# 20th Fighter Group by Ron MacKay, Don Greer

By Ron MacKay, Don Greer

The 20 th Fighter staff joined the eighth Air strength Command in Dec of '43, flying the P-38 in lengthy diversity bomber escort function. the gang later switched over to the P-51 in July of '44. the gang destroyed a complete of 449 enemy airplane in the course of its strive against travel. Over a hundred and fifty photographs, eight pages of colour, eighty pages.

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10 shows that any p o l y t a b l o i d can be w r i t t e n as a linear c o m b i n a t i o n of s t a n d a r d p o l y t a b l o i d s over any field. There- fore, we can deduce that the s t a n d a r d p o l y t a b l o i d s span S ~ over any field, k n o w i n g only the same i n f o r m a t i o n over Q. 12 ~n o b t a i n e d from S ~ O A p p l y the last Corollary. COROLLARY There is a basis of S ~ f all of w h o s e elements inv- olve a unique s t a n d a r d tabloid. Proof: Let {t I} < {t2} < ...

2 give a unique If D is a c o m p o s i t i o n I m ~. matrix irreducible irreducible top c o m p o s i t i o n factor of S ~ n S ~± If ~ is p - s i n ~ u l a r ~ form D l w i t h is an i m m e d i a t e of the p - m o d u l a r S ~ has all the c o m p o s i t i o n I m ~. corollary of T h e o r e m s of a group records representations representations. 1. 3 The decomposition COROLLARY matrix of ~n for the prime p has the form: D ~ (~ p-regular) I S~(~ p-regular) 1 ! 4 are placed Consider representation.

AI b1 A A a • q b q A s ^ a r Let let xij Z(sgn be the e n t r y ~)~ be in the aGarnir { X l , j + I, . . ,Xq,j+l}. ,Xrj} a)~@ and = O. T, T Z (sgn ~)~ is a l i n e a r the places. be a t a b l e a u +- CTl, have Unless IT I] char All F = 2 and {~TIT ~ ~;'o(l,~) } is a b a s i s the with tableaux involved in Z c T T Z (sgn ~)~ is above. , c T ~ O such described ~ IT], combination (l,j+l)th, and s i n c e T ~ T1 the p l a c e s < aq <.. < a r , we m u s t i n i t i a l c h o i c e of T I. < bq contradicts our i is 2 - s i n g u l a r , for H O m F ~ (SI,M~).