College Algebra Exam Review 437

College Algebra - 447 9.6 SYMMETRIC FUNCTIONS(b(c S Kd Œx1 xn  D Kd Œx1 xn  K S Œx1 xn  is a vector subspace of K S Œx1 xn  S K S Œx1

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Unformatted text preview: 447 9.6. SYMMETRIC FUNCTIONS (b) (c) S Kd Œx1 ; : : : ; xn  D Kd Œx1 ; : : : ; xn  \ K S Œx1 ; : : : ; xn  is a vector subspace of K S Œx1 ; : : : ; xn . S K S Œx1 ; : : : ; xn  is the direct sum of the subspaces Kd Œx1 ; : : : ; xn  for d 0. 9.6.4. Prove Lemma 9.6.3. 9.6.5. Show that every monic monomial , and relate to the multiplicities mi . mn mn 1 n n1 ::: m1 1 in the 9.6.6. Show that if is a partition, then the conjugate partition j gj. j D jfi W i 9.6.7. Show that i is an satisfies is homogeneous of total degree j j. 9.6.8. Show that the monomial symmetric functions m with S and j j D d form a linear basis of Kd Œx1 ; : : : ; xn . D. 1; : : : ; 9.6.9. Show that a symmetric function of degree d is an integer linear combination of monomials x ˛ of degree d and, therefore, an integer linear combination of monomial symmetric functions m with j j D d . (Substitute Zq -linear combinations in case the characteristic is q .) 9.6.10. Show that an upper triangular matrix T with 1’s on the diagonal and integer entries has an inverse of the same type. 9.6.11. Write out the monomial symmetric functions m3;3;1 .x1 ; x2 ; x3 / and m3;2;1 .x1 ; x2 ; x3 /, and note that they have different numbers of summands. 9.6.12. Consult the Mathematica notebook Symmetric- Functions.nb, which is available on my World Wide Web site. Use the Mathematica function monomialSymmetric[ ] to compute the monomial symmetric functions m in n variables for (a) D Œ2; 2; 1; 1, n D 5 (b) D Œ3; 3; 2, n D 5 (c) D Œ3; 1, n D 5 9.6.13. Use the algorithm described in this section to expand the following symmetric polynomials as polynomials in the elementary symmetric functions. (a) x1 2 x2 2 x3 C x1 2 x2 x3 2 C x1 x2 2 x3 2 3 3 3 (b) x1 C x2 C x3 9.6.14. Consult the Mathematica notebook Symmetric- Functions.nb, which is available on my World Wide Web site. Use the Mathematica function elementaryExpand[ ] to compute the expansion of the following symmetric functions as polynomials in the elementary symmetric functions. n/ ...
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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.

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