Alternating multilinear map

In, more specifically in , an alternating multilinear map is a with all arguments belonging to the same space (e.g., a  or a ) that is zero whenever any two adjacent arguments are equal.

The notion of alternatization (or alternatisation in some variants of ) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.

Definition
A multilinear map of the form $$f\colon V^n \to W$$ is said to be alternating if it satisfies any of the following equivalent conditions:
 * 1) whenever there exists $1 \leq i \leq n-1$  such that $$x_i = x_{i+1}$$ then $$f(x_1,\ldots,x_n) = 0$$.
 * 2) whenever there exists $1 \leq i \neq j \leq n$  such that $$x_i = x_j$$ then $$f(x_1,\ldots,x_n) = 0$$.
 * 3) if $$x_1,\ldots,x_n$$ are  then $$f(x_1,\ldots,x_n) = 0$$.

Example

 * In a, the is an alternating bilinear map.


 * The of a matrix is a multilinear alternating map of the rows or columns of the matrix.

Properties

 * If any component xi of an alternating multilinear map is replaced by xi + c xj for any j ≠ i and c in the base ring R, then the value of that map is not changed.
 * Every alternating multilinear map is antisymmetric.
 * If n! is a in the base ring R, then every antisymmetric n-multilinear form is alternating.

Alternatization
Given a multilinear map of the form $$f\colon V^n \to W$$, the alternating multilinear map $$g\colon V^n \to W$$ defined by $$g(x_1, \ldots, x_n) := \sum_{\sigma \in S_n} \sgn(\sigma)f(x_{\sigma(1)}, \ldots, x_{\sigma(n)})$$ is said to be the alternatization of $$f$$.


 * Properties


 * The alternatization of an n-multilinear alternating map is n! times itself.
 * The alternatization of a is zero.
 * The alternatization of a is bilinear. Most notably, the alternatization of any  is bilinear. This fact plays a crucial role in identifying the second  of a  with the group of alternating s on a lattice.