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MacDonald characters of Weyl groups of rank ≤4 Andreassian, Agnes 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  University  of Beirut,  1968  A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF SCIENCE  i n the  department of  MATHEMATICS  V7e a c c e p t required  this  thesis  as  conforming to  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA August  1973  the  In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission f o r extensive copying of this thesis for scholarly purposes may he granted by the Head of my Department or by h i s representatives.  It i s understood that copying or publication  of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of  MoiMruL^AAi- cs  The University of B r i t i s h Columbia Vancouver 8 , Canada  Date  R^dt  53  , 19 73  Supervisor:  Dr.B.Chang  ABSTRACT  In t h i s characters  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  o f W e y l 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 m e t h o d e n a b l e s us t o  find  a n d t h e r e m a i n i n g o n e s may be o b t a i n e d characters reflection  irreducible  and c h a r a c t e r s representation.  almost  a l l the  characters,  by c o m b i n i n g M a c D o n a l d  of e x t e r i o r products  of  the  iii  TABLE OF CONTENTS  INTRODUCTION A  2  A  3  A  4  4 5 7 11  D. B  B  2  4  1  2  18  a n d C^  20  and C  26  and C  4  41 BIBLIOGRAPHY  61  iv  LIST OF TABLES  Table I : C h a r a c t e r Table f o r (A )  .  .  . 4  Table I I .-Character Table f o r (A )  .  .  . 6  Table I I I : C h a r a c t e r T a b l e f o r (A ) 4  .  .  .10  Table IV: C h a r a c t e r Table f o r (D )  .  .  .16  .  .  .19  Table V I : C h a r a c t e r Table f o r ( B ) = ( C )  .  .24  Table V I I : Character Table f o r (B )=(C )  .  .38  Table V I I I : C h a r a c t e r Table f o r (F.)  .  .58  2  3  4  Table V: C h a r a c t e r Table f o r ( B ) = ( C ) 2  2  3  3  4  4  V  ACKNOWLEDGMENT  I would l i k e t o express my g r a t i t u d e t o Dr. B.Chang f o r suggesting t h e t o p i c o f t h i s t h e s i s and f o r a l l t h e help he so r e a d i l y o f f e r e d d u r i n g i t s p r e p a r a t i o n . I would a l s o l i k e t o thank Dr.R.Ree f o r r e a d i n g the t h e s i s .  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 Groups"  representations  i n general  o f a Weyl  not a l l of  field,and  suppose V has  the product of  homogeneous all  the subspace of  g(n(S))  absolutely  The p u r p o s e o f t h i s compute  E spanned  thesis  is  characters  characters  that  obtained  •MacDonald  characters.  U s i n g t h i s m e t h o d we f i n d the case of the groups nor the  the  act  and  n(S) II (S)  the  is. a  symmetric  on E,and l e t  that  t o use of  this  P(S)  P(S)  functions is  an  are  subsystems o f b o t h B.  irreducible characters.In  to rank  c a n be so o b t a i n e d  related  to those  b y t h i s method w i l l  be  and  obtained.The called  a l l the i r r e d u c i b l e characters  (A^).We f i n d  of  construction  t h e Weyl groups o f  that  n e i t h e r the  s u b s y s t e m s o f C^ a l o n e  irreducible characters  (R)  (R)-module.  whether the m i s s i n g  alone  (R)  f i n d out which characters  are  S.Then  by t h e p o l y n o m i a l  the i r r e d u c i b l e characters  <4.We w i s h t o  of  on V i s  (R).MacDonald proves  irreducible  a Weyl group  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  t h e Weyl group  for a l l g i n  inner product.Let V*  roots  rational-valued polynomial functions  denote  rational  R . F o r a g i v e n o r d e r i n g on R , l e t  rational-valued  E=Sym(V*).Let  of  system of  a l l the p o s i t i v e  algebra  irreducible  space over the  a positive-definite  be t h e d u a l o f V . L e t R be a r o o t S be a s u b s y s t e m o f  the  group.  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  denote  Weyl  [5],I.G.MacDonald describes the f o l l o w i n g c o n s t r u c t i o n  which g i v e s many,but  let  of  the group  subsystems  provide a l l  (B^)=(C^).However,  and C . we o b t a i n a l l t h e . ' the case of  (D.),the  method  in  the using  -  -• gives  2 eleven of the t h i r t e e n i r r e d u c i b l e characters.The products  exterior  of the 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 the  characters.In  the case of  (F^)  we f i n d  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  seventeen  products  missing  characters.The  o f two M a c D o n a l d  characters. 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 used and t h e method o f  o b t a i n i n g a l l the subsystems of a g i v e n r o o t Weyl  Dynkin diagrams  of p o s i t i v e roots  g i v e n by N . B o u r b a k i of  and  roots  and one w i t h  to represent  [1] . A s i n  [2] , i n l i s t i n g  The c o n j u g a c y (A^) , (B^)  to f i n d . F o r R.W.Carter  shorter roots  or  (F^)  a n d S one w i t h  c l a s s e s and t h e o r d e r s (C^)  for  we u s e  [2],and  i=2,3,4,and  the conjugacy  of the  (D^)  are  classes.For  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 In the t a b l e s , t h e  as  subsystems long  denote shorter  denotes  staightforward  notation  the sake o f convenience,we  entries  i n the conjugacy  centralizers  c l a s s e s o b t a i n e d by  i n a l l c a s e s we u s e C a r t e r ' s  conjugacy  F o r any r o o t a , w  by u s i n g S t o  the  f o r subsystems of C^.  for  use  the  ,we d i s t i n g u i s h b e t w e e n a d i a g r a m w i t h  r o o t s . W e do t h e o p p o s i t e  elements  r o o t systems and  i n terms of o r t h o n o r m a l b a s i s  a Dynkin diagram w i t h long roots  for  a  group.  We u s e set  system of  also  changes.  u n d e r h ^ d e n o t e t h e numbers  classes.denotes  a  character.  t h e r e f l e c t i o n d e f i n e d by a.We  the n o t a t i o n V ( a , , . . . , a ) to denote 1 ' n determinant of order n,  t h e Vandermonde  of  .  Given  a  n-1 n  a r o o t s y s t e m R,we u s e D y n k i n 1 s m e t h o d  [3] t o  o b t a i n a l l the subsystems,We  start  system R the minimal  ( R ) . T h e d i a g r a m so o b t a i n e d  called  root of  by a d j o i n i n g t o t h e  the extended Dynkin diagram for R.By d e l e t i n g  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  root is  roots  subsystems.  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 s u b s y s t e m s o b t a i n e d we eventually  find  a l l the subsystems of the g i v e n r o o t  system.  l.A . 2  Subsystems o f  ;  The extended Dynkin diagram f o r  is  -(x -x ) 1  x -x 1  3  x  2  2  _ x  From t h i s diagram we o b t a i n the f o l l o w i n g  3 subsystems  (1) A : 0  X  l  X  2  X  2  X  3  (2) A, X  We note t h a t  l  X  2  II(A ) =-V ( x ,x ,x^) where V(x-^,x ,x ) i s 2  1  2  2  3  the Vandermonde determinant o f o r d e r 3. T a b l e I : C h a r a c t e r Table f o r (A ) 2  A, Conjugacy C l a s s Representative $ A  l  A,  \  1  1  2  3  1  -1  0  2  1  1  -1  h.  (1)  x -2x+l  1  (12)  x  -1  (123)  x + .x+1  2  2  l  \  Characteristic Polynomial 2  A  I  5 2.A . 3  S u b s y s t e m s o f A., The e x t e n d e d  Dynkin diagram  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  (1) A ^ :  o x  (2) 2A  l  x  o  2  X  A  2 3 X  l  X  2  .  