\[\begin{split}
\frac{1}{|G|} \sum_{ g \in G } \chi(g^{2})
= \begin{cases}
1 & \mbox{($\Gamma$ is real)} \\
-1 & \mbox{($\Gamma$ is pseudo-real)} \\
0 & \mbox{($\Gamma$ is not equivalent to $\Gamma^{\ast}$)} \\
\end{cases}.
\end{split}\]
PIR of finite group
(1) \(\Gamma\) is real
The following construction is based on Ref. [ITO96].
In this case, since \(\Gamma\) and \(\Gamma^{\ast}\) are equivalent, there exists a symmetric unitary matrix with
\[\begin{split}
\mathbf{\Gamma}(g) \mathbf{U} &= \mathbf{U} \mathbf{\Gamma}(g)^{\ast} \\
\mathbf{U}^{\dagger} \mathbf{U} &= \mathbf{1} \\
\mathbf{U}^{\top} &= \mathbf{U}.
\end{split}\]
We impose \(\det \mathbf{U} = 1\) additionally to specify the intertwiner uniquely.
The intertwiner can be numerically computed as shown in here.
The symmetric unitary matrix \(\mathbf{U}\) can be diagonalized with real orthogonal matrix \(\mathbf{S}\) as \(\mathbf{U} = \mathbf{S}^{-1} \mathbf{\Omega} \mathbf{S}\) .
For this symmetric unitary matrix \(\mathbf{U}\), we can choose its square root with
\[\begin{split}
\mathbf{T} &:= \mathbf{S}^{-1} \mathbf{\Omega}^{1/2} \mathbf{S} \\
\mathbf{T}^{2} &= \mathbf{U} \\
\mathbf{T}^{\dagger} \mathbf{T} &= \mathbf{1} \\
\mathbf{T}^{\top} &= \mathbf{T}.
\end{split}\]
A transformed representation \(\mathbf{\Gamma}'(g) := \mathbf{T}\mathbf{\Gamma}(g)\mathbf{T}^{-1}\) is real because
\[\begin{split}
\mathbf{\Gamma}'(g)
&= \mathbf{T} \mathbf{U}^{-1} \mathbf{\Gamma}(g)^{\ast} \mathbf{U} \mathbf{T}^{-1} \\
&= \mathbf{T}^{-1} \mathbf{\Gamma}(g)^{\ast} \mathbf{T} \\
&= \mathbf{\Gamma}'(g)^{\ast}.
\end{split}\]
(2, 3) \(\Gamma\) is pseudo-real or not equivalent to \(\Gamma^{\ast}\)
In these cases, we can transform conjugated basis pair to real vectors by unitary matrix:
\[\begin{split}
(\mathbf{v}_{1}, \cdots, \mathbf{v}_{d}, \mathbf{v}_{1}^{\ast}, \cdots, \mathbf{v}_{d}^{\ast}) \mathbf{U}
&= \sqrt{2} (\mathrm{Re}\, \mathbf{v}_{1}, \cdots, \mathrm{Re}\, \mathbf{v}_{d}, \mathrm{Im}\, \mathbf{v}_{1}, \cdots, \mathrm{Im}\, \mathbf{v}_{d}) \\
\mathbf{U} &:= \frac{1}{\sqrt{2}}\begin{pmatrix}
\mathbf{1}_{d} & -i \mathbf{1}_{d} \\
\mathbf{1}_{d} & i \mathbf{1}_{d} \\
\end{pmatrix} \quad (\mathrm{Unitary}) \\
\mathbf{U}^{-1}
\begin{pmatrix}
\mathbf{D}(g) & \\
& \mathbf{D}(g)^{\ast}
\end{pmatrix}
\mathbf{U}
&= \begin{pmatrix}
\mathrm{Re}\, \mathbf{D}(g) & \mathrm{Im}\, \mathbf{D}(g) \\
-\mathrm{Im}\, \mathbf{D}(g) & \mathrm{Re}\, \mathbf{D}(g) \\
\end{pmatrix}
\quad (g \in G)
\end{split}\]
PIR of space group \(\mathcal{G}\)
Next, we consider to construct PIR of space group from small representation \(\Gamma^{\mathbf{k}\alpha}\) at \(\mathbf{k}\) [SCC13].
Frobenius-Schur indicator for space-group representations
Let \(\Gamma^{(\mathbf{k}, 0)}\) and \(\Gamma^{(\mathbf{k}, 1)}\) be representations of space group \(\mathcal{G}\) with \(\mathbf{k}\) vector.
Let \(\mathcal{T}\) be a translational subgroup of \(\mathcal{G}\).
Consider coset representatives of little group of \(\mathbf{k}\) over \(\mathcal{T}\):
\[
\mathcal{G}^{\mathbf{k}} = \coprod_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } (\mathbf{S}_{i}, \mathbf{w}_{i}) \mathcal{T}.
