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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.
 Fall '08
 EVERAGE
 Algebra

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