2 fixed typo

In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that

$$\tag{\dagger} {\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits_{\substack{\lambda\vdash n\\\\\ell(\lambda)\le p\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} \mathbb{S}_\lambda V \otimes \mathbb{S}_{\bar\lambda}W$$

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

$$(v_1\otimes w_1)\wedge\ldots\wedge(v_p\otimes w_p) \longmapsto \sum\nolimits_{\substack{\lambda\vdash n\\\ell(\lambda)\le p\\\ell(\lambda)\le n\\\lambda_1\le m }} c_\lambda(v_1\otimes\ldots\otimes v_p) \otimes c_{\bar\lambda}(w_1\otimes\ldots\otimes w_p),$$

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.

1

Question about decomposition of exterior product

In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that

$$\tag{\dagger} {\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits_{\substack{\lambda\vdash n\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} \mathbb{S}_\lambda V \otimes \mathbb{S}_{\bar\lambda}W$$

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

$$(v_1\otimes w_1)\wedge\ldots\wedge(v_p\otimes w_p) \longmapsto \sum\nolimits_{\substack{\lambda\vdash n\\\ell(\lambda)\le n\\\lambda_1\le m }} c_\lambda(v_1\otimes\ldots\otimes v_p) \otimes c_{\bar\lambda}(w_1\otimes\ldots\otimes w_p),$$

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.