Determine all irreducible representations of $A_{4}$ over $\mathbb{C}$ I have to determine all the irreducible representations of the alternating group $A_{4}$ over $\mathbb{C}$. So far I have found three, that are all one-dimensional. I believe there is one more that is three-dimensional, but I'm unsure how to find it so any help would be greatly appreciated! Also I was wondering if there was a way that I could show that these four representations are the only irreducible ones? 
 A: $A_4$ is an index 2 normal subgroup of $S_4$, which has a natural 3-dimensional irreducible representation: $S_4$ acts on $\Bbb{C}^4$ by permuting the vectors in some fixed choice of basis. This is a 4-dimensional representation, but it is not irreducible -- it has an invariant 1-dimensional subspace consisting of those vectors whose coordinates (with respect to the fixed choice of basis) sum to zero. Call the resulting three-dimensional quotient $\sigma$. By Clifford theory, as $A_4$ is normal in $S_4$ of index 2, either $\sigma$ restricts irreducibly to $A_4$, or $\sigma|_{A_4}=\sigma_1\oplus\sigma_2$, where $\sigma_1$ and $\sigma_2$ are two irreducible representations of $A_4$ which are conjugate in $S_4$. If representations are conjugate then they are equidimensional, and as 3 is odd this isn't possible. So $\sigma$ restricts irreducibly to $A_4$ and gives your missing representation.
If you aren't familiar with Clifford theory, you can work out the details pretty easily by using character theory.
