Definition
Let \(M\) be magnetic point group other than type I.
Let \(D\) be a maximal point subgroup of \(M\).
Then, \(M\) can be decomposed as \(M = D \sqcup a_{0} D\), where \(a_{0}\) is an antisymmetry operation.
A co-representation \(\Gamma\) gives linear or anti-linear operator for each element in \(M\) while satisfying
\[\begin{split}
\mathbf{\Gamma}(u) \mathbf{\Gamma}(u') &= \omega(u, u') \mathbf{\Gamma}(uu') \\
\mathbf{\Gamma}(u) \mathbf{\Gamma}(a') &= \omega(u, a') \mathbf{\Gamma}(ua') \\
\mathbf{\Gamma}(a) \mathbf{\Gamma}(u')^{\ast} &= \omega(a, u') \mathbf{\Gamma}(au') \\
\mathbf{\Gamma}(a) \mathbf{\Gamma}(a')^{\ast} &= \omega(a, a') \mathbf{\Gamma}(aa'), \\
\end{split}\]
where \(u, u' \in D\) and \(a, a' \in a_{0}D\).
The factor system \(\omega\) satisfies a cocycle condition:
\[\begin{split}
\omega(u, m) \omega(um, m') &= \omega(u, mm') \omega(m, m') \\
\omega(a, m) \omega(am, m') &= \omega(a, mm') \omega(m, m')^{\ast} \\
\end{split}\]
for \(u \in D\), \(a \in a_{0}D\), and \(m, m \in M\) .
Co-representations \(\Gamma\) and \(\Gamma'\) are equivalent if an invertible matrix \(\mathbf{T}\) exists such that
\[\begin{split}
\mathbf{\Gamma}'(u) &= \mathbf{T}^{-1} \mathbf{\Gamma}(u) \mathbf{T} \quad (u \in D) \\
\mathbf{\Gamma}'(a) &= \mathbf{T}^{-1} \mathbf{\Gamma}(a) \mathbf{T}^{\ast} \quad (a \in a_{0}D) \\
\end{split}\]
Frobenius-Schur indicator for co-representation
Let \((\Gamma, \mathrm{Span}_{\mathbb{C}} \{ \mathbf{\phi}_{i} \}_{i=1}^{d} )\) be one of unitary irreps for \(D\) with factor system \(\omega\).
Then, \(\{ a_{0} \mathbf{v}_{i} \}_{i}\) also form irrep as
\[\begin{split}
\overline{\mathbf{\Gamma}}(u)
&:= \frac{ \omega(u, a_{0}) }{ \omega( a_{0}, a_{0}^{-1} u a_{0} ) } \mathbf{\Gamma}( a_{0}^{-1} u a_{0} )^{\ast} \quad (u \in D) \\
u a_{0} \mathbf{\phi}_{j} &= \sum_{j} a_{0} \mathbf{\phi}_{j} \overline{\mathbf{\Gamma}}(u)_{ij}.
\end{split}\]
The following Frobenius-Schur indicator should be one of \(\{ -1, 0, 1 \}\):
\[\begin{split}
\xi^{\alpha} &:= \frac{1}{|D|} \sum_{ u \in D } \omega(a_{0}u, a_{0}u) \chi( (a_{0}u)^{2} ) \\
\chi(u) &:= \mathrm{Tr}\, \mathbf{\Gamma}(u) \quad (u \in D).
\end{split}\]
This indicator can check if \(\Gamma\) and \(\overline{\Gamma}\) are equivalent.
Case: \(\xi^{\alpha} = 1\)
In this case, \(\Gamma\) and \(\overline{\Gamma}\) are equivalent.
Let \(\mathbf{U}\) be a unitary intertwiner between \(\Gamma\) and \(\overline{\Gamma}\):
\[
\overline{\mathbf{\Gamma}}(u) = \mathbf{U}^{-1} \mathbf{\Gamma}(u) \mathbf{U} \quad (u \in D).
\]
Then, a transformed basis \(\{ \mathbf{\psi}_{i} := \sum_{j} a_{0} \mathbf{\phi}_{j} [\mathbf{U}^{\dagger}]_{ji} \}_{i}\) also forms \(\Gamma\).
A new basis \(\{ \frac{1}{\sqrt{2}}(\phi_{i} + \psi_{i}) \}_{i}\) gives the following irrep :
\[\begin{split}
\tilde{\mathbf{\Gamma}}(u) &= \mathbf{\Gamma}(u) \\
\tilde{\mathbf{\Gamma}}(a_{0}) &= \mathbf{U} \\
\tilde{\mathbf{\Gamma}}(a_{0}u) &= \omega(a_{0}, u)^{\ast} \tilde{\mathbf{\Gamma}}(a_{0}) \tilde{\mathbf{\Gamma}}(u) \\
\mathbf{U}\mathbf{U}^{\ast} &= \omega(a_{0}, a_{0})\Gamma(a_{0}^{2}).
