x4^' b 5 = x 1 x 4 ( x 2 - x 2 ) ( x 2 - x 2 ) , H 6 = X 2 X 3 { X l ~ X l ) ( X 2 _ X 3 } - The (B^)-orbit of V(x^,x 2,x 3) gives us three more l i n e a r l y independent polynomials,completing the basis for P( A 3 ) : b 7 = V ^ x l ' x 2 ' X 3 ^ ' b 8 = V ( x 2 f x 2 , x 2 ) , b 9 = V ( x l ' x 3 ' x 4 ) ' P ( A 2 + A X ) : We f i n d II (A 2+A 1)=%(x 1+x 2+x 3+x 4)V(x 2,x 3,x 4) , wr (n (A 2+A 1) ) =-n (A 2+A 1) , W x w r ( n ( A 2 + A i ^ = - x 4 V ( ~ x l ' x 2 , x 3 ^ ' 4 We have seen i n P ( A 2 + A ^ ) of (B̂ -) that the span of the (B 4)- o r b i t of x 4V(x^,x 2,x 3) contains the following l i n e a r l y independent polynomials: b l = X l X 2 ( x 3 - X 4 ) ' b 2 = X l X 3 (X2~XV ' b 3 = x 1 x 4 ( x 2 - x 2 ) , b 4 = X 2 x 3 ( x ! ~ X 4 ) ' V X 2 X 4 ( x l _ x 3 ) ' bg~ x^x. (x — x«) • D 5 4 X Z. We can complete t h i s to a basis of P(A 2+A 1) by considering the ( B 4 ) - o r b i t of IT (A2+A-^) .Let Q = 2 w x -x ( n ( A 2 + A i ) ) " ( x i + x 2 + x 3 + X 4 ) V ( x l ' x 2 ' X 3 ) ' R = w x (Q)=(x 1+x 2+x 3-x 4)V(x 1,x 2,x 3). Then and Q+R=2(x 1+x 2+x 3)V(x 1,x 2,x 3) 1 1 1 = 2 X l X2 X3 x 3  X l x 3 X2 X3 (I+wx ) (Q+R)=-4x 1x 2(x 2-x 2). Therefore,a basis for P(A2+A^) i s given by b-̂ , and . b,. above b b 7 = X l X 2 ( X l - X 2 ) ' b 8=x 1x 3(x 2-x^), V X 1 X 4 ( X 1 - X 4 ) ' b10 : = X2 X3 ' b l l = X 2 X 4 ( X 2 ~ X 4 ) ' b 1 2 = X 3 X 4 ( X 3 " X 4 ) ' P(A 2+A 1): We have n (A 2+A 1)=jx 4 ( x 2 - X 3 ) [ ( x 1 + x 2 + x 3 ) 2 ~ x l ~ l ' and (I-w ) (n(A 2+A 1) )=x xx 4 ( x 2 - x 2 ) , showing that P (A2+A1)=P (A 2+A 1) P(B 2+A 1): We f i n d IT (B 2+A 1)=x 3x (x 1-x 2).(x 3-x 4) -wr (n(B 2+A 1) )=wx w r(n(B 2+A 1) = h [(x 1+x 2) 2 - (x 3-x 4) 2] (x 2-x 2) (x 3-x 4) . Therefore, the ( B ) - o r b i t s of II ( B ^ A ^ and wx wr (II ( B ^ A ^ ) 4 span P(B2+A^).We have already seen that the following polynomials form a basis for P(B0+A,) of (B ): Let b l = X 1 X 2 X 3 (x 2- x 2) b2 = X 1 X 2 X 3 (x 2- x 2) b3 = X 1 X 2 X 4 (x 2- x 2) t b4 = X 1 X 2 X 4 (x 2- X4> b5 = X 1 X 3 X 4 (x 2- l x l x 2) r b6 = X 1 X 3 X 4 X l > i b7 = X 2 X 3 X 4 (x|-x 2) t b8 = X 2 X 3 X 4 fx 2 -l X2 X4> • Q= (xj-xj) (x 3- X4> - We have 4wx w r(H(B 2+A 1))=Q+2b 1-2b 3. Therefore we may consider Q instead of w w (II (B„+A ) ) . x4 r - s i We f i n d h ( Q + W x (Q) ) =x3 (xj-x*) - (x 2-x 2) (x 3 + 3x 3x 2 ) =x 3(x 2-x 2)(x£+x 2-x 2-3x 2). We can now complete the basis for P(B2+A^) by the following l i n e a r l y independent polynomials: b 9=x x(x|- x 3 > ( • x 2 _ x 2 _ 2_o 2 N KX1 x2 3 4 ;' b i o = x i ( X2 -*v (x.