The Cauchy identity states that
$$
\prod\_{i,j} \frac{1}{1-x\_i y\_j} = \sum\_\lambda s\_\lambda(x) s\_\lambda(y),
$$
where $s\_\lambda(x)$ is the Schur function.


Is there a known decomposition of the product
$$
\prod\_{i,j,k} \frac{1}{1-x\_i y\_j z\_k}
$$
as a sum of Schur functions? 


Essentially equivalently, let $U$, $V$ and $W$ be finite dimensional complex vector spaces. Is the irreducible decomposition of $Sym(U \otimes V \otimes W)$ known as a $GL(U) \times GL(V) \times GL(W)$-module?



Cauchy identity in three sets of variables?
Yes, up to the hard problem of determining Kronecker coefficients. Let $\Delta^\lambda$ be the Schur functor for the partition $\lambda$ of $r$ and let $S^\lambda$ be the corresponding irreducible representation of $S\_r$ with character $\chi^\lambda$. Schur–Weyl duality states that 


$$ U^{\otimes r} \cong \bigoplus\_{\lambda \in \mathrm{Par}(r)} \Delta^\lambda(U) \otimes S^\lambda $$ 


as a representation of $\mathrm{GL}(U) \times S\_r$.


Take $\dim U, \dim V, \dim W \ge r$. It follows by taking $S\_r$ invariants in $U^{\otimes r} \otimes V^{\otimes r} \otimes W^{\otimes r}$ that the multiplicity of the irreducible $\mathrm{GL}(U) \times \mathrm{GL}(V) \times \mathrm{GL}(W)$-module $\Delta^\lambda(U) \otimes \Delta^\mu(V) \otimes \Delta^\nu(W)$ in $\mathrm{Sym}^r (U \otimes V \otimes W)$ is the Kronecker coefficient $g\_{\lambda\mu\nu} = \langle \chi^\lambda\chi^\mu\chi^\nu, 1\_{S\_n} \rangle$. Restated in symmetric polynomials, this says


$$ \prod\_{ijk}\frac{1}{1-x\_iy\_jz\_k} = \sum\_{\lambda\mu\nu}g\_{\lambda\mu\nu} s\_\lambda(x)s\_\mu(y)s\_\nu(z)$$