X  l~ 2 x  X  3~ 4 X  3  _ X  4  2 3 X  1  X  P(A^):  X  :  2  X  (4)  o  X  X  (3) A  subsystems:  l  x  2  We h a v e II (A^ ) =V (x^ , x 2 , x 3 , x ^ ) , t h e Vandermonde  of  o r d e r 4.  P(2A^): I n t h i s in  determinant  terms  case,it  i s convenient to write the polynomials  of the roots  the x^'s.We  (where  a  i~xi  - x  j_+i)  rather  than  have II (2A^)-a-^OL^ r  and s o P(2A^) i s s p a n n e d b y : b1=a1a3, ^2~ 2( n a  a  + a  2  + a  3^'  b 3 =(a-^+a 2 ) ( a + a ) =b-^+b2 . 2  P(A2):  The g r o u p  following  3  ( A 3 ) a c t i n g on H ( A 2 ) g i v e s r i s e  polynomials:  to the  b-^V ( x ,x ,x ) , 1  2  3  b =V(x ,x ,x ) , 2  2  3  4  b =V(x ,x ,x ), 3  1  3  4  b =V(x ,x ,x ). 4  1  2  4  Then 1  1  1  1  1  1  1  1  x  l  x  x  2  x  3  x  X  2  x  3  x  4 x 4  2  2  shows b1-b2+b3-b4=o  and { b , b , b } may be taken as a b a s i s f o r P ( A ) . 1  2  3  2  Table I I : C h a r a c t e r Table f o r  (A-,) A  Conjugacy C l a s s C h a r a c t e r i s t i c Polynomial Representative $ A  l  A, A  3  2A  X  3  2A  h. l  *o  Xl  x  X  A,  A  l  x  2  (1)  x -3x +3x-l  1  1  1  2  3  3  (12)  x - x - x+1  6  1  -1  0  -1  1  (123)  x  -1  8  1  1  -1  0  0  (1234)  x + x + x+1  6  1  -1  0  1  -1  (12)(34)  x + x - x-1  3  1  1  2  -1  -1  3  3  2  2  3  3  3  2  2  4  3 .A, Subsystems  o f A^:  The extended Dynkin diagram f o r A^ i s (x x ) r  5  From t h i s diagram we o b t a i n the f o l l o w i n g (i) A. :  _  n  o  0  x  l~ 2  x  2  x  3  X  3~ 4  X  l  X  2  X  2  X  3  x  3~ 4  X  l  X  2  X  2  X  3  X  4~ 5  X  l  X  2  X  3  X  4  X  l  x  2  X  2~ 3  x  subsystems  X  X  4  X  5  (2) A : 3  (3) A + A 2  (4)  x  r  X  2A : ±  (5) A, X  A,  (6)  l  X  X  2  P ( A ^ ) : We have n(A4)=V(X ,X ,X ,X ,x ). 1  P ( A ) : The ( A 4 ) - o r b i t  2  3  4  o f II ( A 3 ) c o n s i s t s o f +b  3  where b =V(x ,x ,x ,x ) , 1  1  2  3  4  b =V(x ,x ,x ,x ) , 2  b b  2  = 3  4  3  4  5  V ( x , x , x , x^) , 3  = V ( x  5  4  l' 2' 4 X  X  b ^ V (x-^,X x =  2f  3 f  , X  5* '  x^) .  ±  (i=l,...,5)  As i n the case o f P ( A 2 ) o f  and  { b ^ , b  P(A2+A1):  2  , b  }  3  (A^),we have  forms a b a s i s  for P(A3)  We h a v e (x4-x5)V(x1,x2,x3)  n (A +A )=2  1  Hence P ( A 2 + A ^ ) i s b  l  (x4- X5 V ( x x , x 2 rx3)  =  V b  b  3 4  =  X  5  (x2-  X  5 V (x  X  5  =  V V  V(x1,x2  (x3-  (x  b  spanned b y :  r  X  4  (x2-  X  4  (x3-  X  4  (x  X  3  r  l  x  f  3  rX ) 4  V(x2,x3 )V(x2,x3 .x5) V ( x 1 , x 3 rX ) 5  V ( x 1 , x 2 rX ) 5  >V(x2,x4 rX ) 5  9= ( x 2 - 3 ) V ( x l f x 4 ' 5 x  x  ho  X  2  )  ) V ( x 3 , x 4 rX ) 5  We h a v e b -b b -b =o 1  2 +  3  4  w h i c h c a n be s e e n f r o m t h e X  1"X5 1  Similarly,we  x  •  X  l  xX  2  l  have  2  X  5  1 x  2  x|  identity: X  3  x  1  5  x4-x 1  x  3  x  4  X  3  x  4  10 1_  1_  1_  T_  2  5  6  1—  1' 2' 3' 5' 6 a r e e a s i l y  seen t o be l i n e a r l y  independent,  and hence form a b a s i s f o r P(A +A^). 2  P(2A^): W r i t i n g t h e polynomials i n terms o f the r o o t s a where a.=x.-x.,,,we have IT (2A ) '=a 3 • a  1  1  I t i s easy t o see t h a t P(2A^) i s spanned by the f o l l o w i n g linearly  independent a  l 3'  a  l 4'  elements:  a  a  a a , 2  4  a  2 ^ l  + a  2  a  3 ( 2  + a  3  a  a  + a  + a  3^' 4) •  P ( A ) : A b a s i s f o r P ( A ) i s g i v e n by t h e p o l y n o m i a l s : 2  2  -IT (A )=V ( x , x , x ) , 2  1  V(x ,x ,x ) , 1  2  4  V(x ,x ,x ), 1  2  5  V(x ,x ,x ), 1  V (x  3  i r  x  3  4  ,x )', 5  V(x ,x ,x ). 1  4  5  2  3  •  10 T a b l e I I I : C h a r a c t e r T a b l e f o r (A.)  A  4  A  3  A  l  A+A 2A 2 1  X  A  2  h. l  *0  Xl  (1)  x'*-4x + 6x -4x+l  1  1  1  4  4  5  5  6  (12)  x"-2x  3  +2x-l  10  1  -1  -2  2  -1  1  0  (123)  x *- x  3  - x+1  20  1  1  1  1  -1  -1  0  (1234)  x*  -1  30  1  -1  0  0  1  -1  0  (12345)  x"*+ x + x +  x+1  24  1  1  -1  -1  0  0  1  (12)(34)  x"  +1  15  1  1  0  0  1  1 -2  (123) (45)  x + x  - x-1  20  1  -1  1  -1  -1  2  A  3  A  4  A  A  Conjugacy C l a s s C h a r a c t e r i s t i c Polynomial Representative  l  A  <S>  2  +  A  l  3  1  3  lf  2  2  -2x 3  2  x  2  *3  x  4  *5  1  0  4.D  4  Subsystems o f D 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 D^ i s Q *  x  x  r  2  o-  *  3  4  o x +x. 3 4 0  x -x 2  3  - (x +x ) 1  2  F r o m 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 c 3 4 > X  (1)  D4  D  v X  4A : 1  A 3  A  1" 2 X  Al:  X  x +x 3  1" 2 X  x  l  4  0  0  o 1~ 2 X  l" 2 X  A": •j x +x  +x  2  X  2" 3  X  2" 3  x  3A : X  X  x -x 3  2 -x  4  1~ 2  X  3~ 4  X  2" 3  3  o X  x  :  X  2A : ±  o 1~ 2 X  x  0 X  2A£:  0  1~ 2 X  X  3  0 X  + X  4  0  1~ 2 X  3~ 4 x  o x +x 3  x  l  + X  4  \j  x  o X  x  '"" '" \J 4  3  A 2  2" 3  /->  X  A  */  :  X  (7)  X  0 X  (3)  X  i*•  :  subsystems:  2  x  3~ 4 x  o x +x 3  4  x +x 3  4  12 (10)  2A": x  o  o  xx-x2 (11)  A,: 1  P(D^):  x3-x4  o X 1~X2  We h a v e n(D4)=V(x?L,x|,x|,x|)  •P(4A1): A basis n  f o r P (4AX) c o n s i s t s  The  of  (4A1) = (x21-x|) ( x 2 - x 2 ) ,  (x'-x2)  P(A3):  .  (x'-x*).  ( D 4 ) - o r b i t of  ^ A ^ consists  of  + b.^  shows  that  where b1=V(x1,x2,x3,x4), b2=V(x-^fx2,—x3,—x4), b3= V ( x 1 , - x 2 , x 3 , - x 4 ) , b  4  = V ( x  l'  - X  2'~X3'X4*'  A simple c a l c u l a t i o n of determinants b1+b2+b3-b =o 4  and b ^ , b 2 , b 3  are  linearly  independent.  P ( A l ) : We h a v e II ( A ^ ) = V ( x 1 , x 2 , x 3 , - x 4 ) , a n d so P(-& 3 ) i s  spanned  by:  b^—V (x-j^, x 2 , x 3 , —x4 ) , b  2  =  b3~V  V (x-^ , x 2 , —x3 , x 4 ) , (x-j^, — x 2  , x3  , x4  ) ,  4  (i=l,,..,4)  b4=V(-x1,x2,x3,x4). As above,a  s i m p l e c a l c u l a t i o n shows  that  b1+b2+b3+b =0 4  and {b-^,b2,,b3}  forms  a basis  for  P(A3).  P (A 3 ') : We f i n d  This case i s  n(A3')=-V(x2,x|,x|)  .  analogous to that  of P(A2) of  obtain a basis consisting  ( A 3 ) , a n d we  of:  V(x2,x2,x2), V(x*,x2,x*), V(xJ,x*,x|).  p (3A^) : We h a v e II (3A ) = ( x 1 - x 2 ) ( x 3 - x 4 ) . 1  The e l e m e n t  (12) [-1 1 - 1 l ] (xx+x2)  Then is  (x1~x2)  ( x  x  +  ( x  l  + x  2  )  i n P ( 3 A ) . S i n c e P(3A ) i s 1  1  ( D 4 ) - o r b i t of x 1 ( x | - x 4 ) P ( 3 A 1 ) . W e c a n now e a s i l y 2 _  X, X.  2  2  xn X  x  X. x  )  2.  3~ 4 ' X  ;  l" 3 ' 2 2 x  J  3~ 4 ' x  ;  X.  1~ 2 ' x 2 _ l 22 > ' 2 4 '  X  x,  X  x  x  '(D 4 )  a c t i n g o n n(3A ) 1  gives  (x3-x4).  3~ 4)  x.  of  )  ;  ;  ( x  3~ 4)=2xl x  ( x  3~ 4 x  }  i r r e d u c i b l e , w e may t a k e  instead  of  that of  II(3A ) t o  w r i t e down a b a s i s a s  1  the span  follows:  X  4 ^X2  X  P "  P ( A 2 ) : We h a v e H (A2)=-v(x1,x2,x3) We n o t e t h a t  N  ( A 2 ) is  i n P(3A^)  above, 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 (A2)=P(3A1) .  P(2A1):  We h a v e n(2AX)=(x1-x2)(x3+x4).  Writing this for  i=l,2,3  i n terms of  the roots  of D^,where  and a ^ x ^ + x ^ w e can e a s i l y f i n d  P(2A1),such  a  a basis  -j_  = x  i  - x  i+i  for  as a  a  la4' 3 ^al+2a2+a3+a4) i  a (a +a +a ). 2  P(2A|):  1  A basis  2  for P(2A|)  2  is  g i v e n by  2  x  l~x2'  x  l  X  l  2  2 x  3'  X  4'  2  P(2A£):  4  2  We h a v e n (2A^) = ( x 1 - x 2 ) ( x 3 - x 4 ) •  Again,it  i s more c o n v e n i e n t t o u s e  We may t a k e a  la3'  the  rather, than the x  a  4 ( 2. a  a (a 2  as a b a s i s f o r  1 +  + 2 a  a  2 +  2  + a  3  + a  4^ '  a ) 3  P(2A^).  