Problem: I want to do some calculations with the character projection operator to investigate the irreducible representations of wave functions. Until now, I did these calculations for simple materials like Copper, where spin-orbit-coupling is not needed, and the corresponding groups are the normal point groups. Their charactertables can be calculated with several methods. One of these could be the method of Dixon and Schneider which is used by GAP. Now I want to calculate systems with spin-orbit-coupling, which results in the need for double groups. Unfortunately, the characters of the double groups are not the same, I calculate from rules from the literature (e.g. Group Theory: Application to the Physics of Condensed Matter by Dresselhaus et al.) compared with the results from GAP.
Question: Are double groups something special, so that none of the known algorithms used by GAP show the right result?
The rules I mentioned are, at first, the orthogonality relations for the rows and columns of the character table. ($h$ is the order of group $\mathcal{G}$, $N_k$ is the number of elements in class $\mathcal{C}_k$ and $\chi_i(R)$ is the character of element $R$ in the $i$-th irreducible representation) $$\begin{align} \sum_R \chi_i(R)\chi_j(R)^* &= h \delta_{i,j}\\ \sum_i \chi_i(\mathcal{C}_k)\chi_i(\mathcal{C}_l)^* &= \frac{h}{N_k}\delta_{k,l} \end{align}$$ Furthermore, it is said, that if the elements of $\mathcal{G}$ and $\bar{\mathcal{G}}$ (which is not a group because of the missing unit element) form separate classes in the double group $\mathcal{D}$ (which is now the union of $\mathcal{G}$ and $\bar{\mathcal{G}}$) their characters have two relationships: In the "old" irreducible representations the characters of the classes are equal ($\chi_i(R) = \chi_i(\bar{R})$). In the "new" irreducible representations, their characters are of opposite sign ($\chi_i(R) = -\chi_i(\bar{R})$).