In their paper "[New lower bounds for the border rank of matrix multiplication](http://arxiv.org/abs/1112.6007)", Landsberg and Ottaviani make use of the fact that 


$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits\_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda\_1\le m }} \mathbb{S}\_\lambda V \otimes \mathbb{S}\_{\bar\lambda}W$$


where $\bar\lambda$ denotes the conjugate partition of $\lambda$. This isomorphism is basically Exercise 6.11 in Fulton & Harris, so there is no doubt about it. However, from what I gather, in Lemma 3.1 of the paper, they use the fact that the above isomorphism is given by the map


$$ (v\_1\otimes w\_1)\wedge\ldots\wedge(v\_p\otimes w\_p) \longmapsto \sum\nolimits\_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda\_1\le m }} c\_\lambda(v\_1\otimes\ldots\otimes v\_p) \otimes c\_{\bar\lambda}(w\_1\otimes\ldots\otimes w\_p), $$


where $c\_\lambda$ denotes the Young symmetrizer corresponding to the partition $\lambda$. I cannot find a proof for this. Can someone explain to me why the above map defines 


   **a)** a morphism of $\mathfrak{S}\_p$-modules and   

   **b)** a bijection? 


Since all vector spaces involved are of finite dimension and by $(\dagger)$, it would certainly suffice to show that it is either injective or surjective. 


Also, if I misunderstood the proof of Lemma 3.1 and the isomorphism is given by another elementary rule, please tell me what it is.



Question about decomposition of exterior product
This is well-known in the theory of symmetric functions and is one of two Cauchy identities.
You can find this in most books, for example, MacDonald Chapter I, Section 4. Orthogonality
equation (4.3').


Knuth's extension of the Robinson-Schensted correspondence gives bijective proofs of both Cauchy identities, for example see Stanley Enumerative Combinatorics 7.14.3 Theorem.


