Tensor product of two irreducible representations is not irreducible

Question

I have seen that if $\rho: G \longrightarrow \text{GL}_{\mathbb{C}}(V)$ and $\alpha: G \longrightarrow \text{GL}_{\mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its tensor product representation $\rho \otimes \alpha: G \longrightarrow \text{GL}_{\mathbb{C}}(V \otimes W)$ is usually not irreducible.

Can anyone tell me an example?

Answer 1

Take the standard representation $\rho$ of $S_3$ on $V=\{(z_1,z_2,z_3)\in\mathbb{C}^3\,|\,z_1+z_2+z_3=0\}$:$$\rho(\sigma)(z_1,z_2,z_3)=(z_{\sigma^{-1}(1)},z_{\sigma^{-1}(2)},z_{\sigma^{-1}(3)}).$$Then $\rho$ is irreducible, but $\rho\otimes\rho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).