In c a l c u l a t i n g the characters,the us  t o s i m p l i f y our The  group  corresponding  (D^)  following  fact  enables  computation. contains  a central  t o 4 A ^ ) . T h e n f o r any  involution  z(the  c h a r a c t e r x and  element  element  g o f t h e group,we have X(zg)=ex(g) where Xd) 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 e v e n o r  odd  degree. D e n o t e by  [x] an  element i n the conjugacy c l a s s  by X i n t h e t a b l e . T h e n  we  have the  relations: z=[  4 A  i] ,  [2A|1 ~ z [ 2 A j ] , [Aj  [3Aj ,  ~Z  L 4 l>] D  (a  ~s[Vl>] ' a  [A ] ~ z [ D ] , 2  4  [2A").z[2Ag, [2Aj [AJ]  [2AJ ,  ~Z  [A'] ,  ~Z  [ 3]~ [ 3]' A  Z  A  [A-] -z [A'] .  following  designated  conjugacy  Using the subsystems o f  we o b t a i n a l l but two o f the  i r r e d u c i b l e c h a r a c t e r s o f (D.).  T a b l e IV: C h a r a c t e r T a b l e f o r  (D„). $  Conjugacy C l a s s Representatives $  4 A  1  l  A  A  l) A" J 3 A 3  C h a r a c t e r i s t i c h. X Polynomial  A  4  Xl  4 A  1  x  2  2 A  x  1 3  2A^ 2A^ x  4  X  5  (1) [1111]  x - 4 x + 6 x -4x+l  1  1  1  2  3  3  3  (1) [-1-111]  x^  +1  6  1  1  2 -1  3  -1  (1) [-1-1-1-1]  x' + 4x + 6x +4x+l  1  1  1  2  3  3  3  (12) [1111]  x"-2x  +2x-l 12  1  -1  0  1  1  1  -1 24  1  -1  0 -1  1  -1  12  1  -1  0  1  1  1  3  - x+1 32  1  1  -1  0  0  0  3  + x+1 32  1  1  -1  0  0  0  4  3  2  -2x  t  3  2  2  3  (12) [-11-11]  3'  D  (12) [-1-1-1-1]  x"+2x  (123) [ l l l l ]  x"- x  -2x-l  3  J  2  D  (123) [-1-1-1-1] x + h  4  2 A  2A  x  -2x  2  +1  6  1  1  2 -1  -1  3  (12) ( 3 4 ) [ - l - l l l ] x  -2x  2  +1  6  1  1  2  3  -1  -1  +1 12  1  1  2 -1  -1  -1  (12) (34) [ l l l l ]  1  h  h  1  D ( a ) (12) ( 3 4 ) [ - l l - l l ] x 4  x  <t  2  +2x  x  <• A  3  A  3  (1234) [ l l l l ]  x  -1 24  1  -1  0 -1  -1  1  (1234)[-l-lll]  X  -1 24  1  -1  0  1  -1  -1  Table I V : Character Table for A  X  3 6  A  1  l  a"' 3 l ) Aj J 3  A  A  2  D  4  2A£ 2AX  A3 A  3  X  3 7  A  X  3 8  A  3A,  l  x  9  X  10  x  l l  A  2  X  12  3  3  4  4  6  8  -1 -1  3  0  0  -2  0  3 -4  -4  6  -8  2 -2  0  0  0  0  0  0  -1 -1 -1 -2  2  1  3  4 A  A  (D.) C o n t i n u e d .  3  3  -1 -1 -1 1  1 -1  0  0  0  0  1  0  -1  0  0  0 -1 -1  0  1  0  0  -2  0  3 -1  0  0  -2  0  -1 -1 -1  0  0  2  0  -1  3 -1 -1 -1  1  1  0  0  0  0  1 -1  1  0  0  0  0  5. B  and  2  Subsystems  C  2  of  B : 2  The extended Dynkin diagram f o r B  2  is -JO  x  l*~ 2 x  x  From t h i s diagram we o b t a i n ( 1 )  B  2  (x +x ) ±  the f o l l o w i n g  1~ 2 X  X  2  2A  1  x  r  x  -(x  2  1 +  x ) 2  (3) A : x  X  (4) A  l" 2 X  1  x  P (B ) : We 2  2  have H (B )=x x (x£-x|) 2  1  2  P (2A ) : P ( 2 A ) i s spanned 1  by  1  n(2A )=-(x -x ) . 2  2  1  Subsystems  of  C : 2  The extended Dynkin diagram f o r C  2  is •3  -2x  We o b t a i n  x 1  the f o l l o w i n g  l~ 2 x  subsystems:  2  subsystems  ,,  :  X  (2)  2  2 x  2  X  2x  1~" 2 X  2  (2) 2A : X  9  9  -2x^ (3) A  o  1  X  l~ 2 x  (4) A : x  2x  2  Combining the P(S) f o r the subsystems S o f B we o b t a i n a l l the c h a r a c t e r s o f  and o f C ,  0  0  (B )=(C ). 2  2  T a b l e V: C h a r a c t e r Table f o r ( B ) = ( C ) . 0  o  B  $  C  2  2A  A  l  A  l  A  l  1  2A  2  1  l  A  Conjugacy C l a s s Representative 2 c B  l 2A  B  h.  (1) [11]  x -2x+l  1  1  1  1  1  2  (1) [-11]  x  2  1  -1  1  -1  0  (1) [-1-1]  x +2x+l  1 .1  1  1  1  -2  (12) [11]  x  2  -1  2  1  -1  -1  1  0  (12) [-11]  x  2  +1  2  1  1  -1  -1  0  l o x  X  l  x  2  X  3  X  4  2  A  A  Characteristic Polynomial  l  A  l  2  C  2  1  2  2  -1  2  Note:The f i r s t l i n e o f the t a b l e g i v e s the subsystems o f B_ and t h e second t h e subsystems o f C r  20 6.  and  C^.  Subsystems o f B.,: The extended Dynkin diagram f o r B^ i s ?-(x  X  1~ 2 X  We o b t a i n the f o l l o w i n g (1)  B : 0  x ) 2  2~ 3 X  subsystems:  oX  (2)  X  1 +  x  l" 2 X  2  X  3  A : 3  -o x -x 1  (3) 2A +A 1  x  2  2  x  -(x  3  1  X  l  x  (4) A.  -(x  2  l~ 2  x  x  (5) B,  o.. -—-7 x  2  X  3  X  l  x  2  x )  x.  2  o  0 x  1 +  *  —  2~ 3 x  ® X,  A +A :  (6)  1  (7)  1  X  2A  o  n  (8) A  3  X  l  X  2  x  l  x  2  -(x +x ) 1  2  n  (9) AX.  P(B ): 3  We  have n ( B  3  ) =  ~ l 2 3 X  X  X  V ( x  i  f X  2' 3 * x  )  1 +  x ) 2  P(A3):  It  is  e a s y t o see  that P(^3)  is  spanned  by  IT ( A ) = V ( x , x , x ) 2  2  2  3  P ( 2 A 1 + A 1 ) : We h a v e n (2A1+A1)=-x3 (x^-x2) . Hence P(2A^+A^) x  l  X  2 ^ i  X  which are  P(A2):  is  i  by:  2~X3)'  ( x  3  spanned  2  -  _  1  ^ '  5  2  x  2  linearly  \  '  independent.  We h a v e (A2)=-V(xlrx2,x3)  n and  V(xlfx2,x3)-wx^ (V(x1,x2,x3) ) =V(x1/x0,x^)-V(x1,x2,-x3) •l'"2'*3 1 1 0 =2  x  l  x  2  x  x  !  x  2  0  3  =2x3(x2-x2) =-2n(2A1+A1) This  P ( B  2  shows t h a t  ) :  P(A2)=P(2A1+A1)  We f i n d n (B )=X X3(x2~x3) . 2  Thus P ( B ) 2  is  2  s p a n n e d by t h e f o l l o w i n g  elements: X  2 3 X  ( x  2~ 3 ' x  )  linearly  independent  X  1~ 2  1 2 X  {X  X  }  P ( A + A ) : The ( B ) - o r b i t o f I l f A ^ A ^ ) c o n s i s t s o f 1  1  3  x^Xj-x^)  and +x.(x.+x, ) where i , j , k a r e d i s t i n c t elements o f {1,2,3) — 1 j K Therefore X  1 2'  X  1 3'  X  2 3  X  X  X  form a b a s i s f o r P ( A + A ) . 1  P (2A ) : We  1  have  1  n(2A )=-(x - 2) , 2  1  x  and a b a s i s f o r P(2A^) i s g i v e n by: 2 2 l 2' 2 2 l 3* of C x  x  Subsystems  _  x  x  3  The extended Dynkin diagram f o r C  3  is  3= 2  x  l  x  l2 x  X  2 3 x  2 x  From t h i s diagram we o b t a i n the f o l l o w i n g (1) 3  l~ 2 x  X  2 2~ ~ 33 X X  (2) C2+A-L  2 x  3  u •2x  (3) A  subsystems  —o  o x  x  2~ 3  X  2  1  x  2  X  l" 2 x  X  3  3  2 x  3  x  2 3 X  2 x  3  <>V 5  2  -2x-^ <> 6  l  A  (7) A  +  A  l  2x^  :  i :  2x„ (8) A : 2  l  X  X  2  We next c o n s i d e r subsystems  o f t h e above.From t h e extended  Dynkin diagram o f C^+A^, 9  •2x  w  >  ~^ 2 x  1  Q x  /  2~ 3 x  -n 2 x  3  we o b t a i n t h e subsystem (9) 3A^:  &  0  "~2x^  ^^3  P (C +A ) : We have 2  1  II (  C  + A 2  i^  =  -  8  x  i 2 3 ^ 2~ 3^ * X  X  X  X  Hence a b a s i s f o r P (C^+A-^) i s g i v e n by; X-j^X^X^ (x^—:X ) 2  1 2 3  X  X  P ( 3 A ) : P (3A ) 1  1  X  ( x  e  l~ 3 ' X  )  i s spanned by  IT (3A ) = 8 x x x . 1  We e note t h a t X 3  1  A  N  (  2  3  3 x ^ f O ^ t a i n e d from subsystems  o f C^,  cannot be o b t a i n e d u s i n g subsystems o f B^,whereas x  an< 2  ^ X5  o b t a i n e d from subsystems of B^,cannot be o b t a i n e d from C^. The group  (B^) c o n t a i n s a n o n - t r i v i a l c e n t r a l  the element c o r r e s p o n d i n g  element,  t o 2A^+A^.As i n the case o f (D^),  denoting t h i s c e n t r a l  element by z and an element i n the  conjugacy  [x] ,we observe  c l a s s X by  the f o l l o w i n g  conjugacy  relations: [A-J]  ~Z [2Aj  [Aj  ~Z [  [Aj  ~Z [ B J .  A  ,  I + A J  ,  Table V I : C h a r a c t e r Table f o r (B-) = (C.J  Conjugacy C l a s s Representative 3  B  $  l  A  2A  2A  X  2A +A 1  A  B  ] 3 A  l  A  l  2  C  2  A  l  A  3  A B  l  A  2  3  +  A  l  1  1  A  l  +  A  l  C  2  +  A  l  A C  B  3  <D  C  3  X  l  h. X 0 X  A3 3 A  x  2  X  1  3  3  C  $  Characteristic Polynomial  $  2  3  (1) [111!  