\end{split}\]
Case: \(\xi^{\alpha} = -1\)
In this case, \(\Gamma\) and \(\overline{\Gamma}\) are equivalent.
Let \(\mathbf{U}\) be a unitary intertwiner between \(\Gamma\) and \(\overline{\Gamma}\):
\[\begin{split}
\overline{\Gamma}(u) &= \mathbf{U}^{-1} \Gamma(u) \mathbf{U} \quad (u \in D) \\
\mathbf{U}\mathbf{U}^{\ast} &= -\omega(a_{0}, a_{0})\Gamma(a_{0}^{2}).
\end{split}\]
We can take \((\mathbf{\phi}_{1}, \cdots, \mathbf{\phi}_{d}, \mathbf{\psi}_{1}, \cdots, \mathbf{\psi}_{d})\) as basis and they form irrep of \(M\) as
\[\begin{split}
\tilde{\mathbf{\Gamma}}(u) &=
\begin{pmatrix}
\mathbf{\Gamma}(u) & \mathbf{0} \\
\mathbf{0} & \overline{\mathbf{\Gamma}}(u) \\
\end{pmatrix} \\
\tilde{\mathbf{\Gamma}}(a_{0}) &=
\begin{pmatrix}
\mathbf{0} & -\mathbf{U} \\
\mathbf{U} & \mathbf{0} \\
\end{pmatrix} \\
\tilde{\mathbf{\Gamma}}(a_{0}u) &= \omega(a_{0}, u)^{\ast} \tilde{\mathbf{\Gamma}}(a_{0}) \tilde{\mathbf{\Gamma}}(u)
\end{split}\]
Case: \(\xi^{\alpha} = 0\)
In this case, \(\Gamma\) and \(\overline{\Gamma}\) are not equivalent.
We can take \((\mathbf{\phi}_{1}, \cdots, \mathbf{\phi}_{d}, a_{0}\mathbf{\phi}_{1}, \cdots, a_{0}\mathbf{\phi}_{d})\) as basis and they form irrep of \(M\) as
\[\begin{split}
\tilde{\mathbf{\Gamma}}(u) &=
\begin{pmatrix}
\mathbf{\Gamma}(u) & \mathbf{0} \\
\mathbf{0} & \overline{\mathbf{\Gamma}}(u) \\
\end{pmatrix} \\
\tilde{\mathbf{\Gamma}}(a_{0}) &=
\begin{pmatrix}
\mathbf{0} & \omega(a_{0}, a_{0}) \mathbf{\Gamma}(a_{0}^{2}) \\
\mathbf{1} & \mathbf{0} \\
\end{pmatrix} \\
\tilde{\mathbf{\Gamma}}(a_{0}u) &= \omega(a_{0}, u)^{\ast} \tilde{\mathbf{\Gamma}}(a_{0}) \tilde{\mathbf{\Gamma}}(u)
\end{split}\]
Convention of anti-linear operators in spgrep
Let \(M\) be magnetic point group other than type I.
Let \(D\) be a maximal point subgroup of \(M\).
Then, \(M\) can be decomposed as \(M = D \sqcup D a_{0}\), where \(a_{0} = \mathbf{S}_{0} 1'\) is an antisymmetry operation.
We choose the following convention for choosing unitary or anti-unitary matrices for \(M\):
\[\begin{split}
D \ni \mathbf{S} &\mapsto \mathbf{U}(\mathbf{S}) \\
D a_{0} \ni \mathbf{S}1' &\mapsto \mathbf{U}(\mathbf{S}) \left( -i \sigma_{y} \right) K,
% a_{0} &\mapsto \mathbf{U}(\mathbf{S}_{0}) \left( -i \sigma_{y} \right) K \\
% a_{0} D \ni a_{0} \mathbf{S} &\mapsto \mathbf{U}(\mathbf{S}_{0}) \left( -i \sigma_{y} \right) \mathbf{U}(\mathbf{S})^{\ast} K,
\end{split}\]
where \(\mathbf{U}: \mathrm{O}(3) \to \mathrm{SU}(2)\) is defined here.
\(\sigma_{y}\) is the Pauli matrix.
\(K\) is an anti-linear operator that takes complex conjugate.