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College Algebra Exam Review 437

College Algebra Exam Review 437 - 447 9.6 SYMMETRIC...

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9.6. SYMMETRIC FUNCTIONS 447 (b) K S d OEx 1 ; : : : ; x n Ł D K d OEx 1 ; : : : ; x n Ł \ K S OEx 1 ; : : : ; x n Ł is a vector subspace of K S OEx 1 ; : : : ; x n Ł . (c) K S OEx 1 ; : : : ; x n Ł is the direct sum of the subspaces K S d OEx 1 ; : : : ; x n Ł for d 0 . 9.6.4. Prove Lemma 9.6.3 . 9.6.5. Show that every monic monomial m n n m n 1 n 1 : : : m 1 1 in the i is an , and relate to the multiplicities m i . 9.6.6. Show that if is a partition, then the conjugate partition satisfies j D jf i W i j gj . 9.6.7. Show that is homogeneous of total degree j j . 9.6.8. Show that the monomial symmetric functions m with D . 1 ; : : : ; n / and j j D d form a linear basis of K S d OEx 1 ; : : : ; x n Ł . 9.6.9. Show that a symmetric function of degree d is an integer lin- ear combination of monomials x ˛ of degree d and, therefore, an integer linear combination of monomial symmetric functions m with j j D d . (Substitute Z q -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 m 3;3;1 .x 1 ; x 2 ; x 3 / and m 3;2;1 .x 1 ; x 2 ; x
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