1. ## [SOLVED] Irreducible Representations

Let V be a finite dimensional representation of a finite group G. Consider Hom(V,V) = Set of all Linear Transformations from V to V. Its clearly a vector space (under the natural operations).

Prove that : Hom(V,V) is an irreducible representation of G $\Leftrightarrow$ dim V = 1

The problem for me is that I cant understand how the group GL(Hom(V,V)). It looks extremely complicated.

Trying to prove: $\Leftarrow$
I assume ( $\phi$,V) is the representation over the field F, dim V = 1. Then $Hom(V,V) \cong F$.
F is a one-dimensional vector space over itself and thus Hom(V,V) is one dimensional and hence irreducible.

To prove: $\Rightarrow$
I am struggling here. So any ideas?

Thank you,
Srikanth

2. Hello ppl,

Well... I solved it anyway. But I am still open to comments >_<

Proof of $\Rightarrow$

$\psi: G \to \text{GL(Hom(V,V))}, \psi_g(T) = \phi_g \circ T \circ \phi_{g}^{-1}$

Now consider the subspace of scalar matrices W. We can easily see that $\psi_g(\lambda \mathbb{I}_V) = \phi_g \circ \lambda \mathbb{I}_V \circ \phi_{g}^{-1} = \lambda \mathbb{I}_V \in W$. Thus W is non zero G-invariant subspace of Hom(V,V). But data says Hom(V,V) is irreducible and thus W = Hom(V,V). But W is clearly 1 dimensional ( $\mathbb{I}_V$ is a basis)