Alternating $k$-linear maps

Let $F$ be a field and let $V,W$ be $F$-vector spaces (Vector space over a field). For $k\ge 0$, a map

$f:V^k⟶W$

is $k$-linear (multilinear) when it is linear in each of its $k$ arguments separately, with the other arguments held fixed. It is alternating when

$f(v_1,\dots ,v_k)=0\text{whenever }{v}_i=v_j\text{ for some }1\le i<j\le k\mathrm{.}$

For $k=0$, a $0$-linear map is a choice of one element of $W$; for $k=1$, a $1$-linear map is a linear map. Both are alternating vacuously, because there is no pair of arguments to compare.

When $2\mathrm{\ne }0$ in $F$, alternation is equivalent to the sign rule $f(\dots ,v_i,\dots ,v_j,\dots )=-f(\dots ,v_j,\dots ,v_i,\dots )$: expand $f(\dots ,v_i+v_j,\dots ,v_i+v_j,\dots )=0$ and use $2\mathrm{\ne }0$. In characteristic two the sign rule collapses to the tautology $x=-x$ and does not imply alternation; the published witness is Over $\mathbb{Z}\mathrm{/}2$, an antisymmetric bilinear form need not be alternating. This page therefore works directly with the repeated-argument condition, never with a division by $k!$.