Let $G_i$ be groups (in my case, compact simple and Lie), and $R_i$ finite-dimensional representations thereof. Also, $S^n$ and $\wedge^n$ denote symmetrized/anti-symmetrized tensor powers of a given representation.
We consider a representation of $G_1\times G_2\times\cdots$ of the form $(R_1,R_2,\dots)$. I am trying to reduce tensor powers of the form $$ S^n(R_1,R_2,\dots),\qquad \wedge^n(R_1,R_2,\dots) $$ into powers in the individual components. For example it is clear that $$ \begin{aligned} S^2(R_1,R_2)&=(S^2R_1,S^2R_2)\oplus (\wedge^2R_1,\wedge^2R_2)\\ \wedge^2(R_1,R_2)&=(S^2R_1,\wedge^2R_2)\oplus (\wedge^2R_1,S^2R_2) \end{aligned} $$ Higher powers are trickier to me. For example I expect something like $$ S^n(R_1,R_2)\overset?=\sum_{\lambda,\lambda'}(S^\lambda R_1,S^{\lambda'}R_2) $$ where $\lambda,\lambda'$ are suitable partitions of $n$ and $S^\lambda$ is the corresponding symmetrizer.
How can these products be reduced in general? Where can I find an (elementary, if possible) introduction to this problem? In particular, I would like to see how $S^3(R_1,R_2)$ and $\wedge^3(R_1,R_2)$ decompose. I might be able to guess the general structure given these two cases.