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 polynomial.

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?

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?