Unformatted text preview: j ± 1 ;x;x j C 1 ;:::;x n / is an R –module homomorphism. It is easy to check that the set of all multilinear maps ' W M 1 ² ³³³ ² M n ´! N is an abelian group under addition; see Exercise 8.3.1 . We will be interested in the special case that all the M i are equal. In this case we can consider the behavior of ' under permutation of the variables. Deﬁnition 8.3.2. (a) A multilinear function ' W M n ´! N is said to be symmetric if '.x ±.1/ ;:::;x ±.n/ / D '.x 1 ;:::;x n / for all x 1 ;:::;x n 2 M and all ± 2 S n . (b) A multilinear function ' W M n ´! N is said to be skew– symmetric if '.x ±.1/ ;:::;x ±.n/ / D ².±/'.x 1 ;:::;x n /...
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 Fall '08
 EVERAGE
 Algebra, Determinant, Ring, MULTILINEAR MAPS

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