By Donald Knutson
Knutson D. Lambda-Rings and the illustration concept of the Symmetric staff (LNM0308, Springer, 2007)(ISBN 3540061843)(1s)_Mln_-o
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On the other function Since of the for i~j). h(x) , the same product is also of divided Hence ~ det) 1 l-xiY j so equal to a constant the equation w r i t t e n ~. To in the form ~(x). 7]- (l_xiYj)de t 1-xiYj Similarly of the x's and y's, To find the value in x. column. ~--~,(1-xiYj) (x) A (y) l, j evaluate to degree ¥fI~- 1} "~v'-~' in the x's. 2 (0) = But it in the second sign if x. and x. are switched, l J is independent is independent the x's. = - - , l Yi that ~=i. Ykt for y = y l + Y 2 + .
Note that since 1 is the i d e n t i t y map, the m a p Y:R ~ R ~ is one-one. , following p. pre-~-ring. Sul]pose t h e r e are o p e r a t i o n s co ~d logkt(x) = ~ - ( - 1 ) n y n + l ( x ) t n, a l l x£R. n=l , we can c a l c u l a t e l(x) 1 , I~ y2 n. k n(x) = de yl (x) (x) yl(xI n(x) T h e n we s u p p o s e that b y the d e t e r m i n a n t so t h a t kn(x) that this for each n > i, and each x 6 R the e l e m e n t on the right h a n d is d e f i n e d . ) Note b y n: defined in R, guarentees that if 51 R contains possible a field of c h a r a c t e r i s t i c so the kn's As usual, automatically the p o w e r series division is always exist.
Now, integer k chosen Fl, k. 's: l for s o m e p o l y n o m i a l (k = the n u m b e r it w i l l b e F k , k = F k , k + I. +Xk_iXk , x = X l + X 2 + . + 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.