^—x^-3x.^-~x^) / b l l = x 2 ( x i -x|) (-xj +x 2-x|-3x 2) r b 1 2 = X 2 ( x 2  X l -*4> (-x 2+x 2-3x 2-x 2) r b 1 3=x 3( X 2 x l -x 2) X 2 J (-x 2-x 2 +x 2-3x 2) i b 1 4 = X 3 ( X 2 x l -x 2) (-x2-3x|+x2-x|) f b15 = X4< X 2 1 -x 2^ x 2; (-x 2-x 2-3x 2+x 2) t b 1 6 = X 4 ( x 2 x l - x 2 l X 3 J (-x 2-3x 2-x 2 +x 2) • P(A 2): As i n the case of (B4),we have and n(A 2)=-v(x 2,x 3,x 4) P(A 2)=P(2A 1+A 1) . P(A 2): We f i n d n ( A 2 ) - | x 4 f ( X l + x 2 + x 3 ) 2 - x 2 ] , -wr (n(A 2) )=wx w r(n(A 2) )=n(A 2). Therefore the ( F 4 ) - o r b i t of n(A 2) i s the same as i t s (B 4)- or b i t . L e t Hence Q=(I-wx ) (H (A 2) )=x 1x 4 ( x 2 + x 3 ) (I+wx )(Q)=2x 1x 2x 4 Also, (I+w +wx +wx ) (II (A 2) ) =x4 (x 2+x 2+x 2-x 2) . It can be seen that II (A 2) and any polynomial i n i t s ( B 4 ) - o r b i t may be expressed as a l i n e a r combination of polynomials of the form x ( X 2 + X 2 + X 2 T X 2 ) and x.x.x, m x D k m' x j k Therefore a basis for P(A 2) i s given by: x 1(x 2+x 2+x 2-x 2), x 2(x 2+x 2+x 2-x 2), x 3(x 2+x 2+x 2-x 2), x 4(x 2+x 2+x 2-x 2), X l X 2 X 3 f X 1 X 2 X 4 ' X 1 X 3 X 4 ' X2 X3 X4* P (2A^) : We have n ( 2 A 1 ) = (x 1-x 2) (x 3-x 4) , w r ( n ( 2 A 1 ) )=n(2A 1) , wx w r(n(2A ) )=-(x 2-x 2) . 4 We observe that (x^-x 2) (x^-x^) i s II(2A|) of (B^) and -(x 2-x 2) i s II (2A^) of (B^).We f i n d that the union of the bases for P ( 2 A 1 ) and P ( 2 A | ) of (B 4) provides a basis for P ( 2 A 1 ) of (F 4) x l X 2 ' X 1 X 3 ' X 1 X 4 ' X 2 X 3 ' X 2 X 4 ' X 3 X 4 ' 2 2 X l ~ X 2 ' 2 2 x l ~ X 3 ' 2 2 x i " X 4 * P (Aj+A.^) : We have n(A 1+A 1)=-J 2 (x 3-x 4) (x 1+x 2+x 3+x 4) , wr (n (A 1+A 1) ) =-n (A 1+A 1) , W x W r ^ n ( A i + A i ) ) =~ x4 ( x i + x 2 ^ • Since -x^(x^+x 2) i s contained i n P ( 2 A ^ ) , P ( A 1 + A ] L ) = P ( 2 A 1 ) . P(B ) : We f i n d n ( B 2 ) = X 3 X 4 ( x 3 - x 4 ) . This i s one of the elements i n the basis of P(A 2+A^).Therefore P ( B 2 ) = P ( A 2 + A 1 ) . • P(D^): As in (E^),P{D^) i s spanned by one element n(D 4)=V(x 2,x 2,x 2,x 2). P (B2+2A-L) : We find n(B 2+2A 1)=x 3x 4(x 2-x 2)(x 2-x 2). But this i s i n P(A3),hence P(B2+2A1)=P(A3). P(3A X): We have n(3A1) = (x 2-x 2) (x 3-x 4) . We observe that this i s in P(2A^+A^),hence P(3A1)=P(2A1+A1). P (4A1) : We have n(4A 1)=-(x 2-x 2) (x 2-x 2) and w (n(4A1))=w w (II (4A.. ) )=n (4A ) . 