By "Cauchy decomposition" I mean the following identity, both sides in which are representations of $GL_n(\mathbb C)\times GL_m(\mathbb C)$: $$\mathrm{Sym}^p(V\otimes W)=\bigoplus_{\lambda\vdash p} V^\lambda\otimes W^\lambda.$$ In the above $V=\mathbb C^n$ with $GL_n(\mathbb C)$ acting on in the natural way, similarly $W=\mathbb C^m$. The sum is over all Young diagrams with $p$ squares and of height no more than $\mathrm{min}(n,m)$. Finally, $V^\lambda$ and $W^\lambda$ denote the irreducible representation of the corresponding $GL$ with its highest weight given by $\lambda$. (Is there a more appropriate name for this fact?)

Well, an analogous identity also holds: $$\mathrm{Sym}^p(V^*\otimes W)=\bigoplus_{\lambda\vdash p} (V^\lambda)^*\otimes W^\lambda.$$ Here the asterisk simply denotes the dual of a module.

What would be the best (or any) paper to cite in this case? Of course, I'm hoping to see precisely this dual version written down somewhere.