College Algebra Exam Review 348

College Algebra Exam Review 348 - D x ±².j Thus ±.².x...

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358 8. MODULES for all x 1 ; : : : ; x n 2 M and all 2 S n . (c) A multilinear function ' W M n ! N is said to be alternating if '.x 1 ; : : : ; x n / D 0 whenever x i D x j for some i ¤ j . Lemma 8.3.3. The symmetric group acts S n on the set of multilinear func- tions from M n to N by the formula '.x 1 ; : : : ; x n / D '.x .1/ ; : : : ; x .n/ /: The set of symmetric (resp. skew–symmetric, alternating) multilinear func- tions is invariant under the action of S n . Proof. We leave it to the reader to check that ' is multilinear if ' is multilinear, and also that if ' is symmetric (resp. skew–symmetric, alter- nating), then ' satisfies the same condition. See Exercise 8.3.2 . To check that S n acts on ˚ n , we have to show that . /' D . '/ . Note that . '/.x 1 ; : : : ; x n / D . '/.x .1/ ; : : : ; x .n/ /: Now write y i D x .i/ for each i . Then also y .j /
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Unformatted text preview: D x ±².j/ . Thus, ±.²'/.x 1 ;:::;x n / D .²'/.y 1 ;:::;y n / D '.y ².1/ ;:::;y ².n/ / D '.x ±.².1// ;:::;x ±.².1// / D '.x ±².1/ :::;x ±².n/ / D .±²/'.x 1 ;:::;x n /: n Note that that a multilinear function is symmetric if, and only if ±' D ' for all ± 2 S n and skew–symmetric if, and only if, ±' D ³.±/' for all ± 2 S n . Lemma 8.3.4. An alternating multilinear function ' W M n ±! N is skew–symmetric. Proof. Fix any pair of indices i < j , and any elements x k 2 M for k different from i;j . Define ´.x;y/ W M 2 ±! N by ´.x;y/ D '.x 1 ;:::;x i ± 1 ;x;x i C 1 ;:::;x j ± 1 ;y;x j C 1 ;:::;x n /...
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