Numerically obtain intertwiner between equivalent representations#
Intertwiner between two projective representations of finite group#
Let \(\Delta^{(0)}\) and \(\Delta^{(1)}\) be projective representations of a finite group \(H\). An intertwiner of \(\Delta^{(0)}\) and \(\Delta^{(1)}\) is a matrix \(\mathbf{U}\) satisfying
\[
\mathbf{\Delta}^{(0)}(S) \mathbf{U} = \mathbf{U} \mathbf{\Delta}^{(1)}(S)
\quad (\forall S \in H)
\]
The intertwiner is unique up to scalar multiplication: if \(\mathbf{U}\) and \(\mathbf{U}'\) are intertwiner between \(\Delta^{(0)}\) and \(\Delta^{(1)}\), \(\mathbf{U}^{-1}\mathbf{U}'\) should be written as \(c\mathbf{I}\) with some complex number \(c\) from Schur’s lemma.
In particular, when \(\Delta^{(0)}\) and \(\Delta^{(1)}\) are unitary irreps, we can choose \(\mathbf{U}\) as a unitary matrix [1].
The following matrix is an intertwiner for these projective representations:
\[
\mathbf{U} = \sum_{S \in H} \mathbf{\Delta}^{(0)}(S) \mathbf{B} \mathbf{\Delta}^{(1)}(S)^{-1},
\]
where \(\mathbf{B}\) is any matrix.
Intertwiner between two space-group representations of space group \(\mathcal{G}\)#
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 \(\mathcal{T}\):
\[
\mathcal{G}^{\mathbf{k}} = \coprod_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } (\mathbf{S}_{i}, \mathbf{w}_{i}) \mathcal{T}.
\]
Consider a similar matrix as the above finite-group case and take summation over lattice points beforehand:
\[\begin{split}
\mathbf{U}
&:= \frac{1}{N} \sum_{ \mathbf{t} } \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{(\mathbf{k}, 0)}\left( (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{(\mathbf{k}, 1)}\left( (\mathbf{E}, \mathbf{t})(\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{-1} \\
&= \left(
\frac{1}{N} \sum_{ \mathbf{t} } 1
\right)
\sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{(\mathbf{k}, 0)}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{(\mathbf{k}, 1)}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{-1} \\
&= \sum_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} }
\mathbf{\Gamma}^{(\mathbf{k}, 0)}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)
\mathbf{B}
\mathbf{\Gamma}^{(\mathbf{k}, 1)}\left( (\mathbf{S}_{i}, \mathbf{w}_{i}) \right)^{-1} \\
\end{split}\]