λ-Rings and the Representation Theory of the Symmetric Group by Donald Knutson
By Donald Knutson
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The contents of this e-book were used in classes given by means of the writer. the 1st was once a one-semester path for seniors on the collage of British Columbia; it was once transparent that stable undergraduates have been completely in a position to dealing with straight forward staff idea and its software to easy quantum chemical difficulties.
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R w is c e r t a i n l y isomorphic a k-algebra, and under the hypothesis, is to l + R [ [ t ~ +. The Proof of the original R has k-operations, R is a k-ring theorem is n o w accomplished. and hence Y-operations, iff R is a Y-ring. A useful If and is torsion-free, restatement of the theorem is the following proposition. Proposition: Let R b e a t o r s i o n - f r e e Suppose there is a ring homomorphism. There and S b e any ring. is given a map of sets ~:S--~> l + R [ ~ t ~ +. Then is a ring h o m o m o r p h i s m k-structure ring, iff the c o m p o s i t e map S --~ I + R [ [ t ~ + - ~ R If S is a pre-k-ring, ~ preserves the iff the c o m p o s i t e map L~ does.
26 on the category of ~-rings is a natural transformation from the underlying set identity functor to itself. That is, we have an assignment to each k-ring R, a map (of sets) ~R:R ~ > R such that for any map of ~-rings f:R ---~ S, f~R = ~S f :R--~ S. operations is defined b y for multiplication Proposition: isomorphic A d d i t i o n of natural (~R+~R) (r)= ~R(r)+~R(r) and similarly and k-operations. The set of natural operations is a i-ring, to the free i-ring on one generator, Proof: Let ~ be an operation.
N. an with is t h e n u m b e r of p a r t i t i o n s rl2r 2 of the n u m b e r n. Indeed, to e a c h p a r t i t i o n rI w e can a s s o c i a t e be denoted a . n of n. an This monomial will An is a free a b e l i a n g r o u p w i t h b a s i s ~ a partition An rn the 29 At this p o i n t notation it is c o n v e n i e n t on p a r t i t i o n s . Given is any s u m n = n l + n 2 + . . >0. , the p a r t s in d e c r e a s i n g n l ~ n 2 ~ ... >_nk. associated with n squares, = Given ~ = in k rows, any p a r t i t i o n b e the lengths graph notation or Y o u n q (nl,n 2, ....