# Relation between alternating and symmetric multilinear maps

Denote the set of all alternating multilinear mappings and the set of all symmetric multilinear maps from $V^{n}$ to $W$ by $\operatorname{Alt}_{F}(V^{n},W)$ and $\operatorname{Sym}_{F}(V^{n},W)$, respectively. The book I'm reading says:

Note that $\operatorname{Sym}_{F}(V^{n},W)\supseteq\operatorname{Alt}_{F}(V^{n},W)$ whenever $F\supseteq\mathbb{F}_{2}$.

According to the book, for every $\sigma\in S_{n}$, an alternating multilinear map $\phi$ would satisfy $$\phi(\alpha_{\sigma(1)},\ldots,\alpha_{\sigma(n)}) = \operatorname{sgn}(\sigma)\phi(\alpha_{1},\ldots,\alpha_{n})$$ while a symmetric multilinear map would satisfy $$\phi(\alpha_{\sigma(1)},\ldots,\alpha_{\sigma(n)}) = \phi(\alpha_{1},\ldots,\alpha_{n}).$$

Their definition is clearly different. What special property does $F$ exhibit when $F\supseteq\mathbb{F}_{2}$ such that alternating $F$-multilinear maps become symmetric?

## 1 Answer

Oh well, it really is… trivial.

Let $1_{F}$ be the multiplicative identity in field $F$.

If $F\supseteq\mathbb{F}_{2}$, then $\mathbb{F}_{2}$ is isomorphic to the prime subfield of $F$, so $F$ has characteristic $2$, then by definition we have $1_{F}+1_{F}=0$. Rearrange the terms to get $1_{F}=-1_{F}$.

The value of $\operatorname{sgn}(\sigma)$ is commonly written as $\pm1$, but this is just a shorthand for $\pm 1_{F}$. Because $1_{F}=-1_{F}$, $\operatorname{sgn}(\sigma)$ always is $1_{F}$, so for an alternating map $\phi$ over $F\supseteq\mathbb{F}_{2}$, $$\operatorname{sgn}(\sigma)\phi(\alpha_{1},\ldots,\alpha_{n}) = 1_{F}\cdot\phi(\alpha_{1},\ldots,\alpha_{n}) = \phi(\alpha_{1},\ldots,\alpha_{n}),$$ making $\phi$ a symmetric multilinear map.