College Algebra Exam Review 433

College Algebra Exam Review 433 - 9.6. SYMMETRIC FUNCTIONS...

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Unformatted text preview: 9.6. SYMMETRIC FUNCTIONS 443 and N 1 ; r s D 1 if r Ä k and s Ä r , and r s D 0 otherwise. Here is a matrix representing the partition D .5; 4; 4; 2/: 2 3 111110 61 1 1 1 0 0 7 6 7 61 1 1 1 0 0 7 6 7: D6 7 61 1 0 0 0 0 7 40 0 0 0 0 0 5 000000 P The size of a partition is j j D are i . The nonzero entries in referred to as the parts of . The number of parts is called the length of . The conjugate partition is that represented by the transposed matrix . Note that r D jfi W i r gj, and . / D ; see Exercise 9.6.6. For D .5; 4; 4; 2/, we have D .4; 4; 3; 3; 1/, corresponding to the matrix 2 3 111100 61 1 1 1 0 0 7 6 7 61 1 1 0 0 0 7 7 D6 61 1 1 0 0 0 7 : 7 6 41 0 0 0 0 0 5 000000 For D. 1; 2; : : : ; s /, define Y .x1 ; : : : ; xn / D i .x1 ; : : : ; xn /: i For example, .5;4;4;2/ D . 5 /. 4 /2 . 2 /. Note that .x1 ; : : : ; xn / D 0 if 1 > n. We will show that the set of with j j D d and 1 Ä n is a basis of S Kd Œx1 ; : : : ; xn . In order to do this, we first produce a more obvious basis. For a partition D . 1 ; : : : n / with no more than n nonzero parts, define the monomial symmetric function X m .x1 ; : : : ; xn / D .1=f / .x /; 2Sn where f is the size of the stabilizer of x under the action of the symmetric group. (Thus, m is a sum of monic monomials, each occurring exactly once.) For example, 5442 5424 5244 m.5;4;4;2/ .x1 ; : : : ; x4 / D x1 x2 x3 x4 C x1 x2 x3 x4 C x1 x2 x3 x4 C : : : ; a sum of 12 monic monomials. In the Exercises, you are asked to check that the monomial symmetric functions m with D . 1 ; : : : ; n / and j j D d form a linear basis of S Kd Œx1 ; : : : ; xn . ...
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