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By Manning W.A.

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Of x n, w e o b t a i n problems of n w h e r e the d e s i r e d of c o n v e r g e n c e equality is r e f e r r e d (The to [3]). 2 does not require p to be prime, and it is only w h e n we come to r e p r e s e n t a t i o n theory that we m u s t not allow p to be composite. 3 ~ the integer g~ d e f i n e d by = g . c . d . { < e t , e t , > l e t and et, are p o l y t a b l o i d s in S~}. The i m p o r t a n c e of this n u m b e r is that it is the g r e a t e s t common d i v i s o r of the entries in the G r a m m a t r i x w i t h respect to the s t a n d a r d basis of the Specht module.

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~). n Proof: Suppose @ is a n o n - z e r o element of H o m F ~ n ( S i , M ~) . 10, {t}

3). - et4 F is the l± ~ ~ SF . Proof: 5 (cf. 14 has -I p, w e obtain a s e t of v e c t o r s the corres~R e d u c i n g in M~, the % last m - k which S F1 • are of w h i c h linearly are the standard independent basis and o r t h o g o n a l and the to the f i r s t k of standard basis of Since d i m S FIi = d i m MFI - d i m S F1 = we h a v e when of S~, constructed the tabloid Now, any one combination all i n t e g e r s to zero, a basis are of S 11 ~ whose coefficients are of o u r b a s i s of l - t a b l o i d s , reduced as r e q u i r e d .

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