3T X IT X X Therefore,as i n (B 4), ( x 2 - x 2 ) ( x 2 - x 2 ) , (x 2-x 2) (xj-x*) form a basis for P(4A^). Using the above (F4)-modules,we obtain seventeen of the irreducible characters of (F4).The characters that cannot be obtained by the above method can be found by using the following relations: x 1 0 = x 4 x 5 / X 1 1 = X 1 X 8 ' X 1 2 = X 1 X 9 ' X 1 4 = X 2 X 1 3 , X 2 2 = X 1 X 2 0 ' The following conjugacy re l a t i o n s are used i n c a l c u l a t i n g the characters of (F^): z = [ 4 A J , [ A J - Z ^ + A J , [ 2 A J . z [ 2 A j , [ A 1 + A j - z [ A X + A J , [ A j . z f D j , [ A j - z ^ + A j , [ B J - z [ A 3 + A J , [ A 3 ] ~ z [ A 3 ] , [ B 2 + A j - . z [ B 2 + A j , [ C 3 ] ~ 4 V A l ] ' [ B 3 ] . z [ A 2 + i j , [ A 2 + A 2 - J - . z [ P 4 ( A I ) ] , [ D 4 ( a 1 ) ] ~ z [ D 4 ( a 1 ) ] , 5 8 Table VIII: Character Table for (F.). F4 D4 B4 Conjugacy Class Representative C h a r a c t e r i s t i c . Polynomial h, l X0 X l x 2 X3 x 4 (i) [ m i ] x"*-4x3+6x2-4x+l 1 .1 1 1 1 2 A l (12) [ l l l l ] x 4-2x 3 +2x-l 12 1 -1 -1 1 -2 A l (i) [ - 1 1 1 1 ] x"-2x 3 +2x-l 12 1 -1 1 -1 0 2A 1 (i) [ - 1 - 1 1 1 ] x1* -2x 2 +1 18 1 1 1 1 2 A l + A l (12) [11-11] x" -2x 2 +1 72 1 1 -1 -1 0 A2 (123) [ l l l l ] x1*- x 3 - x+1 32 1 1 1 1 2 A2 x 1*- .x3 - x+1 32 1 1 1 1 -1 B2 (12) [-1111] x"-2x 3+2x 2-2x+l 36 1 1 -1 -1 0 3A 1 (12) [-1-1-1-1] x"+2x3 -2x-l 12 1 -1 -1 1 -2 2 A 1 + A 1 (1) [-1-1-11] x"+2x3 -2x-l 12 1 -1 1 -1 0 A3 (1234) [ l l l l ] x" -1 72 1 -1 -1 1 -2 B 2 + A l (12) (34) [-1111] x 4 -1 72 1 -1 1 -1 0 C 3 x 4- x 3 + x-1 96 1 -1 -1 1 1 B3 (123) [ - l l l l ] x1*- x 3 + x-1 96 1 -1 1 -1 0 A 2 + A l xk + x 3 - x-1 96 1 -1 -1 1 1 A 2 + A l (123) [ l l l - l ] xk + x 3 - x-1 96 1 -1 1 -1 0 4 A 1 (1) [ - l - l - l - l ] x'*+4x3 + 6x2+4x+l 1 1 1 1 1 2 A 2 + A 2 x'* + 2x 3 + 3x 2+2x+l 16 1 1 1 1 -1 A 3 + A l (12) [-11-1-1] x' ,+2x3 + 2x 2 + 2x+l 36 1 1 -1 -1 0 C 3 + A l x"*+ x 3 + x+1 32 1 1 1 1 -1 D4 (123) [-1-1-1-1] x V x 3 +X+1 32 1 1 1 1 2 D 4(a x) (12) (34) [-11-11] x k +2x2 +1 12 1 1 1 1 2 B4 (1234) [-1111] xH +1 144 1 1 -1 -1 0 F 4 ( a i ) x"-2x 3+3x 2-2x+l 16 1 1 1 1 -1 F4 • x* - x 2 +1 96 1 1 1 1 -1 4 A 1 A l A l A 3 + A l A 2 + A 2 B3 C3 X5 X6 X7 X8 x 9 X10 X l l X12 X13 X14' X15 X16 2 2 2 4 4 4 4 4 6 6 8 8 A l 0 0 2 2 -2 0 -2 2 0 0 -4 0 A l 2 -2 0 2 2 0 -2 -2 0 0 0 -4 2A 1 2 2 2 0 0 4 0 0 -2 -2 0 0 A 1 + A 1 0 0 0 0 0 0 0 0 2 -2 0 0 A2 -1 -1 2 1 1 -2 1 1 0 0 2 -1 A2 2 2 -1 1 1 -2 1 1 0 0 -1 2 B2 0 0 0 2 -2 0 2 -2 -2 2 0 0 0 0 2 -2 2 0 2 -2 0 0 4 0 2A 1 +A 1 2 -2 0 -2 -2 0 2 2 0 0 0 4 A 3 0 0 2 0 0 0 0 0 0 0 0 0 B 2 + A l 2 -2 0 0 0 0 0 0 0 0 0 0 C 3 0 0 -1 1 -1 0 -1 1 0 0 1 0 B3 -1 1 0 1 1 0 -1 -1 0 0 0 1 A 2 + A l 0 0 -1 -1 1 0 1 -1 0 0 -1 0 A 2 + A l -1 1 0 -1 -1 0 1 1 0 0 0 -1 4 A 1 2 2 2 -4 -4 4 -4 -4 6 6 -8 -8 A 2 + A 2 -1 -1 -1 -2 -2 1 -2 -2 3 3 2 2 A 3 + A l 0 0 0 -2 2 0 -2 2 -2 2 0 0 C 3 + A l 2 2 -1 -1 -1 -2 -1 -1 0 0 1 -2 D4 -1 -1 2 -1 -1 -2 -1 -1 0 0 -2 1 2 2 2 0 0 4 0 0 2 2 0 0 B4 0 0 0 0 0 0 0 0 0 0 0 0 P 4 ( a l } -1 -1 -1 2 2 1 - 2 2 3 3 -2 -2 F4 -1 -1 -1 0 0 1 0 0 -1 -1 0 0 ^2 B2 3AX A2 2A 1 A 3 C 3 + A 1 A 2 + A l B 2 + A 1 2 A 1 + A 1 A l + A l B 2 +2A 1 A 2 + A l X17 X18 X19 X20 X21 X22 X23 X24 8 8 9 9 9 9 12 16 A l 0 » 4 3 -3 -3 3 0 0 A l 4 0 3 3 -3 -3 0 0 0 . 0 1 1 1 1 -4 0 A l + A l 0 0 1 -1 1 -1 0 0 A2 -1 2 0 0 0 0 0 -2 A2 2 -1 0 0 0 0 0 -2 B2 0 0 . 1 -1 1 -1 0 0 3A 1 0 -4 3 -3 -3 3 0 0 2A 1 +A 1 -4 0 3 3 -3 -3 0 0 A 3 0 0 -1 1 1 -1 0 0 B 2 + A l 0 0 -1 -1 1 1 0 0 C3 0 -1 0 0 0 0 0 0 B3 -1 0 0 0 0 0 0 0 A 2 + A l 0 1 0 0 0 0 0 0 A 2 + A l 1 0 0 0 0 0 0 0 4 A 1 -8 -8 9 9 9 9 12 -16 A 2 + A 2 2 2 0 0 0 0 -3 -2 A 3 + A l 0 0 1 -1 1 -1 0 0 C 3 + A l -2 1 0 0 0 0 0 2 D4 1 -2 0 0 0 0 0 2 D 4(a x) 0 0 -3. -3 -3 -3 4 0 B4 0 0 -1 1 -1 1 0 0 F 4 ( a l > -2 -2 0 0 0 0 -3 2 F4 0 0 0 0 0 0 1 0 BIBLIOGRAPHY [l] N.Bourbaki ,Groupes et Algebres de L i e ,Chapitres 4,5 et 6, Hermann,Paris,1968 . [2] R.W.Carter,Conjugacy Classes i n the Weyl Group,Seminar on Algebraic Groups and Related F i n i t e Groups,Springer Lecture Notes No.131,1970. [3] E.B.Dynkin,Semisimple subalgebras of semisimple L i e Algebras.Mat. SbornikN.S. 30 (72),349-462 (1952). (Russian).(Amer. Math. Soc. Transl.,(2) 6 (1965), 111-244). [4]T.Kondo, Characters of the Weyl Group of Type F ^ J . F a c . • S c i . Univ. Tokyo, Sect. I, 11 (1965), 145-163. [ 5 ] I.G.MacDonald,Some Irreducible Representations of Weyl Groups, B u l l . London Math. S o c , 4 (1972), 148-150.

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