x -3x +3x-l  1  1  1  1  1  (1) [ l l - l ]  x -  x - x+1 2  3  1  -1  1  -1  (1) [1-1-1]  x + x - x-1 2  3  1  1  1  1  (1) [-1-1-1]  x +3x +3x+l  1  1  -1  1  -1  (12) [111]  x 3  x - x+1  6  1  -1  -1  1  (12) [-111]  x -  x + x-1 2  6  1  1  -1  -1  (12) [ l l - l ]  x + x - x-1  6  1  1  -1  -1  (12) [ l - l - l ]  x + x +  x+1  6  1  -1  -1  1  (123) [111]  x  -1  8  1  1  1  1  +1  8  1  -1  1  -1  (123) [-111]  3  3  2  3  3  3  3  3  3  2  2  2  2  25 Table V I : Continued  2A  C  2  x (1) [111] (1) [11-1] (1) [1-1-1] (1) [-1-1-1]  +  A  4  l X  5  1  A  l  A  l  A  l  A  l  A  X  6  X  A  2  2A A  B  2  2  C  2  7  X  8  1 +  1  • V  A  l  x  +  1  A  A  l  9  2  2  3  3  3  3  -2  2  1  1  -1  -1  2  2  -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 o f the t a b l e g i v e s the subsystems o f B and the second the subsystems o f C-..  7. B  and C .  4  4  Subsystems  of B : 4  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 B  4  i s  9-(x +x ) 1  l 2  X  X  2  x  2  3  X  3  X  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  (2) A + A : 3  1~ 2 X  x  2  X  2~ 3  X  3  3  X  X  4  ^  1  l 2  X  x  (3) D,  (x +x )  X  1  2  9- ( x + x ) 1  2  -o x  (4)B„+2A, : 2. 1  l~ 2 x  O X  1~ 2 X  (5) B : 3  2~ 3  X  X  -(x  1 +  x ) 2  X  3  X  3~ 4  X  4  X  -Cr x  2  3  x  (6) B +A : 2  1  2  3~ 4 X  x  az X  (7) A + A  X  l 2 X  X  3~ 4 X  x ©  1  X  1~ 2-  X  l 2  X  X  2  X  3  X  x  2  X  3  l 2  x  2  x  3  X  x,  (8) A. -(x  1 +  x ) 2  (9) A'  (10) 2 A + A : 1  1  x  X  Q  X  1~ 2 X  - (x +x ) 1  2  3  x  4  27 (11)  3A :  (12) A  (13)  Q  ±  2  X  1"X2  X  2"X3  "  ( X  1  + X  X  3  X  2  )  X  3  "X4  : 4  2A : x  r  x  ~(x + 2 x  2  )  1  (14) 2A| l  x  (15)  A1+A1:  X  X  4  3~ 4 X  X  4  ±  X  (18) A  3~X4  ^  Q  X  A  X  2~ 3  B2:  (17)  2  D  X  (16)  X  x  l"X2  : X  4  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 B2+2A-L, O  O  x1-x2  Cn  -(x1+x2)  x  3~x4  x  4  -(x3+x4)  we o b t a i n t h e s u b s y s t e m : (19) 4A  1 x  P(B4):  l~x2  ~(x1+x2)  P ( B 4 ) i s spanned by H(B4)=x1x2x3x4V(x2fx2,x3,x2)  P(A3+A^):  We h a v e  x3-x4  -(  x  3  + x  4  )  n (A +A )=x V(x _,x ,,x ) , 2  3  1  2  2  4  and so PCA^+A^ i s spanned  by:  xjV(x '^3'^4^' 2  x V(x ,x*,x ), 2  2  2  x^V (X-^ , x f x^) , 2  x^V (x-^ x , x^) . f  2  I t i s easy t o see t h a t these are l i n e a r l y  P ( D ) : P(D^)  i s spanned  4  by one  independent.  element:  n(D )=V(x ,x ,x ,x ). 2  2  2  2  4  P ( B + 2 A ) : We 2  find  1  IT ( B + 2 A ) = - x x (x£-x ) ( x - x ) . 2  2  1  3  4  2  4  I t can be seen t h a t the f o l l o w i n g elements are independent and span P (B +2A^); 2  x.^x (x^—x ) ( x ~ x ) , 4  2  3  4  x.^x ( x ^ x ) ( x x ) f —  _  2  2  3  4  ^2 4 ^ l ~ ~ 3 ^ ^ 2~~ 4 ^ ' X  x  X  x  X  x^x (x^—x ) (x^ x ) , -  4  X  P (B ) : We  4  2 3 ^ l X  X  3  4 ^ ^ 2~ 3 ^ *  — x  x  x  find  3  ir(B )=x x x v(x 3  hence P ( B ) 3  2  3  4  i s spanned  2  2,x ), 2  / X  by:  X  2 3 4^^ 2' 3' 4^'  X  1 3 4^^ l' 3' 4^'  X  2. 2 4^ ^ l ' 2 ' 4 ^ '  X  X  X  X  X  X  X  x  X  X  X  X  X  X  X  linearly  x-j^x2^ 3 V ( x i , x2/• x^) • We o b s e r v e t h a t t h e s e  P(B2+A1):  are l i n e a r l y independent.  We h a v e n (B2+A1)=x3x4(x|-x4)  (x1~x2).  Since (I-wx P(B2+A^)  ) (x3x4(x3~x4) (x1-x2)=2x1x3x4(x3-x4)  is  spanned by t h e  ( B 4 ) - o r b i t of  We c a n now e a s i l y w r i t e down a b a s i s as  P(A2+A1);  X  1 X 2 X 3 <*i - * 2  X  1X2X3  X  1X2X4  -x|  x  ix2X4  -4  X  1X3X4  "*3  X  1X3X4  -4  X  2 X 3 X 4 (x| "*3  X  2 X 3 X 4 (x»  ,  x^x^x^(x^-xp follows:  "«!  We h a v e 11 ( A 2 + A 1 ) = x 4 V ( x 1 , x 2 , x 3 )  In t h i s case i t i s  easier  to consider the  ( B 4 ) - o r b i t of  (I-wx )(n(A +A ))=2x1x4(x2-x2) 2  r a t h e r than t h a t of  II (A^+A^)  given by: X  1X2  X  1X3(x2~X4)'  X  1X4(x2~X3)'  X  2X3  ( x  ( x  3"x4''  l  _ x  4)'  1  .Then a b a s i s  for P(A2+A1)  is  X  2  X  4  (  1  X  X  3  )  '  x^x^ (x.^—x )• 2  P ( A ) : We  find  3  n ( A ) = V ( x ,x ,x ) . 2  2  2  3  ( B ) - 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  The  4  1T(A2)  w i t h each x^ r e p l a c e d by x? . T h e r e f o r e , a b a s i s f o r  P(A3)  i s g i v e n by: V(x ,x ,x )., 2  2  2  Vfx* ,x*,x ) , 2  VCx^x  P ( A  3  ) :  ,x ) .  2  2  We have n (A3)=V(X ,X ,X ,X ) . 1  2  3  4  Let Q=(I-w  ) (n(A') ) ,  v  3  X  R=(I-w  ) (Q) ,  v  X  S=(I+v/ X  4  ) (R) .  2  W r i t i n g these i n determinant form we see t h a t S=8II ( B 2 + 2 A 1 ) .  Therefore  P ( 2 A  1  + A  1  linearly  P(A^)=P(B2+2A1).  ) :  I t can be seen t h a t the f o l l o w i n g elements a r e  independent and t h a t they span -n(2A1+A )=x (x -x ), 2  1  • X  X  x  4  (x-^— ) x  4  l l^  3  ^2~*l) X  2*~  X  f  ' 4 ^ '  2  P ( 2 A  1  + A  1  ) :  31 x  2  X  2 ^Xl~X4 ^ '  (  l x3* '  x  (x.^—x2) / x^  (x-^""X^) .  P ( 3 A 1 ) : We h a v e n(3AX)=-(x^-x2)(x3-x4), and .2  (I-wx  „2  )(n(3A1))=2x4(x£-x^) =7-211 ( 2 A 1 + A 1 ) .  T h e r e f o r e P (3A1)=P (2A1+A]L) .  P(A2):  We h a v e n(A2)=-V(x2,x3,x4),  and ) ( w X i - X 3 (n ( A 2 }  (I-w  ) ) = 2 x  4  ( x  l  _ x  2  )  =-211 ( 2 A 1 + A 1 ) H e n c e P ( A 2 ) =P ( 2 A 1 + A ] L ) .  P(2A-^): A b a s i s  f o r P ( 2 A ^ ) i s g i v e n by  -n (2A )=xJ-x , 2  1  2 x  2  l~x3' 2  2  x -x . 1  P(2A|):  4  Let c1=II(2A|)=(x1-x2) ( x 3 ~ x 4 ) , .C2 C  3  = W  =  W  X X  2  4  ( C  1  )  =  ( X  ( C  1  )  =  ( X  1+X2)  ( X  3~X4) '  1~X2) <  x  + x 3  4 ' }  C  We  4  =  1  + C  x  W  (  3  c  2  )  =  (  l  X  +  2  X  )  (  X  3  +  X  4  }  have C  2  3  + C  4  + C  = 4 X  1 3' X  a n d 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£):  P(A +A ): 1  1  x  l 2'  X  1 3'  X  l 4  X  2 3 '  X  2 4'  X  3 4*  We  x  X  X  f  X  X  X  have n ( A 1 + A ) =x 1  showing t h a t  P ( B ) : We 2  4  (x -x ) , 2  3  P(A-j+A^)=P(2A£)  find n(B )=x x (x -x ), 2  2  3  4  4  and P ( B ) i s spanned b y : 2  X  1 2  ( x  X  l  l  X  1 4 ^ 1~ 4  X  2 3  X  2 4  x  X  X  3  l~ 2 X  i  _  X  X  X  3 4 X  2  2  l  X  X  X  ( X  3 X  2~ 3 X  ( X  2  (  2  l X  3  _ X  _  4 2  X  4  I t i s c l e a r that these are l i n e a r l y  P ( 4 A ) : We 1  have n ( 4 A ) = (xJ-x|) ( x - x ) , 2  1  3  independent,  V  and the  ~<±=1,2 , 3) ,where  ( B ) - o r b i t o f IT(4A ) c o n s i s t s o f 4  1  2  b  = ( x  2" 3 x  )  ( x  i~ 4 ' X  )  b =(xj-x5)(x|-xj). 3  As i n P(2A )  of A ,  X  3  b =b 3  and {b^,b }  1 +  b  2  forms a b a s i s f o r P (4A^)  2  Subsystems o f  :  The extended Dynkin diagram f o r  2  x  l  l2  x  x  X  is  2~ 3 X  X  From t h i s diagram we o b t a i n the f o l l o w i n g  3" 4 X  2 x  subsystems:  (1) C, '4  —=*\ x  l  x  2  (2) 2 C : ~ 3  2  i :  x  l  (4)A +2A :  x  (5) C +A : 1  C +A 2  1~ 2  l  X  2 3  "4  X  x  a  © **™2JC-J^  -JC  1~ 2  2X^  2x  ~4  X  x  1  O  2x-^ (7) A :  x ~x 3  x  l~ 2 x  x  2x  4  2 3 X  4  6S> 4  ^  3  -i  x  3 4 x  -  2x  s  *s  4  X  Q  X  (6)  o  —<j  2  2  X  o  1  3~ 4 X  '"'  e  2  1  2 3  :  X  ^  2  (3) C + A  x  4  2x  4  2x  4  s  N  34 (8) C  (9)  x  2~ 3  X  3  X  1~ 2  X  2~ 3  X  3  x  2x,  4  X  A +A : 2  1  X  2x,  X  (10) 2A. X  l  (11) A ^ A ^  4  X  0  X  (12)  2  X  2x,  2~ 3 X  2A : 1  2x  -2x^  4  (13) A,  u < x  2~ 3  X  x  (14) C.  3~ 4  o X  X  BI 2x,  3~ 4 X  (15) A, x  l  x  2  (16) A, 2x. Next,we c o n s i d e r  subsystems o f the above.From the extended  Dynkin diagram f o r 2 C  1 2  o -2x-  x  l  x  2  2x.  -2x.  x  *C 2x,  3~ 4 x  we o b t a i n (17) 4A, :  0  -2x, '1  2x, ""2  -2x. 3  From the extended Dynkin diagram f o r C^+A^  E  2x,  SJC^  2 ~~ ^ 3  ^  ^*3^4  ^ ^* 4  we o b t a i n (18) C +2A : . 2  3  o  Q  ^^2  25C-^  <  ^3 "^4  2x  F i n a l l y , f r o m the extended Dynkin diagram f o r C„+A, Jt—A.  •2x^  ~  2 x  3  X  3  4  - x  we o b t a i n (19)  3A  i :  -2x^  - 2 x  3  2 x  4  P (2C ) : We have 2  n (2C )=-16x x x x (x|-x|) 2  1  2  3  4  (x -x ) 2  2  As i n P ( 4 A ) o f ( B ) , 1  4  X-.X«X~X. ( X -T ~ X ~ ) ( X - — x. ) / 1 2 3 4 :  X  X  X  ^1^2^3^4 (x ~x^) (x^—x ) 2  4  i s a basis f o r P(2C ). 2  P (C +A ) : We 3  find  1  II (C +A ) = 1 6 x x x x V ( x ,x ,x ) . 3  1  1  2  3  4  2  3  4  As i n P ( A ) o f (B ) ,a b a s i s f o r P(C +A-^) i s g i v e n by: 3  4  3  X  1 2 3 4  X  1 2 3 4^^ 2' 3' 4^'  X  1 2 3 4  X  X  X  X  X  X  X  X  P (A +2A ) : We have 1  1  1' 2' 3 '  V ( X  X  X  X  V ( X  X  X  )  X  1' 3' 4 X  X  K  „, 2  x  36 JI ( A + 2 A ) = - 4 x x 1  and  we  1  1  (x -x ) ,  4  2  3  find (I+w  )(n(A +2A ))=-8x x x  x  1  1  1  2  4  Thus X  1 2 3'  X  1 2 4'  X  1 3 4' '  X  2 3 4  X  X  X  X  X  X  X  X  form a b a s i s f o r P(A^+2A^)  P ( 4 A ) : P(4A ) i s spanned by 1  II (4A )  =16x x x x  1  P (C +2A ) : We 2  1  2  3  4  find  1  n(C +2A )=16x x x x (x|-x|) 2  1  1  2  3  4  A b a s i s f o r P (C +2A^) i s g i v e n by: 2  X  1 2 3 4 ^ l " * ! * ' X  X  X  x x x (x -x ), 2  X l  X  2  3  2  4  1 2 3 4( l~ l* ' X  X  X  x  x  P ( 3 A ) : We have 1  II ( 3 A ) = 8 x x x 1  and  1  3  4  P(3A )=P(A +2A ). 1  1  1  We note t h a t X 3 , X 5 / Xg^ X ,and x 9  1 3  o b t a i n e d from subsystems  of C , c a n n o t be o b t a i n e d u s i n g subsystems o f B , w h i l e x ' X 4 » 4  Xg,  4  X-jr & an  from C..  X-^Q  robtained  2  from subsystems o f B ,cannot be o b t a i n e d 4  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 u s e t h e relations: z= [ 4 A j , [ A j - z [2A +A ] , 1  ]  [2AJ ~ z [ 2 A j , [ A j ~Z [3A.J  ,  [ B J ~z [B2+2AJ , [A +Aj x  - Z ^ + A J  ,  [A ].z[A ], 3  3  [ 2]~ f J'' A  Z  D  [B^-zfA^Aj , [2A-] ~ z [ 2 A j ] , [B +Aj -z[B +Aj , 2  2  [ D ( ) ] -^VM 4  a i  [A ]. [A j, 3  [  B  Z  ]~ [ Z  4  3  B 4  1 •  ,  following  3.8 T a b l e V I I : C h a r a c t e r Table f o r (B.) = (C ,). /  $  Characteristic Polynomial  Conjugacy C l a s s R e p r e s e n t a t i v e C  *  4  $ A  4  C  4  X  l  D  4 4 A  X  2  X  1 3  x -4x +6x -4x+l  1  1  1  1  1  (1) [-1111]  x -2x  +2x-l  4  1  -1  1  -1  +1  6  1  1  1  1  (i)  l  h. i  B  [1111]  4  3  4  2  3  2A  1  (1) [-1-111]  x  2A +A 3A  X  (1) [-1-1-11]  x +2x  -2x-l  4  1  -1  1  -1  (1) [-1-1-1-1]  x' +4x + 6x +4x+l  1  1  1  1  1  (12) [ m i l  x"-2x  12  1  -1  -1  1  (12) [-1111]  x"-2x +2x -2x+l  12  1  1  -1  -1  (12) [11-11]  x  +1  24  1  1  -1  -1  (12) [-11-11]  x"  -1  24  1  -1  -1  1  (12) [-11-1-1]  x'* + 2x + 2x + 2x+l  12  1  1  -1  -1  (12) [-1-1-1-1]  x"+2x  3  -2x-l  12  1  -1  -1  1  (123) [ m i ]  x - x  3  - x+1  32  1  1  1  1  (123) [-1111]  x - x  3  + x-1  32  1  -1  1  -1  (123) [ l l l - l ]  x + x  3  - x-1  32  1  -1  1  -1  (123) [-1-1-1-1]  x + x  3  + x+1  32  1  1  1  1  (12) (34) [1111]  x  4  +1  12  1  1  1  1  d 2 ) (34) [-1111]  x  4  -1  24  1  -1  1  -1  (12) (34) [-11-11] x  4  +1  12  1  1  1  1  (1234) [ l l l l ]  x  4  -1  48  1  -1  -1  1  (1234) [-1111]  x  4  +1  48  1  1  -1  -1  1  1  4 A  A  B  A  l  2  C  2  A  l  +  A  l  C  2  +  A  l  l  A  3 2 3  A  1  4 A  l  A  B  1  +  3A  A  l  A  +  2  + A  i C 1J  3  A  2  A  1  D  4  A  l  2A| 2  B  V  a  +  A  2 +  1  A  +  4  3  f  2  3  2  +2x-l  3  3  2  -2x  4  2  A  2  B  -2x  4  2A  + 2 A  C  3  A  2  +  A  l  C  3  +  A  l  C  1  2  '  2A£ l  1  2  +  l > 2C  A  3  A  3  B  4  C  4  2  A  l  3  4  4  4  4  2  -2x  +2x  2  2  Table V I I : Continued  4 A  2A  1 2C  x  4  X  l  A  3 3 L  2  5  +A  X  6  X  7  X  8  x  9  X  1Q  A  l  B  3  A  l  C  3  A  l  A  l \  X  3A +  2  A  1  l  l l 12 , 13 K  X  (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  40  Table V I I : Continued A  B  2  +  2  A  A1+A  3 1  A  2  +  A  2A.  l  A 3A|  1  .  B  2  2A1 AX+A: A  X  3  14  A  2  + A  X  1  1 5  2A.  X  16  .  C  X  2  17  B  2  +  A  l  C  2  +  A  l  C  2  +  A  l  X  18  2  A  1  +  X  [im]  6  6  6  6  (i)  [-1111]  0  0  0  0  (i)  [-l-iii]  -2  -2  -2  -2  0  (i)  [-1-1-11]  0  0  0  0  4  -4  (i)  [-1-1-1-1]  6  6  6  6  -8  -8  0  2  0  0  0  2  0  0  0  0  -2  -4  4 0  [1111]  (12)  [-1111]  0  -2  0  (12)  [11-11]  0  2  0  (12)  [-11-11]  2  0  -2  0  0  0  (12)  [-11-1-1]  0  0  2  0  0  (12)  [-1-1-1-1]  0  0  -2  -2  0  2  0'  (123)  [llll]  0  0  0  0  -1  -1  (123)  [-1111]  0  0  0  0  1  -1  (123)  [lll-l]  0  0  0  0  -1  1  (123)  [-1-1-1-1]  0  0  0  0  1  1  0  0  -2  1 9  8  (12)  -2  1  V  (i)  8  A  (12)  (34)  [1111]  2  -2  2  (12)  (34)  [-1111]  0  0  0  0  0  0  (12)  (34)  [-11-11  -2  2  -2  2  0  0  (1234)  [llll]  0  0  0  0  0  0  (1234)  [-1111]  0  0  0  0  0  0  Note: The f i r s t l i n e i n the t a b l e g i v e s the subsystems o f and the second the subsystems o f C..  Subsystems o f  :  extended Dynkin diagram f o r F. i s  The  O  Ox  l~ 2 x  o  2~ 3  X  X  X  13  3~ 4 X  x  From t h i s diagram we o b t a i n t h e f o l l o w i n g (1) F : 4  2  B  x  (3) A + A : 3  1  2  2  x  C  3  +  A  l  2  X  x  r  x  X  3 4  Q  :  o  x -x  2  _ X  2  2  X  2  o  :  x  3  S  >  3  o  x  2 3  x  3~ 4  X  x  x  l S  X  1+  4  x  1+  (10) A + A : 2  X  (11) A + A 2  i :  2  >  x  x  4  4  -%(x +x +x +x ) 1  2  3  4  X  3~ 4  ~ %(  X  x  + x 1  + x 2  3  + x  4)  2  x  2 3 X  X  3~ 4 x  3  x  2~ 3  x  - X  3" 4 x  (x +x +x +x ) 1  2  3  4  0  x  2  1  ^  1  3  4  3  4  -h(x +x +x +x )  4  3  x  2  -J2(x x +x +x )  x  4  3~ 4  x  l" 2 •  l  4  @  (9) A : X  3  -J5(x x +x +x )  4  0  X  4  0  3  3  x -x  2  Q  A  + X  4  #  _ X  2  3  1 +  3  -J* X  x -x  3  1  (7) C :  (8) 2 A  + X  o  x -x (6) B :  2  + X  -3 (x +x +x +x )  4  X  Q  1  1  h( X  subsystems:  o  2 3  0  X  x  4  o  l  x  ( 5 )  3  Q  (4) A + A :  ~  ^  x -x  3  o  : 4  4  o  Q  x -x (2)  —O  x  4  -%(x +x +x +x ) 1  2  3  4  2  3  4  (13) A  x  l 2  x  2" 3  x  X  3 4 X  X  4  2  x  X  3  X  4  (14) A : 2  x (15) 2 A  -h ( x + x + x + x )  4  i :  1  2  3  4  0  X  l 2  '  X  X  3  X  4  (16) A ^ A - ^ x -x 3  < ) 17  B  x  3 4  2  3  4  X  X  4  1  X  (19)  1  :-  : 2  (18) A  -%(x +x +x +x )  4  l 2 X  A. ±  ®  x, 4 Repeating t h e process w i t h the above systems o f r o o t s we f i n d f o u r subsystems t h a t a r e not congruent under ( F ) 4  t o any o f the above,as seen below. The extended Dynkin diagram f o r B  4  is  o - (x +x ) 1  2  -o x  l 2 x  x  2 3 x  x  from which we o b t a i n (20) D.:  o -(x  1 +  x ) 2  3 4 X  x  4  The extended Dynkin diagram f o r C^+A^ i s  x  l~ 2 x  x  3" 4 x  x  4  l  ( x  + X  2  + X  From t h i s diagram we o b t a i n the f o l l o w i n g (21) B + 2 A 2  (22) 3A  i:  Q  3  + X  l~ 2  X  x  l~ 2  X  x  X  two new "—  Q  x  4^  l  +  X  2  X  l  +  X  2  subsystems >  3 4 X  X  4  X  x  3~ 4 X  l  x  +  x  2  F i n a l l y , from the extended Dynkin diagram f o r B2+2A-^ o x  o  j ~ 2 X  x  o  l"^~ 2 x  X  3  — x  >  4  X  4  —  ^ 3"^ 4^ X  X  we o b t a i n (23) 4 A : 1  x  l~ 2 x  In the group  l  x  +  x  2  X  3~ 4 x  ( F ^ ) , we have the f o l l o w i n g  (F ) = (B )U(B )w U(B )w ^w 4  4  4  r  4  x  where r=-h(x^+X2+x +x ).Consequently 3  ( x  3  + X  4  )  c o s e t decomposition  r  the ( F ) - o r b i t of a  4  4  p o l y n o m i a l II (S) i s t h e union o f t h e ( B ) - o r b i t s o f 4  n (S) ,w (H (S) ) and w r x  f  In some c a s e s we f i n d t h a t  v  4  w^ (H (s) ) . n  x  are  i s the same as i t s ( B ) - o r b i t . 