Symmetry And Group

## K-loops from the General Linear Groups over Rings by Kiechle H.

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By Kiechle H.

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The g e n e r a l linear simple e [Y(lr~) is modules a basis a n d the n u m b e r s for HornF ~ are n o n - d e c r e a s i n g (MXrM ~) . ,T k from each that of the set 0 is a n e l e m e n t constructed by row equivalence follows f r o m the of HOmF~ taking class one of definition ~(l,~). of 8 T. (MX,M ~) • If T a n d T' are n row equivalent, t h e n T' < {t}0,T' = Tz > = < {t}0,T~ = Hence and since for some < {t}@,T z in Rt, a n d so > = < {t}0~-I,T > > {t}@ = ~ < { t } @ , T i > { t } 0 T i i=l M 1 is a c y c l i c as r e q u i r e d module, 0 is a l i n e a r : k @ = ~ < {t}@,T i > i=l @Ti combination o f 0T.

I)T b e i n t. the entry Let (iw'l)T ~n in T w h i c h act on ~(l,~) (i ~ i ~ n, T ~ ~ ( l , ~ ) , n action forced occurs o f ~ is t h e r e f o r e t h a t o f a p l a c e -I to t a k e ~ in the d e f i n i t i o n to m a k e same ~ ~n ) . permutation, the in the by and we ~-action are well- defined. 2 EXAMPLE If t = 1 3 4 5 and T = 2 2 1 1 2 T(I 2) = 1 2 1 1 and T(I 2 3) = 2 1 1 1 . 3 eT in M ~ in a w a y w h i c h column) to ~(l,~) . depends ~(l,U), equivalent we for o f the If T E ~ ( l , U } , define to v e r i f y that the m a p to T } S eT b e l o n g s why we t h a t T 1 a n d T 2 are r o w stabilizer : {t}S + ~ { T I [ T 1 is r o w e q u i v a l e n t It is e a s y say soon emerge I a n d U.

It is standard tabloid than one form a basis 2 4). polytabloids for the S p e c h t field. have independence the may involves the (3,2)-tableaux listed. a polytabloid We p r o v e The tableau 5 standard are to i n c r e a s e that (In E x a m p l e linear in the polytabloids have module, columns MODULE if t is s t a n d a r d . 10. 3 LEMMA {t} order lower than is the on in e t s a t i s f y x. 15, of t' {t'} shows all If alv I + . . = a m = O. involved that in e t w h e n the standard go for a s t r o n g e r down {t'} ~ a non-iden~ty induction {tm}.