TITLE: Exterior and symmetric  powers of external tensor products of representations    
QUESTION [10 upvotes]: Let us assume that $\pi: G\to Aut(V)$  and $\rho : K\to Aut(W)$ are two finite-dimensional representations of two Lie groups $G$ and $K$, and   consider the representation
  $\pi\hat{\otimes}\rho : G\times K\to  Aut(V\otimes W)$, the so called external tensor product of $\pi$ and $\rho$, given by
$$
(\pi\hat{\otimes}\rho)(g, k)(v\otimes  w):= \pi(g)v\otimes \rho(k)w,
$$
for any $g\in G$, $k\in K$, $v\in V$ and $w\in W$. For the second exterior power of the representation  $\pi\hat{\otimes}\rho$ is known the following 
isomorphism:
$$
\Lambda^{2}(\pi\hat{\otimes}\rho)=(\Lambda^{2}\pi \  \hat{\otimes}\ Sym^{2}\rho) \ \oplus  (Sym^{2}\pi \ \hat{\otimes} \ \Lambda^{2}\rho).
$$
Similarly, for the second symmetric power it holds that
$$
Sym^{2}(\pi\hat{\otimes}\rho)=(Sym^{2}\pi \  \hat{\otimes}\ Sym^{2}\rho) \ \oplus  (\Lambda^{2}\pi \ \hat{\otimes} \ \Lambda^{2}\rho).
$$
I would like to understand  how these formulas can be  generalized for exterior and symmetric powers of bigger degree.  For example, what we can say about 
$$
\Lambda^{3}(\pi\hat{\otimes}\rho), \ \Lambda^{4}(\pi\hat{\otimes}\rho),  \ Sym^{3}(\pi\hat{\otimes}\rho), \ Sym^{4}(\pi\hat{\otimes}\rho), \ \dots \ ?
$$
For the isomorphisms above, you can see for example the link
An isomorphism of 2-Schur modules
Thank you!

REPLY [11 votes]: One has
$$
Sym^k(\pi\otimes\rho) = \bigoplus_{|\alpha|=k} \Sigma^\alpha(\pi)\otimes\Sigma^\alpha(\rho),
$$
the sum is over all Young diagrams with $k$ boxes, $\Sigma^\alpha$ is the Schur functor. Similarly,
$$
\Lambda^k(\pi\otimes\rho) = \bigoplus_{|\alpha|=k} \Sigma^\alpha(\pi)\otimes\Sigma^{\alpha^T}(\rho),
$$
where $\alpha^T$ is the transposed diagram.