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!
Best Answer
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.