Unformatted text preview: 3 ± 3´ .2;1/ C 3´ 3 . Of course, such computations can be automated. A program in Mathematica for expanding symmetric polynomials in elementary symmetric functions is available on my World Wide Web site. 1 Exercises 9.6 9.6.1. (a) Show that the set K d Œx 1 ;:::;x n Ł of polynomials that are homogeneous of total degree d or identically zero is a ﬁnite–dimensional vector subspace of the Kvector space KŒx 1 ;:::;x n Ł . (b) Find the dimension of K d Œx 1 ;:::;x n Ł . (c) Show that KŒx 1 ;:::;x n Ł is the direct sum of K d Œx 1 ;:::;x n Ł , where d ranges over the nonnegative integers. 9.6.2. Prove Lemma 9.6.1 . 9.6.3. (a) For each d , K d Œx 1 ;:::;x n Ł is invariant under the action of S n . 1 www.math.uiowa.edu/~ goodman...
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 Fall '08
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 Algebra, Elementary symmetric polynomial, Symmetric polynomial, homogeneous symmetric polynomial, Kd Œx1

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