Tensor product of irreducible representations

Question

Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $V,W$ two irreducible finite dimensional representations. When is $V\otimes W$ irreducible?

Answer 1

The answer is: almost never.

In fact, I believe this only happens when either $V$ or $W$ is the trivial module. For $\mathfrak{sl}_2$, this can be seen immediately by considering the Clebsch-Gordon formula. In general, I expect one can see this from computations using crystals.

Answer 2

Starting with the case $\mathfrak{sl_2}(\mathbb{C})$, the Clebsch-Gordon formula gives $$ V(r)\otimes V(s)=\bigoplus_{i=0}^sV ( r + s − 2 i ), $$ where $s\le r$, and $V(r),V(s)$ are highest-weight modules. The tensor product is irreducible for $s=0$, i.e., in the trivial case. Now this can be extended to the general case using combinatorics (crystals).