Let $k$ be an algebraically closed field. All considered representations are linear. Suppose $G$ is a group, not necessarily finite.
Suppose $V$ and $W$ are irreducible, finite dimensional representations of $G$ over $k$. Is it the case that $V\otimes W$ is irreducible as a representation of $G$? I think the answer is affirmative if we consider the tensor product representation on $G\times G$ as a consequence of the density theorem, but I'm not sure about the question I ask.
Best Answer
If I understand your question right, the answer is definitely no. The task of writing the tensor-product of two irreduciple representations as a sum of irreducible representations is known as the "Clebsch-Gordan problem", and can be quite intricate depending on the group in question.
See here for some general remarks, or here for a detailed discussion in the $SU(2)$ case (which is is important in physics, and therfore very well developed).