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.

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 taking the dual space equipped with the structure of a representation in the canonical way.

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.