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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 Fall '08
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 Algebra, 2 m, Multilinear, multilinear function

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