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!