Homework Help: Wedge Product as an (r+s) form

1. Oct 11, 2008

cathalcummins

1. The problem statement, all variables and given/known data

If $\omega \in \Lambda^{r}(V^*)$ and $\nu \in \Lambda^{s}(V^*)$ show that $\omega \wedge \nu \in \Lambda^{r+s}(V^*)$

2. Relevant equations

$\omega \wedge \nu (v_1,... ,v_r,v_{r+1},... ,v_{r+s})= \frac{1}{(r+s)!} \Sigma_{\sigma}(-1)^\sigma \omega(v_{\sigma(1)},v_{\sigma(2)},...,v_{\sigma(r)}) \nu(v_{\sigma(r+1)},v_{\sigma(r+2)},...,v_{\sigma(r+s)})$

3. The attempt at a solution

It would take really long to write out my solution thus far, I broke the sum on the right into even and odd sums. Then proceeded to show (I hope) that as $\omega$ and $\nu$ behave like $\omega$(odd interchange) $= - \omega$(no interchange) and the same for $\nu$.

Then the "odd sum" will have three negatives, one from the $(-1)^\sigma$ and two from the above relations. This leaves a net (-1) as per definition of an (r+s) form, similar reasoning is used for the even sum.

Combining these two, we have, up to a scaling factor, that this new object behaves just as an (r+s) form.

Cheers, it's mainly the permutation thing is bugging me.