Complex Representations of A4

Question

Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number of conjugacy classes , so there are 4, but how do i obtain they re dimension? I also know that $|G|= \sum_{i=1}^{n} n_i^2$ where the $n_i$ are the dimension but i dont see how that helps me. Thanks.

Answer 1

Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] \cong A_4 / (\Bbb Z_2 \times \Bbb Z_2) \cong \Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.

Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = \sum_{i = 1}^4 n_i^2 ,$$ where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.