Hi , I misunderstood the question sorry about that . I am new to group theory and quantum mechanics. Anyway,
I did the problem and got the symmetrized basis functions .
Next step would be to find the eigenvalues. The matrix I have is in block diagonal form.
\left(
\begin{array}{ccccc}
\left(
\begin{array}{c}
\beta
\end{array}
\right) & \left(
\begin{array}{c}
-\delta
\end{array}
\right) & \left(
\begin{array}{c}
-\sqrt{3} \gamma
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) \\
\left(
\begin{array}{c}
-\delta
\end{array}
\right) & \left(
\begin{array}{c}
\epsilon
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) \\
\left(
\begin{array}{c}
-\sqrt{3} \gamma
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
-2 \alpha
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) \\
\left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
\alpha
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) \\
\left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
0
\end{array}
\right) & \left(
\begin{array}{c}
\alpha
\end{array}
\right)
\end{array}
\right)
My question would be ? The basis functions from the 2-D representation looks like they are eigenfunctions but what about the 3*3 matrix that are in off diagonal form in the 5*5 matrix ? What do I do with that ?