irreductibility of induced representation

Question

Let $\rho: G \longrightarrow \text{GL}_{\mathbb{C}}(V)$ and $\alpha: G \longrightarrow \text{GL}_{\mathbb{C}}(W)$ be two irreducible representations of a finite gr0up $G$.

Is true that the induced representation $G \longrightarrow \text{GL}_{\mathbb{C}}\big(\text{Hom}(V,W)\big)$ is always irreducible?

I have been trying to prove it by contradiction but I am lost. Any help?

Answer 1

I assume that by $Hom(V,W)$ you understand the set of all linear maps $V\to W$ (well it doesn't make much sense for it to mean anything else). Then the answer is "no". If $\rho$ is of maximal dimension with $\dim_{\mathbb{C}} V>1$ and $\alpha=\rho$ then

$$\dim_{\mathbb{C}} Hom(V,V)=(\dim_{\mathbb{C}} V)^2>\dim_{\mathbb{C}} V$$

is just too big and thus it has to be reducible.

Such an example is given by $S_3$ which has $3$ irreducible representations over $\mathbb{C}$: two of dimension $1$ and one of (maximal) dimension $2$. You can read more about it here.

Answer 2

A nice direct counter-example is that for $V=W$, the space $Hom(V,V)$ always contains the one-dimensional invariant subspace spanned by the identity map.