Why is an alternating multilinear map called "alternating"?

Question

I know it's just a small detail, but it's kind of bothering me... A multilinear map is called "alternating" if having two equal arguments makes it zero.

Why use the word "alternating" then? What precisely is alternating?

Answer 1

As mentioned in the comment, if the characteristic of the coefficient field is different from $2$, then the following statements are equivalent:

• $f(x, y, \dots) = -f(y, x, \dots)$ for any $x, y$;

• $f(x, x, \dots) = 0$ for any $x$.

The proof is an easy exercise.

In your question, you take the second statement as definition. But the first one is more suitable for the name "alternating".

Answer 2

A $k$-Tensor $\omega\in\mathcal{T}^{k}(V)$ is called alternating if $$\omega(v_{1},\cdots ,v_i,\cdots v_j,\cdots, v_n)=-\omega(v_{1},\cdots ,v_j,\cdots v_i,\cdots, v_n)$$ for $v_1,\cdots v_{n} \quad in \quad V$.

As such, that is the definition that it is altered that the multilinear application changed the position coordinates and fixed the rest