Alternating multilinear map

Let $V,W$ be vector spaces over a field $K$ , and let $k\ge1$ . A multilinear map $f:V^k\to W$ is alternating if

f(v_1,\ldots,v_k)=0 \quad\text{whenever }v_i=v_j\text{ for some }i\ne j.

Signs and characteristic

Swapping two arguments negates the value. The converse holds in characteristic different from $2$ , but not in characteristic $2$ : there, the bilinear map $(x,y)\mapsto xy$ on $K$ is skew-symmetric because $-1=1$ , yet its value at $(1,1)$ is nonzero.

For $k=1$ , the alternation condition is vacuous. Scalar-valued alternating multilinear maps are called alternating forms.

Example

The map $((x_1,x_2),(y_1,y_2))\mapsto x_1y_2-x_2y_1$ is an alternating bilinear form on $K^2$ .

References

Bernhard Leeb, Some multilinear algebra, 2020, §§1.1, 1.4 and 2.3.