Hello I am trying to construct the 5 dimensional irreducible representation of $A_5$.
Here is my attempt:
Take the irreducible character $\phi$ of $A_4$ that has the values:
$\phi(1)=1, \phi(1,2)=1, \phi((1,2),(3,4))=\omega$ and $\phi(1,2,3,4)=\omega^2$.
Consider $\text{Ind}(\phi)$.
It is 5-dimensional, since $[A_5:A_4]=5$.
By Forbenius Reciprocity we have
$$\langle \text{Ind}(\phi),\text{Ind}(\phi) \rangle_{A_5} =\langle \phi,\text{Res(Ind)}(\phi) \rangle_{A_4}=\langle \phi,\phi \rangle_{A_4}=1,$$
by the irreducibility of $\phi$. Hence, $\text{Ind}(\phi)$ is a 5-dimensional irreducible representation of $A_5$.
Is this correct?