By John Stanaway

Shaped with the easiest to be had fighter pilots within the Southwest Pacific, the 475th Fighter crew used to be the puppy undertaking of 5th Air strength leader, normal George C Kenney. From the time the crowd entered strive against in August 1943 until eventually the tip of the conflict it was once the quickest scoring staff within the Pacific and remained one of many crack fighter devices within the whole US military Air Forces with a last overall of a few 550 credited aerial victories. among its pilots have been the best American aces of all time, Dick Bong and Tom McGuire, with high-scoring pilots Danny Roberts and John Loisel additionally serving with the 475th. one of the campaigns and battles special during this quantity are such recognized names as Dobodura, the Huon Gulf, Oro Bay, Rabaul, Hollandia, the Philippines and Luzon.

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Shaped with the simplest on hand fighter pilots within the Southwest Pacific, the 475th Fighter staff used to be the puppy undertaking of 5th Air strength leader, basic George C Kenney. From the time the crowd entered strive against in August 1943 till the tip of the struggle it was once the quickest scoring workforce within the Pacific and remained one of many crack fighter devices within the whole US military Air Forces with a last overall of a few 550 credited aerial victories.

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V2, ~ ) the set of maps V 1 --~ V 2 is the map gv for F ( g ) (v). of G on V. f : V l - - ~ > V 2 satisfying, Let HomG(VI,V2) g~: F:G and w ~W. A map of G-modules called to write it from a p e r m u t a t i o n V is called of G on V, for all g ~ G and to write representation ring). on V as /(g) simpler (V, p ) is also of a map an action an action a fixed point, for F ( g ) , (to d i s t i n g u i s h commutative to give it is u s u a l l y terminology, representation below) But operation all v ~ V I.

Possible in K(S) nm 9n:K(S)~) K(S) As in the theorem, for e v e r y by, for f : S ~ ) K, and sES, s i n c e d i v i s i o n b y n: is i n t e g e r n, we can d e f i n e o p e r a t i o n s in by kn _ 1 n: det 91 1 0 ,2 ~I 2 ~n and w e can c o n c l u d e to the o p e r a t i o n s ~-ring, k t h a t the n th ~n are the A d a m s o p e r a t i o n s c o r r e s p o n d i n g since K(S) is c l e a r l y a it is a l s o a k-ring. one is g i v e n n 0 41 By the theorem, This construction of G. / a n power applies a g r o u p G, is the m a p in G.

X k, Thus symmetric k ~ n, F with integer c o u l d d e p e n d on the o r i g i n a l of x.! 's), of the if k~n, so w e s h o u l d w r i t e interpretation and k ~ n , of t h e s e then is i n v i t e d to p r o v e t h i s d i r e c t l y . ) of k, even t h o u g h to c a l c u l a t e a particular it for number k~n must be chosen and fixed. Exercise: Schur been {1 n} = a functions, n , In] = h . also c a l l e d S - f u n c t i o n s d e f i n e d and s t u d i e d in a n u m b e r of r e p r e s e n t a t i o n theory other see L i t t l e w o o d approaches, n as we of ways, in the l i t e r a t u r e , not shall do in C h a p t e r (£29~), R e a d just in t e r m s III.