I'm self studying some group representation theory and I've hit a wall with Exercise 3.10 (Computing the character table of $G = SL_2(\mathbb{Z}/3)$) in Fulton and Harris' book. Any advice or hints would be really appreciated!
Here's what I know so far:
(1) The representatives of the conjugacy classes can be chosen to be:
$I, -I, A_1 = \left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right), A_2 = \left(\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array} \right), B_1 = \left(\begin{array}{cc} -1 & 1 \\ 0 & -1 \end{array} \right), B_2 = \left(\begin{array}{cc} -1 & -1 \\ 0 & -1 \end{array} \right), \text{ and } C = \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$
Thus there are 7 irreducible representations of $G$.
There are 1, 1, 4, 4, 4, 4, and 6 conjugates for each of these representatives respectively (see this post).
(2) $G/\{\pm I\}$ has a faithful action on the lines in the $\mathbb{Z}/3$ plane, of which there are 4, thus $G/\{\pm I\}$ is a subgroup of $S_4$ of size 12, thus it must be $A_4$. This allows us to pullback the irreducible representations of $A_4$ to get irreducible representations of $G$. By doing this we can obtain 4 rows of the character table:
\begin{equation*} \begin{array}{|c|c|c|c|c|c|c|c|} \hline & 1 & 1 & 4 & 4 & 4 & 4 & 6 \\ \hline SL_2(\mathbb{Z}/3) & I & -I & A_1 & A_2 & B_1 & B_2 & C \\ \hline U & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline U' & 1 & 1 & \omega & \omega^2 & \omega^2 & \omega & 1 \\ \hline U'' & 1 & 1 & \omega^2 & \omega & \omega & \omega^2 & 1 \\ \hline V & 3 & 3 & 0 & 0 & 0 & 0 & -1 \\ \hline W & d_1 & ? & ? & ? & ? & ? & ? \\ \hline W' & d_2 & ? & ? & ? & ? & ? & ? \\ \hline W'' & d_2 & ? & ? & ? & ? & ? & ? \\ \hline \end{array} \end{equation*}
(3) For the remaining three, using the fact that the sum of the squared dimensions of the irreducible representations is $|G| = 24$, we can deduce that $d_1^2+d_2^2+d_3^2 = 12$, implying that we must have $d_1 = d_2 = d_3 = 2$.
My question is, how can we determine the remaining three characters of the 2 dimensional representations? I've tried looking at tensor products of the first 4 representations among other things and nothing seems to work.
Note:
- This is over $\mathbb{C}$.
- The person in this post was considering a similar problem, but ran into the same issue.
Thanks in advance!