ligand pi system irreducible representation for trigonal bipyramidal transition metal complex I am considering a D$_{3h}$ trigonal bipyramidal transition metal complex MX$_5$.  I have found that the irreducible representation for the ligand s and $\sigma$ systems is 2A$_1'$+A$_2''$+E$'$.  I have
\begin{array}{|c|c|c|c|c|c|}
\hline
&amp;E&amp; 2C_3 &amp; 3C_2 &amp; \sigma_h &amp; 2S_3&amp;3\sigma_v\\ \hline
 \Gamma_s&amp;5&amp; 2&amp; 1&amp;3&amp;0&amp;3\\ \hline
 \Gamma_\sigma&amp;5&amp; 2&amp; 1&amp;3&amp;0&amp;3\\ \hline
 \Gamma_{\pi+\pi'}&amp;10&amp;0&amp;-2&amp;0&amp;0&amp;0\\ \hline
\end{array}
However, I think that my $\Gamma_{\pi+\pi'}$ is incorrect since I get fractional values for coefficients for the irreducible representations when I try to reduce it to irreducible representations.  But from sketching the trigonal pyramidal complex, I cannot see how my $\chi(R)$ are wrong for the pi system representation, $\Gamma_{\pi+\pi'}$.  Any help would be appreciated.
I believe you have the wrong character for the $\hat{C_3}$ rotation. None of the p orbitals in the equatorial positions remain in place so they all contribute zero. The axial p orbitals are rotated by $120^°$ which means they each contribute $\cos(120^°)=-\frac{1}{2}$. Since there are two of them at each of the two axial position, $\chi(\hat{C_3})=-2$. Using this new value for the character (assuming all the others are right, which they seemed to be), I obtained: $a_2',a_2'',2e',2e''$