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Concept# Alternating multilinear map

Summary

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module over a commutative ring.
The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
Definition
Let R be a commutative ring and V, W be modules over R. A multilinear map of the form f\colon V^n \to W is said to be alternating if it satisfies the following equivalent conditions:
# whenever there exists 1 \leq i \leq n-1 such that x_i = x_{i+1} then f(x_1,\ldots,x_n) = 0.

# whenever there exists 1 \leq i \neq j \leq n

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