Unformatted text preview: S n action. The set of symmetric polynomials is denoted K S Œx 1 ;:::;x n Ł , and the set of symmetric rational functions is denoted K S .x 1 ;:::;x n / . Note that for each d , K d Œx 1 ;:::;x n Ł is invariant under the action of S n , and K S Œx 1 ;:::;x n Ł is the direct sum of the vector subspaces K S d Œx 1 ;:::;x n Ł D K d Œx 1 ;:::;x n Ł \ K S Œx 1 ;:::;x n Ł for d ² . See Exercise 9.6.3 . Lemma 9.6.1. (a) The action of S n on KŒx 1 ;:::;x n Ł is an action by ring automorphisms; the action of of S n on K.x 1 ;:::;x n / is an action by by ﬁeld automorphisms. (b) K S Œx 1 ;:::;x n Ł is a subring of KŒx 1 ;:::;x n Ł and K S .x 1 ;:::;x n / is a subﬁeld of K.x 1 ;:::;x n / . (c) The ﬁeld of symmetric rational functions is the ﬁeld of fractions of the ring of symmetric polynomials in nvariables....
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Xn, Kd Œx1

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