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Note: I have updated my work section of this since I asked the question yesterday to reflect my understanding to this point.

Let $V$ and $W$ be two vector spaces. We denote by $S_2V$ the second symmetric power of $V$ and by $\bigwedge^2 V$ the second exterior power of $V$. Prove there exists a natural linear map $$ \phi\colon S_2(V\otimes W)\to S_2 V\otimes S_2 W $$ defined by the formula $$ \phi(v_1\otimes w_1)(v_2\otimes w_2) = (v_1 v_2\otimes w_1 w_2) $$ Show that the kernel of $\phi$ is functorially isomorphic to $\bigwedge^2 V\otimes \bigwedge^2 W$.

I think I understand how to prove the existence of a natural linear map using the universality property. If you saw my previous version, I was mistakenly trying to construct mappings from $V\otimes W$ when what is actually required is to construct the mappings from $(V\otimes W)\times(V\otimes W)$. It's fairly straightforward to set everything up and to show that the resulting map is natural.

I'm still confused about how to show that $\ker\phi$ is functorially isomorphic to $\bigwedge^2 V\otimes \bigwedge^2 W$.

I think this means that I need to construct a map $\bigwedge^2 V\otimes \bigwedge^2 W \to S_2(V\otimes W)$ and then understand what happens when I apply $\phi$. I'm not really sure about this approach, and I would appreciate any help you could offer.

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  • $\begingroup$ Your mapping $\psi_1$ is not linear (I am not sure it is even well defined, in fact!) $\endgroup$ – Mariano Suárez-Álvarez Oct 31 '12 at 5:01
  • $\begingroup$ I had a feeling that it might not be. As I mentioned, I'm not really sure how to interpret the spaces I'm looking at so it's kind of hard to define sensible mappings. $\endgroup$ – chris Oct 31 '12 at 5:04
  • $\begingroup$ Have you looked in a textbook dealing with this subject? (The 3rd chapter of Bourbaki's Algebra is a pretty meticulous treatement of all this) Reading up a bit is the best known cure for incredibly weak understanding of anything; this site is not particularly well adapted to fixing that, in fact. In any case: why do you have to prove this? Is this a homework problem? $\endgroup$ – Mariano Suárez-Álvarez Oct 31 '12 at 5:06
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    $\begingroup$ I have read over the textbook material several times, looking at different texts. I have not looked at the reference you provided. I thought I should try to work out some problems rather than assuming I'm understanding what I've read. This is not a homework problem. I'm preparing for an upcoming exam, and this problem was given last year. I don't think I said at any point that I must prove this, I would just like to understand it. $\endgroup$ – chris Oct 31 '12 at 5:28

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