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Unformatted text preview: S n action. The set of symmetric polynomials is denoted K S x 1 ;:::;x n , and the set of symmetric rational functions is denoted K S .x 1 ;:::;x n / . Note that for each d , K d x 1 ;:::;x n is invariant under the action of S n , and K S x 1 ;:::;x n is the direct sum of the vector subspaces K S d x 1 ;:::;x n D K d x 1 ;:::;x n \ K S x 1 ;:::;x n for d . See Exercise 9.6.3 . Lemma 9.6.1. (a) The action of S n on Kx 1 ;:::;x n is an action by ring automor-phisms; the action of of S n on K.x 1 ;:::;x n / is an action by by eld automorphisms. (b) K S x 1 ;:::;x n is a subring of Kx 1 ;:::;x n and K S .x 1 ;:::;x n / is a subeld of K.x 1 ;:::;x n / . (c) The eld of symmetric rational functions is the eld of fractions of the ring of symmetric polynomials in n-variables....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08