Let $\rho : SL(2,\mathbb{C}) \rightarrow S^{k}(\mathbb{C}^{2})$ be the symmetric powers ($Sym^{k}(V)$) representation.
Given $\xi_{\rho}(t) = tr(\rho \left( \begin{bmatrix} t & 0 \\ 0 & t^{-1} \end{bmatrix}\right)) = \sum_{i=0}^{k} t^{2i-k}$, and $ \xi_{\rho \otimes \rho'}(t) = \xi_{\rho}(t) \xi_{\rho'}(t)$ for $\rho : SL(2,\mathbb{C}) \rightarrow S^{k}(\mathbb{C}^{2})$ and $\rho' : SL(2,\mathbb{C}) \rightarrow S^{m}(\mathbb{C}^{2})$
Determine the decomposition of $S^{k}(\mathbb{C}^{2}) \otimes S^{m}(\mathbb{C}^{2})$ into irreducible representations.
I am not really sure how to use the fact that $\xi_{\rho \otimes \rho'}(t) = \xi_{\rho}(t) \xi_{\rho'}(t)$, and don't know where to go. Any suggestions would be appreciated.
Find how to write $\xi_{\rho}(t)\xi_{\rho'}(t)$ as a sum of $\xi_{\rho''}(t)$'s. Try some small examples first like $\rho=\rho'=2$. Then you will be able to find a pattern and thus rediscover a result by Paul Gordan and Alfred Clebsch from the 1860's and 1870's.