# Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (e.g., 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

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 ${\textstyle 1\leq i\leq n-1}$  such that $x_{i}=x_{i+1}$  then $f(x_{1},\ldots ,x_{n})=0$ .
2. whenever there exists ${\textstyle 1\leq i\neq j\leq n}$  such that $x_{i}=x_{j}$  then $f(x_{1},\ldots ,x_{n})=0$ .
3. if $x_{1},\ldots ,x_{n}$  are linearly dependent then $f(x_{1},\ldots ,x_{n})=0$ .

## Example

• The determinant 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 ji 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 unit 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 $g(x_{1},\ldots ,x_{n}):=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)})$  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 symmetric map is zero.
• The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating bilinear forms on a lattice.