4  V7e f i n d t h i s t o be the case f o r t h e subsystems , A  out  2  + A  1 2 2 4 ' 1 , A  that  , A  , D  3 A  a n c  ^ 4A^.In  either w  or  o f H(S).  4  4  3  r  polynomials that are already i n the ( B ) - o r b i t  Then the ( F ) - o r b i t o f n(s)  C  (n(S))  ±w ( ( S ) ) and ±w w r 4  r  (n(S) ) = ± n ( S )  9  1  1  t  n  e  A2+A ,C +A^, 2  3  o t h e r cases i t t u r n s  44 (n (s) ) = ± w  w r and  v  x  4  w (n ( s ) ) ,  s o a b a s i s f o r P ( S ) c a n be f o u n d 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 ( i i ( s ) ) ,w w„(n(s)). x x i v  4  P (F^) :: We h a v e n(F„)=:. 1  ¥096"  V ( y ,y ,y , y ^ ) V ( x ,x ,x ,x^) J-  ^  -a  «  x  ^  J  *  where =x  y i  1 +  x , 2  2 l~ 2' y =x +x , y  =  x  x  3  3  4  y =x -x . 4  3  4  P ( B ) : We h a v e 4  II ( B ) = x x x x V ( x 4  We f i n d  1  2  3  , x , x ,x|)  2  2  4  2  that X  3_ 2 3 4^ ^ l ' 2 ' 3' 4 ^ ' X  X  X  x  X  X  -16w ( n ( B ) )=16w r  X  w  4  (II ( B ) )  r  4  -• £ ( x - x ) - ( x + x ) ] [ ( x + x f - ( x - x ) ] V ( x , x , x , x ) 2  1  2  3  2  4  1  2  3  2  2  4  4  form a b a s i s f o r P ( B ) . 4  P ( A + A ) : We h a v e 3  1  IT (A +A )=-32 ( x + x + x + x ) V ( x , x , x , x ) 3  We  1  1  2  3  4  1  2  3  4  find w (n(A +A ))=-n(A +A ), r  3  1  3  1  w ^w (H(A +A ))=x V(x|,x ,x ). 2  x  r  3  x  2  4  A s i n (B ),we o b t a i n f r o m x V ( x , x , x ) 2  4  linearly  4  independent polynomials:  2  2  2  the following  four  b  l  b  2  b  X  l  = X  2  =  * 2  V  X  3  4) ,  , X  l' 3' 4 ) ,  V ( x  x  ^ l x  3  , X  X  , x  2  4) ,  , X  b =x V(x*,x ,x* ) . 2  4  4  We have  l  x  3  i l  x  -2(n(A +A ))= 1  x  =-b l  + b  1  1  1  1  X  2  X  X  3  2  X  3  *2  X  3  2  -b  2  3 +  b  X  4  l  X  X  4  •  4  We note t h a t any o t h e r p o l y n o m i a l i n the ( B ) - o r b i t o f 4  H(A +A ) can be o b t a i n e d from t h i s by an a p p r o p r i a t e change 3  1  i n signs.We c a n , t h e r e f o r e , e x p r e s s each o f these as a l i n e a r combination  o f b^,...,b  polynomials  by making t h e  4  c o r r e s p o n d i n g change i n s i g n s i n b^ , . . . b . Thus (b-^ ,b ,b ,b } 4  2  i s a b a s i s f o r P(A +A^)3  P ( A + A ) : We 2  find  2  n ( A  2  + A  2  ) =  4 4 [ X  ( x  l  + x  2  + x  3  )2 _ x  4 ] V(x ,x ,x ), 2  1  2  3  and -w  w ( n ( A 2 + A 2 ) )=IT(A +A ) .  (II(A +A ) )=w 2  2  r  2  2  4 T h e r e f o r e t h e ( F ) - o r b i t o f II (A +A ) i s the same as i t s (B )-orbit.Let X  4  2  2  4  Q= ( l - w ) (n (A +A ) )., x  R=(I+w X  v  3  2  2  ) (Q) .  Then R=x x (x|-x ) (x -x -x +x ). 2  1  The  4  3  2  3  4  ( B ) - o r b i t o f R g i v e s us the f o l l o w i n g l i n e a r l y 4  independent  polynomials  spanning  P(A +A ): 2  2  3  4  3 " 4> ( x + x - x - x ) , 2  2  2  2  X  1 2  X  l 3  X  1 4  X  2 3  X  2 4  <*;-x|>  X  3 4  (x - x l ) ( - x - x + x + x ) .  ( X  X  X  (x  x  (x -x +x -x ),  2  2  - 4^ X  2  2  1V  (X  X  X  2  2  x ),  2  2  +  (-x +x +x -x ), 2  i  ( - X  2  X  2  -x ) ( x - x - x 2  ( X  X  2  2  + x  2  2~ 3 x  2  2  2  + x  4 ' )  2  2  P (C^+A^ : We have n(C +A )=3  x x ( x - x ) (x -x ) 2  1  3  2  2  [(x -x ) -(x  2  2  4  1  2  x ) J 2  3 +  4  [(x -x ) -(x -x ) } 2  1  We  r  3  =w  4  w (n(c +A ))  1  r  3  1  4  w  (n(C-+A )). n  T h e r e f o r e we o n l y2 ^need t o c o n s i d e r the x""*^3 4 We f i n d t h a t P  /  3 4  b  2  = X  3 4  ^ l " ^ tx  b  3  = x  l 2  ^ l  b  4  = x  l 2  b  5  = X  l 4  6  = X  b  7  b  x  )  l x  x  /  ^  -  ( x  X  l"" 2 X  2~ 3  ( x  x  l 4  ( X  = X  2 3 X  ( x  8  = X  2 3  ( x  9  = X  X  2  _ X  3  )  { x  X  /  2  2 \  12  = X  ( X  [(x  1 +  x ) - (x x )  2  [(x  1 +  x ) - (x -x )  2  [(x  1 +  2  2  3 +  4  2  2  3  4  2  2  3  2  x ) -  1 +  x ) - x +x )  2  [(x  2 \  2  2 \  ' 2~ 3  4  X  X  )  2  4  2  [(x x ) 4  2  3  2  ) 2  - (x -x ) ] , 3  x ) - ( x +x ) 4  2  3  4  2  i" 3 x  1~ 3 X  3  2  2  3  ][ ][ ( x  x ) ~(x -x ) ], 2  r  )[(x  )  3  2  ( X  4  x ) - ( x - x ). l 2  2  3  2  4  x ) - (x +x ) J 2  1 +  3  2  2  4  4  2  x ) ] , 2  3 +  4  2  2  r  x )  2  4  3  ~(x +x ) ] , 2  2  3  -(x -x ) ] ,  2  2  2  3  x ) -(x -x ) ] , 2  1 +  2  4  2  3  -(x -x ) ] , 2  4  2  3  1 +  (x|-x| ) [ ( x  substitute  2  2  -(x -x ) ] ,  ] [; ( x  4  1  4  2  <x|-x" ) [(x +x ) - ( x + x ) ]  }  2  2  V  2  ]  4  x ) "(x -x ) ] ,  1 +  2  3  2  1 +  ][ ( x ] '[( x  1  2  X  x +x ) J [ " ( x ^ )  2  [(x  X  2  4  1 +  r  X  of n ( c + A )  x ) - (x +x ) J [ ( x x ) - ( x  1  b  l 3  2  4  4  2  2  =  x  3 +  ( 2 2 \ 3 ^ i x - x ; [(x ~x ) - ( x +x )  ll  I f we  J  }  b  (  2  2  - X  ( X  l 3  2  1  4  x  10 2 4 I- 3  X  4  (x -x )  b  X  2 \ X  r  X  2 \  x  X  [ ( x - x ) - C x x ) J[( 1" 2  4  2  /  X  X  x ;  2~ 3* ( x x )  2 4 ^ l = X  2 \  3  3  / X  J  1  r  )  4  i s spanned by:  i 2 2\ (x x ) [(x  }  2~ 3  }  2 \ X  2  / X  1  3 4  , X  + A  2  = x  ( x  l~ 2  3  l  X  (B ) - o r b i t  3  ( C  b  b  3  find  -w ( n ( c + A ) )=w  b  2  2  1~ 3 X  ) 2  2  4  [(x x ) -(x -x ) ] , 2  1 +  2  3  2  4  [(x x ) -(x -x ) J , 2  r  2  2  3  [(x -x ) - ( x  4  2  1  3  x ) ] . 2  2 +  4  4 =x +x  2  y =x -x  2  y =x  4  y i  2  2  1  3  Y  4  i n b^,b ,b ,fc> 2  3  = X  3 +  x  3~ 4 X  we o b t a i n the s i m p l e r  4  b.=-%y y y.y v(y ,y 2  n  b  4  r =  7  0  /1  y  y ) ,  2  2  1  -^iy2 3 4 Y  i' 2' 3  V ( Y  forms  Y  Y  )  From t h i s we can see t h a t  V l" 2 b  Similarly,we  b  + b  3  find  W  W  12 9" 10 ll' I t can be seen t h a t {b^ ,b ,b^ ,br-,bg ,b^ ,b^ ,b^^ ,b^^} b  s = b  b  + b  forms  2  a linearly  P (B ) : We 3  independent s e t .  find n (B )=-x x x V(x ,x ,x ) , 2  3  -w  2  3  (IT (B.J )=w  r  2  2  4  w (II (B ) )  1 = . — (x -x +x +x ) (x +x ~x +x ) (x +x +x -x )V(-x ,x ,x ,x ) x  Four l i n e a r l y (B ) - o r b i t 4  2  3  4  1  b^ x x x^V( =  2  = 2  3  x 2  ,x iX ), 3  4  x ^ x x V(x^,x ,x^), 3  3  b -x x x V(x ,x ,x ), 2  3  1  2  2  2  4  b ~x^x X V(x^,x ,x ). 4  Let  3  4  1  2  3  4  independent p o l y n o m i a l s a r e found from the  o f JI (B ) ;  b  2  2  3  2  3  1  2  3  4  48 Q=8w  (w -(n(B-)))  v  = (x,+x„-x_-x.)(x.-x^+x-.-x.)(x.-x,-x.+x.)V(x,,x ,x,,x ) 0  4  4  4 x  . ? i.T? i  =  (  x  X  jt?£ i j k x  i^j  x  l  x l x l  2  x  x  2  x 2 X  x 2  1  1  l  x  x  ) V ( x  l' 2' 3' 4 x  x  3  *3  X  X  4  E I t£ i jt£ X  =  x  4 i=l  x  xj = l ift  J  X  ij,k=l i<j<k  i  X  x j  k  )V(x ,x ,x ,x ), 1  X  X  i  1  2  X  3  x  4 4 4 = ( & ? x . + 2 r ; x.x x ) V ( x , x , x , x )  2  x 3  X  2  x 3  <  1  1  X  5  5 X  x  1  1 x  x 2 2  X  X  x 3  2  *2  xj = l  *4  J  3j,k=l  i^j  1  2  J  i<j<k  2  X  x 3 3  X  X  x" 3 X  *4 x 4 x* 4 3  X  :  (Z] iik=l  x x x  i<j<k  X  We have Q=A-2B+3C, Let R=(I+w x  S=(I+w  2  ) (Q) ,  ) (R) , 3 T=(I+w ) ( S ) . v  X  x  4  Then  3  3  3  x  2  2  J  6  X  3  x l  )  i n terms o f the f o l l o w i n g determinants  k  x l x l  x  1  2  x  6  B=  1  1  x  i<j<k  T h i s can be w r i t t e n 1  x  i i k  ) ( l' 2' 3' 4 * v  x  x  x  x  )  3  4  4  4  49 1  x  T=<  1  1  *3  x 4  *3  x 4  2  x 2  X  6  \  + 3x  2  1  1  *2  *;  3  1  x 4 2  X  x 4 4  6  X  =-8 [ x ( x + x + x ) V ( x , x 2  2  2  X  *3  X  2  ,x )-3x V(x ,x ,x )]  2  2  x  3  2  2  2  =-8x [ ( x + x + x + x ) - 4 x ] V ( x ,x ,x ) . 