\]
Then sum over translations in \(\mathcal{G}^{\mathbf{k}}\):
\[\begin{split}
\frac{1}{|\mathcal{G}^{\mathbf{k}}|} \sum_{ g \in \mathcal{G}^{\mathbf{k}} } \chi^{\mathbf{k}}(g^{2})
&= \frac{1}{N} \sum_{ \mathbf{t} } \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\chi^{\mathbf{k}}\left( (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) \right) \\
&= \frac{1}{N} \sum_{ \mathbf{t} } \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\chi^{\mathbf{k}}\left( (\mathbf{E}, \mathbf{t} + \mathbf{S}_{i}\mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i})^{2} \right) \\
&= \left(
\frac{1}{N} \sum_{ \mathbf{t} } e^{-i \mathbf{k} \cdot (\mathbf{t} + \mathbf{S}_{i}\mathbf{t}) }
\right)
\left(
\sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } \chi^{\mathbf{k}}\left( (\mathbf{S}_{i}, \mathbf{w}_{i})^{2} \right) \\
\right) \\
&= \mathbb{I}\left[ 2\mathbf{k} \equiv \mathbf{0} \right] \cdot
\sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } \chi^{\mathbf{k}}\left( (\mathbf{S}_{i}, \mathbf{w}_{i})^{2} \right),
\end{split}\]
where \(\mathbb{I}[C]\) takes one if the condition \(C\) is true and takes zero otherwise.
Here we use \(\mathbf{S}_{i}^{\top} \in \overline{\mathcal{G}}^{\mathbf{k}}\) as
\[\begin{split}
\frac{1}{N} \sum_{ \mathbf{t} } e^{-i \mathbf{k} \cdot (\mathbf{t} + \mathbf{S}_{i}\mathbf{t}) }
&= \frac{1}{N} \sum_{ \mathbf{t} } e^{-i (\mathbf{E} + \mathbf{S}_{i})^{\top} \mathbf{k} \cdot \mathbf{t} } \\
&= \mathbb{I} \left[ (\mathbf{E} + \mathbf{S}_{i})^{\top} \mathbf{k} \equiv \mathbf{0} \right] \\
&= \mathbb{I} \left[ \mathbf{k} + \mathbf{S}_{i}^{\top} \mathbf{k} \equiv \mathbf{0} \right] \\
&= \mathbb{I} \left[ 2\mathbf{k} \equiv \mathbf{0} \right].
\end{split}\]
(1) \(\Gamma^{\mathbf{k}\alpha}\) is real
At first, we need to find an intertwiner
\[
\mathbf{\Gamma}^{\mathbf{k}\alpha}(g) \mathbf{U} = \mathbf{U} \mathbf{\Gamma}^{\mathbf{k}\alpha}(g)^{\ast}.
\]
Because we can assume \(2\mathbf{k} \equiv \mathbf{0}\) in this case, the following matrix is an intertwiner:
\[\begin{split}
\mathbf{U}
&:= \frac{1}{N} \sum_{ \mathbf{t} } \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{\ast \dagger} \\
&= \left(
\frac{1}{N} \sum_{ \mathbf{t} } e^{2i\mathbf{k}\cdot\mathbf{t}}
\right)
\sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{\ast \dagger} \\
&= \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{\mathbf{k}\alpha}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{\ast \dagger}
\quad (\because 2\mathbf{k} \cdot \mathbf{t} \in \mathbb{Z}) \\
\end{split}\]
(2, 3) \(\Gamma^{\mathbf{k}\alpha}\) is pseudo-real or not equivalent to \(\Gamma^{\mathbf{k}\alpha \ast}\)
In these cases, we can transform conjugated basis pair to real vectors by unitary matrix:
\[\begin{split}
(\mathbf{v}_{1}, \cdots, \mathbf{v}_{d}, \mathbf{v}_{1}^{\ast}, \cdots, \mathbf{v}_{d}^{\ast}) \mathbf{U}
&= \sqrt{2} (\mathrm{Re}\, \mathbf{v}_{1}, \cdots, \mathrm{Re}\, \mathbf{v}_{d}, \mathrm{Im}\, \mathbf{v}_{1}, \cdots, \mathrm{Im}\, \mathbf{v}_{d}) \\
\mathbf{U} &:= \frac{1}{\sqrt{2}}\begin{pmatrix}
\mathbf{1}_{d} & -i \mathbf{1}_{d} \\
\mathbf{1}_{d} & i \mathbf{1}_{d} \\
\end{pmatrix} \quad (\mathrm{Unitary}) \\
\tilde{\mathbf{\Gamma}}^{\mathbf{k}\alpha}(g)
&:=
\mathbf{U}^{-1}
\begin{pmatrix}
\mathbf{\Gamma}^{\mathbf{k}\alpha}(g) & \\
& \mathbf{\Gamma}^{\mathbf{k}\alpha}(g)^{\ast}
\end{pmatrix}
\mathbf{U} \\
&= \begin{pmatrix}
\mathrm{Re}\, \mathbf{\Gamma}^{\mathbf{k}\alpha}(g) & \mathrm{Im}\, \mathbf{\Gamma}^{\mathbf{k}\alpha}(g) \\
-\mathrm{Im}\, \mathbf{\Gamma}^{\mathbf{k}\alpha}(g) & \mathrm{Re}\, \mathbf{\Gamma}^{\mathbf{k}\alpha}(g) \\
\end{pmatrix}
\quad (g \in \mathcal{G})
\end{split}\]
In particular, for translation \((\mathbf{E}, \mathbf{t}) \in \mathcal{G}\),
\[\begin{split}
\tilde{\mathbf{\Gamma}}^{\mathbf{k}\alpha}( (\mathbf{E}, \mathbf{t}) )
&= \begin{pmatrix}
\cos (\mathbf{k} \cdot \mathbf{t}) \mathbf{1}_{d} & -\sin (\mathbf{k} \cdot \mathbf{t}) \mathbf{1}_{d} \\
\sin (\mathbf{k} \cdot \mathbf{t}) \mathbf{1}_{d} & \cos (\mathbf{k} \cdot \mathbf{t}) \mathbf{1}_{d} \\
\end{pmatrix}
\end{split}\]