Prove properties of the alternating multilinear map

Question

Let $V$ be a vector space over a field $K$ (with $\operatorname{char}(K) \neq 2$) and let $F:L^n(V,K)\to L^n(V,K)$ be a map defined by $$F(g)(x_1, \dotsc, x_n) := \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma)g(x_{\sigma(1)},\dotsc, x_{\sigma(n)}).$$ Prove the following:
(a) $F$ is a linear map.
(b) $F(g) = g$ for every $g\in A^n(V,K)$
(c) $\operatorname{im} F = A^n(V,K)$.
(d) $F\circ F = F$.

$A^n(V,K)\subseteq L^n(V,K)$ is the vector space of alternating multilinear maps.

Hi guys,

is anyone able to help me with the problem attached?

I think I managed to solve (a) but I don't really have a clue about (b), (c) and (d).

Thanks for your help in advance. Ralf

Answer 1

Some observations:

• if $g$ is a multilinear alternating map, then $g(x_{\sigma(1)}, \ldots, x_{\sigma(n)}) = \text{sgn}(\sigma)g(x_1, \ldots, x_n)$. Since $\text{sgn}(\sigma)^2 = 1$ for all $\sigma$ and there are $n!$ permutations $\sigma$, substituting this into the definition of $F$ will take care of b)
• b) shows that $A^n(V, K) \subset \text{im}F$, since every multilinear alternating map is mapped by $F$ to itself. Can you show the other direction?
• d) follows immediately from b) and c): since $F$ maps anything to a multilinear alternating map, and $F$ fixes any multilinear alternating map, $F$ is idempotent.