2  2  2  2  2  2  2  2  1  We can now complete t h e b a s i s f o r P(B^) by a p p l y i n g w X . -x.  i  t o T: b =x  [ ( x + x + x + x ) -4x ] V ( x , x , x ) , 2  5  x  3  2  2  2  2  2  2  2  b = x [ ( x + x + x + x ) - 4 x ] v ( x ,x ,x ) , 2  6  2  2  2  2  2  2  2  2  b =x  3  [(x +x +x +x )-4x ]v(x ,x ,x ) ,  b =x  4  [(x +x +x +x )-4x ]v(x ,x ,x ) .  ?  g  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  P ( C ) : In t h i s case i t i s simpler t o w r i t e II (C ) i n terms o f 3  the  3  y^ as d e f i n e d i n P(F ).We 4  n  (  3  c  )  =  have  gT Y Y y 4 V ( y ^ , y , y ) .  ~  1  3  3  4  We note t h a t t h e elements w  ,w , , x. x.+x.,w ,and w w o f (F.) x l j r 4 be expressed i n terms o f w ,w ,w , , . . , y ' y^Yj' -%(y +y +y +y ), c  fr  N  x  can  N  L  and  1  2  3  4  ww,, , , , .,and conversely.Therefore,we can f i n d y (yi y2 3 4 ( F . ) - o r b i t o f II (C ) by a p p l y i n g w ,w . ,w , , , , , 3 y^ y i j (y +y y +y ) w w . . . +  + y  + y  )  4  the  0  ± Y  4  and  y  +  1  2  3  ~h ( y + y + y + y ) t o  4  1  2  3  4  y y y v(y , 2  1  3  4  2 y  ,y ). 2  We observe t h a t t h i s . i s the same as -w  (II (B_) ) w i t h y.  r e p l a c i n g - t h e X £ Hence a b a s i s f o r P ( C ) i s g i v e n by the 3  p o l y n o m i a l s i n t h e b a s i s f o r P(B^) w i t h y^ r e p l a c i n g the x^. P ( 2 A + A ) : We have 1  1  4  w  50 n (2A 1 +A 1 )=-3s ( x 1 - x 2 ) ( x - x ) ( x 1 + x 2 + x 3 + x 4 ) 3  4  We f i n d wr(n(2A +A ))=-n(2A +A ), 1  1  1  1  w „ w_ (II ( 2 A + A ) ) = - x 4 ( x 2 - x 2 ) . 4 As i n ( B 4 ) 1  x  are  4  X  4  X  l  X  l  X  2  X  2  X  3  X  3  (x2-x2), ( x  i"TX3)'  (x2-x2), ( x  2~X4*'  ( xx 2 - xx 2 ); l 3 ' l  (x2-x2), (x2-x2),  l i n e a r l y independent.lt  other of  X  1  r  polynomials  these  P(A3):  in its  c a n be s e e n t h a t  (B4)-orbit  H(2A^+A^) a n d  are a l l linear  combinations  elements.  We f i n d n(A3)=V(x1,x2,x3,x4), w (II(A ))=II<A ) , r  3  3  wx^w (n(A3) )=V(x2,x2,x2) . r  A s shown f o r P ( A 3 ) o f of V ( x ^ , x 2 , x 3 , x 4 ) , w e  (B4),to  find  may c o n s i d e r  t h e span o f t h e  (B4)-orbit  the o r b i t of  x . ^ x 4 (x^—x 2 ) (x^—x 4 ) • We h a v e s e e n i n P ( B 2 + 2 A ^ ) are  linearly  2  = x  (B4) t h a t  independent:  b1=x3x4(x2-x2) b  of  lx2  ( x  l-x2  ) ( x  (x2-x2), 3-x4)'  the following  polynomials  51 t> =x x ( x - x ) (x^-x^) , 2  3  4  b  1  2 4  = X  2  3  X  i~ 3  ( x  x  )  ^ 2" 4^' x  x  b =x x (x -x )(x -x ), 2  5  6  H  The  1  =  2  X  2  2  2  4  3  X  {  l ~  X  X  l  )  (  2  X  _  3  X  }  -  ( B ^ ) - o r b i t o f V ( x ^ , x , x ) g i v e s us three more l i n e a r l y 2  3  independent p o l y n o m i a l s , c o m p l e t i n g the b a s i s f o r P ( A ) : 3  7  b  ^ l' 2' 3^'  = V  x  x  b =V(x  2  8  9  b  P(A +A ): 2  X  We  X  x ,x ), 2  f  2  l' 3' 4 '  = V ( x  x  x  )  find  II ( A + A ) = % ( x + x + x + x ) V ( x , x , x ) , 2  w  1  1  2  3  (n ( A + A ) ) =-n  r  2  w  n  r  A  + A  i^  3  4  (A +A ) ,  1  ( ( 2  2  2  1  3^' 4 We have seen i n P ( A + A ^ ) of (B^-) t h a t the span o f the ( B ) W  x  4  = - x  4 (~ l' 2 V  x  x  , x  4  2  o r b i t o f x V ( x ^ , x , x ) c o n t a i n s the f o l l o w i n g l i n e a r l y independent 4  2  3  polynomials: b  l  = X  l 2  b  2  = X  l 3 ( 2~ V  X  3- 4 '  ( x  X  X  )  '  X  X  b =x x (x -x ), 2  3  b  1  4  V  = X  4  2 3 x  2  X  2  X  !~ 4 '  ( x  4  X  (  x  l  _  )  x  3 ' )  x^x. (x — x«) •  bg~ D  5  4  X  Z.  We can complete t h i s to a b a s i s o f P(A +A ) by c o n s i d e r i n g 2  the  ( B ) - o r b i t o f IT (A +A-^) .Let 4  2  x  Q = 2 w  R  =  w x  Then  1  -x  ( ( n  A  +  A  2  i  ) )  "  ( x  i  + x  2  + x  3  + X  4  ) V ( x  (Q)=(x +x +x -x )V(x ,x ,x ). 1  2  3  4  1  2  3  l' 2' 3 x  X  )  '  Q+R=2(x +x +x )V(x ,x ,x ) 1  =2  2  3  1  1  1  1  l x l  2 x 2  X  3  X  3  X  X  3  2  3  3  X  X  and (I+w Therefore,a  ) (Q+R)=-4x x (x -x ). 2  x  1  2  2  b a s i s f o r P(A +A^) i s given by b-^,  . b,. above b  2  and b  7  =  X  l 2 X  (  l- 2 '  X  X  )  b =x x (x -x^), 2  8  1  3  V 1 4 X  P ( A + A ) : We 2  1  X  ( X  1- 4 ' X  )  b  10  : = X  2 3  '  b  ll  = X  2 4 X  ( X  2~ 4 '  b  12  = X  3 4  ( X  3" 4 '  X  X  X  )  X  )  have n (A +A )=jx 2  1  ( x 4  2  - X  3  [(  )  x  +  x  1  +  x  2  3  )  2  ~  x  l ~ l  '  and (I-w  ) (n(A +A ) ) = x x 2  1  x  (x -x ), 2  4  2  showing t h a t P (A +A )=P (A +A ) 2  P ( B + A ) : We 2  1  1  2  1  find IT ( B + A ) = x x 2  -w  r  1  3  (x -x ).(x -x )  (n(B +A ) )=w 2  1  1  x  2  3  4  w (n(B +A ) r  2  1  = h [(x +x ) - ( x - x ) ] ( x - x ) ( x - x ) . 2  1  2  2  3  2  4  T h e r e f o r e , the ( B ) - o r b i t s of II ( B ^ A ^ span P(B +A^).We have a l r e a d y 2  2  3  and w  4  x  w  r  (II ( B ^ A ^ )  4 seen t h a t the f o l l o w i n g  polynomials  form a b a s i s f o r P(B +A,) o f (B ) : 0  (x - x ) 2  b  l  = X  1 2 3  b  2  = X  1 2 3  b  3  = X  1 2 4  b  4  = X  1 2 4  X  X  2  (x - x ) 2  X  X  2  (x - x ) 2  X  X  2  (x -  t  2  X  X  X  4>  (x - x ) 2  5  = X  b  6  = X  b  7  b  8  b  1 3 4 X  X  l  l  x  X  1 3 4 X  2  X  r  l>  i  x ) 2 3 4 (x|2  = X  X  t  X  fx 2 4> 2  2 3 4  = X  X  X  X  lX  •  Let Q= ( x j - x j ) ( x - 4> X  3  We have 4w  w (H(B +A ))=Q+2b -2b .  x  r  2  1  1  3  T h e r e f o r e we may c o n s i d e r Q i n s t e a d o f w 4  x  We  find h (Q  w (II (B„+A ) ) . - s i r  (Q) ) =x (xj-x*) - ( x - x ) (x + 3 x x )  + W  2  x  2  3  3  2  3  =x (x -x )(x£+x -x -3x ). 2  2  2  2  2  3  We can now complete t h e b a s i s f o r P(B +A^) by the f o l l o w i n g 2  linearly  independent  polynomials:  b =x (x|9  x  b  io  = x  i  (  b  ll  =  2  (  b  12  = X  2  (  b  x  X  x  b  2_o  x  2 N  1 2 3 4 ' (x.^—x^ 3x.^-~x^) / x  ;  -  2  2  +  x  r  (-x +x -3x -x ) l -*4>  r  = x ( X -x ) ( - x - x x - 3 x ) 2  i  X  2  2  3  14  = X  3  (  x  2  l  16  = X  2  2  2  2  2  2  +  J  -x ) (-x -3x|+x -x|) 2  2  2  -x ^ (-x -x -3x +x ) 1 x ; x - x l (-x -3x -x x ) 4 3  15 4< = X  X  l  2  2  X  X  2  2  (  X  x  l  2  2 2  2  b  x  KX  i -x|) ( - x j x - x | - 3 x ) 2  1 3  (• 2_ 2_  3>  2 -*v  x  b  x  2  J  2  2  2  2  2  2  +  f t •  P ( A ) : As i n the case o f (B ),we have 2  4  n(A )=-v(x ,x ,x ) 2  2  3  4  and P(A )=P(2A +A ) . 2  P ( A ) : We 2  1  1  find n(A )-|x f( +x +x ) -x ], -w  2  4  2  X l  2  (n(A ) )=w  r  2  w (n(A ) )=n(A ).  x  T h e r e f o r e the ( F ) - o r b i t  2  3  r  2  2  o f n(A ) i s the same as i t s  4  2  (B )4  orbit.Let Q=(I-w  ) (H (A ) ) = x x (  x  2  1  4  x  + x 2  3  )  Hence (I+w  x  )(Q)=2x x x 1  2  4  Also, (I+w  +w  x  +w  ) (II (A ) ) =x ( x + x + x - x ) . 2  x  2  2  2  2  4  I t can be seen t h a t II (A ) and any polynomial i n i t s 2  (B )4  o r b i t may be expressed as a l i n e a r combination o f p o l y n o m i a l s o f the form x  m  (X +X +X TX ) 2  2  x  D  2  k  2  m'  and x.x.x, x j k  T h e r e f o r e a b a s i s f o r P ( A ) i s g i v e n by: 2  x (x +x +x -x ), 2  2  2  2  1  x (x +x +x -x ), 2  2  2  2  2  x (x +x +x -x ), 2  2  2  2  3  x (x +x +x -x ), 2  2  4  X  l 2 3  X  1 2 4'  X  1 3 4'  X  2 3 4*  X  X  X  X  X  f  X  X  X  2  2  P (2A^) : We  have n (2A ) = (x -x ) (x -x ) , 1  1  2  3  4  w ( n ( 2 A ) )=n(2A ) , r  w  1  1  w (n(2A ) )=-(x -x ) . 2  x  2  r  4  We observe t h a t i s II (2A^) P(2A ) 1  of  (x^-x ) (x^-x^) i s II(2A|) 2  (B^).We f i n d  and P ( 2 A | ) o f x  l  X  1 3'  X  1 4'  X  2 3'  X  2 4'  X  3 4'  X  l ~  x  l ~  x  i" 4*  X  -(x -x ) 2  2  t h a t the union of the bases f o r  (B ) p r o v i d e s 4  a basis for P ( 2 A ) of 1  (F ) 4  X  X  X  X  X  2 X  2 '  X  3 '  2  2  2  :  (B^) and  2 '  2  P (Aj+A.^)  of  2 X  We  have n(A +A )=-J 1  w W  r  x  1  2  ( x - x ) (x +x +x +x ) , 3  4  (n ( A + A ) ) =-n 1  W  r ^  n  A  + A  2  3  4  (A +A ) ,  1  ( i  1  1  i))~ 4 =  x  S i n c e -x^(x^+x ) i s c o n t a i n e d 2  1  ( i x  + x  2^•  in P(2A^),  P (A +A )=P (2A ) . 1  P(B  ) : We  ] L  1  find n (B )=X X (x -x ). 2  3  4  3  4  T h i s i s one o f the elements i n the b a s i s of P ( A + A ^ ) . T h e r e f o r e 2  P(B )=P(A +A ). • 2  2  1  P(D^): As i n  i s spanned by one element  (E^),P{D^)  n(D )=V(x ,x ,x ,x ). 2  2  2  2  4  P (B +2A- ) : We find 2  L  n(B +2A )=x x (x -x )(x -x ). 2  2  1  3  2  2  2  4  But t h i s i s i n P(A ),hence 3  P(B +2A )=P(A ). 2  1  3  P(3A ): We have X  n(3A ) = (x -x ) (x -x ) . 2  2  1  3  4  We observe that t h i s i s i n P(2A^+A^),hence P(3A )=P(2A +A ). 1  1  1  P (4A ) : We have 1  n(4A )=-(x -x ) ( x - x ) 2  2  2  2  1  and w (n(4A ))=w  w (II (4A.. ) )=n (4A ) .  1  X  3T  X  IT  X  Therefore,as i n ( B ) , 4  (x -x )(x -x ), 2  2  2  2  ( x - x ) (xj-x*) 2  2  form a basis f o r P(4A^). Using the above (F )-modules,we 4  obtain seventeen of the  irreducible characters of (F ).The characters that cannot 4  be obtained by the above method can be found by using the following  relations:  x  4  5  X  11  =  X  X  12  =  X  X  14  = X  2 13,  X  22  =  1 20'  The f o l l o w i n g the  =x x /  1 0  1 8' X  1 9' X  X  X  X  conjugacy r e l a t i o n s are used i n c a l c u l a t i n g  characters of ( F ^ ) : z= [ 4 A J ,  [ A J - Z ^ + A J  [2AJ  ,  .z[2Aj ,  [ A  1  +  A j  - z [A +AJ X  ,  [ A j . z f D j , [ A j - z ^ + A j  ,  - z [A  ,  [BJ  AJ  3 +  [A ]~z[A ] 3  ,  3  [B +Aj-.z[B +Aj 2  ,  2  [C ]~4V l]' A  3  [ B  3  ] . z [ A  2  +  i j  ,  [A +A -J-.z[P ( 2  2  4  A I  )] ,  [D (a )]~z[D (a )] , 4  1  4  1  58  Table V I I I : C h a r a c t e r Table f o r  Conjugacy C l a s s Representative  A  l  A  l 2A  A  l  A  2  A  2  B  2  1  +  A  l  C h a r a c t e r i s t i c . h, l Polynomial  1  1  2 A  + A  A  3  B  2  C  3  B  3  A  2  +  A  l  A  2  +  A  l  +  A  1  l  A  2  + A  2  A  3  +  A  l  C  3  +  A  l  D  4  B  F ( 4  F  4  3  x  4  1  2  (12) [ l l l l ]  x -2x  3  +2x-l  12  1  -1  -1  1  -2  (i)  [-1111]  x"-2x  3  +2x-l  12  1  -1  1  -1  0  (i)  [-1-111]  x* 1  -2x  2  +1  18  1  1  1  1  2  (12) [11-11]  x"  -2x  2  +1  72  1  1  -1  -1  0  (123) [ l l l l ]  x *- x  3  - x+1  32  1  1  1  1  2  x *- .x  - x+1  32  1  1  1  1  -1  (12) [-1111]  x"-2x +2x -2x+l  (12) [-1-1-1-1]  4  1  2  3  3  2  36  1  1  -1  -1  0  x"+2x  3  -2x-l  12  1  -1  -1  1  -2  (1) [-1-1-11]  x"+2x  3  -2x-l  12  1  -1  1  -1  0  (1234) [ l l l l ]  x"  -1  72  1  -1  -1  1  -2  (12) (34) [-1111]  x  -1  72  1  -1  1  -1  0  4  x - x  3  + x-1  96  1  -1  -1  1  1  x *- x  3  + x-1  96  1  -1  1  -1  0  1  x  k  +  x  3  - x-1  96  1  -1  -1  1  1  (123) [ l l l - l ]  x  k  +  x  3  - x-1  96  1  -1  1  -1  0  (1) [ - l - l - l - l ]  x'*+4x + 6x +4x+l  1  1  1  1  1  2  x'* + 2x + 3x +2x+l  16  1  1  1  1  -1  x' +2x + 2x + 2x+l  36  1  1  -1  -1  0  x"*+ x  3  + x+1  32  1  1  1  1  -1  xV  3  +X+1  32  1  1  1  1  2  +1  12  1  1  1  1  2  +1  144  1  1  -1  -1  0  16  1  1  1  1  -1  96  1  1  1  1  -1  (12) [-11-1-1]  x  (1234) [-1111] )  X  1  (123) [-1-1-1-1]  a i  2  4  1  3  3  2  3  4  x  B  .1  D ( a ) (12) (34) [-11-11] 4  l  4  1  (123) [ - l l l l ]  1  X  D  x"*-4x +6x -4x+l  4  4 A  0  4  (i) [ m i ]  1  3A  X  F  (F.).  ,  x  2  3  x  2  +2x  k  2  x  H  x"-2x +3x -2x+l 3  • x*  2  - x  2  +1  4 A  X  A  l  A  l 2A  A  1  1 +  A  2  A  2  B  2  2A  A  1  A  1 +  A  3  B  2  C  3  B  3  A  2  +  A  l  A  2  +  A  l  A  l  1  4 A  A  +  2 +  A  2  A  3  +  A  l  C  3  +  A  l  D  4  B  4  P  4  F  4  (  a  l  }  1  1  5  X  6  X  7  A  l  A  l  X  A  3  8 x  +  A  A  l  X  9  10  X  ll  X  12  X  2 +  13  A  2  X  14'  B  3  C  3  X  15  X  16  2  2  2  4  4  4  4  4  6  6  8  8  0  0  2  2 -2  0  -2  2  0  0  -4  0  2  -2  0  2  2  0  -2  -2  0  0  0  -4  2  2  2  0  0  4  0  0  -2  -2  0  0  0  0  0  0  0  0  0  0  2  -2  0  0  -1  -1  2  1  1  -2  1  1  0  0  2  -1  2  2  -1  1  1  -2  1  1  0  0  -1  2  0  0  0  2 -2  0  2  -2  -2  2  0  0  0  0  2  -2  2  0  2  -2  0  0  4  0  2  -2  0  -2 -2  0  2  2  0  0  0  4  0  0  2  0  0  0  0  0  0  0  0  0  2  -2  0  0  0  0  0  0  0  0  0  0  0  0  -1  1 -1  0  -1  1  0  0  1  0  -1  1  0  1  1  0  -1  -1  0  0  0  1  0  0  -1  -1  1  0  1  -1  0  0  -1  0  -1  1  0  -1 -1  0  1  1  0  0  0  -1  2  2  2  -4 -4  4  -4  -4  6  6  -8  -8  -1  -1  -1  -2 -2  1  -2  -2  3  3  2  2  0  0  0  2  0  -2  2  -2  2  0  0  2  2  -1  -1 -1  -2  -1  -1  0  0  1  -2  -1  -1  2  -1 -1  -2  -1  -1  0  0  -2  1  2  2  2  0  0  4  0  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -1  -1  -1  2  2  1  -2  2  3  3  -2  -2  -1  -1  -1  0  0  1  0  0  -1  -1  0  0  -2  ^2 3A 2 A  X  X  1  A  + A  17  2A  2  1  l  A  X  18  1  X  A  +  19  A  l  B X  C  3 2 +  2A  20  3 +  A  1  1  X  21  X  22  B  2  A  2  +  A  l  A  2  +  A  l  X  23  B  X  2 +  24  8  8  9  9  9  9  12  16  0»  4  3  -3  -3  3  0  0  4  0  3  3  -3  -3  0  0  0.  0  1  1  1  1  -4  0  0  0  1  -1  1  -1  0  0  -1  2  0  0  0  0  0  -2  2  -1  0  0  0  0  0  -2  0  0  . 1  -1  1  -1  0  0  0  -4  3  -3  -3  3  0  0  A -4  0  3  3  -3  -3  0  0  0  0  -1  1  1  -1  0  0  0  0  -1  -1  1  1  0  0  0  -1  0  0  0  0  0  0  -1  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  1  0  0  0  0  0  0  0  -8  -8  9  9  9  9  12  2  2  0  0  0  0  -3  -2  0  0  1  -1  1  -1  0  0  -2  1  0  0  0  0  0  2  1  -2  0  0  0  0  0  2  D (a ) 0  0  -3.  -3  -3  -3  4  0  0  0  -1  1  -1  1  0  0  -2  -2  0  0  0  0  -3  2  0  0  0  0  0  0  1  0  A  l  A  l  A  l  A  2  A  2  B  2  +  3A  l  A  1  2A  1 +  1  A  3  B  2  C  3  B  3  A  2  +  A  l  A  2  +  A  l  4 A  +  A  l  1  A  2  + A  2  A  3  +  A  l  C  3  +  A  l  D  4 4  B  4  F  4  F  4  x  ( a  l>  -16  A  1  BIBLIOGRAPHY  [l] N.Bourbaki ,Groupes e t A l g e b r e s de L i e ,Chapitres 4,5 e t 6, Hermann,Paris,1968 .  [2] R.W.Carter,Conjugacy C l a s s e s i n the Weyl Group,Seminar on A l g e b r a i c Groups and R e l a t e d F i n i t e  Groups,Springer  L e c t u r e Notes No.131,1970.  [ ] E.B.Dynkin,Semisimple subalgebras o f semisimple L i e 3  Algebras.Mat. S b o r n i k N . S . 30  (72),349-462  (1952).  (Russian).(Amer. Math. Soc. T r a n s l . , ( 2 ) 6 (1965), 111-244).  [4]T.Kondo, C h a r a c t e r s o f the Weyl Group of Type F ^ J . F a c . • S c i . Univ. Tokyo, S e c t . I, 11  (1965),  145-163.  [ 5 ] I.G.MacDonald,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 Groups, B u l l . London Math. S o c , 4 (1972),  148-150.  

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