On-the-fly irreps generation from solvable group chain#

This page describes an algorithm to generate irreps for solvable groups [Net73].

Subduced and induced representations for solvable group#

Let \(H\) be invariant subgroup of \(G\) such that \(G/H = \mathbb{Z}_{p}\) (\(p\) is prime and \(r^{p}=e\)),

(1)#\[ G = \coprod_{m=0}^{p-1} r^{m} H. \]

Consider projective irrep \(\Delta\) with factor system \(\mu\) for \(H\),

\[ S \psi_{j} = \sum_{i=1}^{d} \psi_{i} \Delta(S)_{ij} \quad (R \in H, i = 1, \dots, d). \]

The following representations are also irrep over \(H\):

\[\begin{split} \mathbf{\Delta}^{(m)}(S) &:= \frac{ \mu(S, r^{m}) }{ \mu(r^{m}, S_{m}) } \mathbf{\Delta}(S_{m}) \quad (S \in H, m = 0, \dots, p-1) \\ S_{m} &:= r^{-m} S r^{m} (\in H) \end{split}\]

There are two cases for \(\{ \Delta^{(m)} \}_{m=0, \dots, p-1}\):

They are mutually inequivalent: \(\Delta \uparrow G \downarrow H \cong \sum_{m=0}^{p-1} \Delta^{(m)}\)
They are all equivalent: \(\Delta \uparrow G \downarrow H \cong p \Delta\)

Case-1: conjugated irreps are mutually inequivalent#

For case-1, the induced representation \(\Delta \uparrow G\) is irrep [1] [2]:

\[\begin{split} \Delta \uparrow G (r^{m} S)_{i:, j:} &= \mathbb{I}[i \equiv j + m] \frac{ \mu(r^{m}S, r^{j}) }{ \mu(r^{i}, S_{j} ) } \mathbf{\Delta}( S_{j} ) \quad (S \in H).\\ \Delta \uparrow G (S) &= \begin{pmatrix} \mathbf{\Delta}^{(0)}(S) & & & \\ & \mathbf{\Delta}^{(1)}(S) & & \\ & & \ddots & \\ & & & \mathbf{\Delta}^{(p-1)}(S) \end{pmatrix} \quad (S \in H) \\ \Delta \uparrow G (r) &= \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \frac{\mu(r, r^{p-1})}{\mu(E,E)} \mathbf{1} \\ \frac{\mu(r, E)}{\mu(r,E)} \mathbf{1} & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \frac{\mu(r, r)}{\mu(r^{2},E)} \mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \cdots & \frac{\mu(r, r^{p-2})}{\mu(r^{p-1},E)} \mathbf{1} & \mathbf{0} \end{pmatrix}. \end{split}\]

Case-2: conjugated irreps are equivalent#

For case-2, the induced representation \(\Delta \uparrow G\) is reducible. Let one of intertwiner between \(\Delta^{(0)}\) and \(\Delta^{(1)}\) be \(\mathbf{U}\),

\[ \mathbf{\Delta}^{(0)}(S) \mathbf{U} = \mathbf{U} \mathbf{\Delta}^{(1)}(S) \quad (\forall S \in H). \]

See this page for numerical way to obtain \(\mathbf{U}\). We scale \(\mathbf{U}\) such that \(\mathbf{U}^{p} = \mathbf{1}\).

The induced representation \(\Delta \uparrow G\) is decomposed to \(p\) irreps \(\{ \Delta_{q} \}_{q=0}^{p-1}\),

\[\begin{split} \mathbf{\Delta}_{q}(S) &= \mathbf{\Delta}(S) \quad (S \in H) \\ \mathbf{\Delta}_{q}(r) &= \frac{1}{\omega_{q}} \mathbf{U}. \end{split}\]

The coefficient \(\omega_{q} \, (q = 0, \cdots, p - 1)\) is determined as follows:

\[\begin{split} \mu(E, E) \mathbf{1} &= \mathbf{\Delta}_{q}(E) \\ &= \mathbf{\Delta}_{q}(r^{p}) \\ &= \frac{1}{ \prod_{m=1}^{p-1} \mu(r, r^{m}) } \mathbf{\Delta}_{q}(r)^{p} \\ &= \frac{1}{ \prod_{m=1}^{p-1} \mu(r, r^{m}) } \frac{1}{\omega_{q}^{p}} \mathbf{1} \quad (\because \mathbf{U}^{p} = \mathbf{1}) \\ \therefore \omega_{q} &:= \frac{1}{ \left( \mu(E, E) \prod_{m=1}^{p-1} \mu(r, r^{m}) \right)^{\frac{1}{p}} } \exp \left( \frac{2 \pi i q}{p} \right). \end{split}\]

Decomposition of crystallographic point groups#

A crystallographic point group \(G\) can always be decomposed as Eq. (1) because it is a solvable group. spgrep adapts the following decomposition [Aro16]:

point_group_chain

References#

[Aro16]

M. I. Aroyo, editor. International Tables for Crystallography. Volume A. International Union of Crystallography, December 2016. URL: https://doi.org/10.1107/97809553602060000114, doi:10.1107/97809553602060000114.

[Net73]

N. Neto. Irreducible representations of space groups. Acta Cryst. A, 29(4):464–472, Jul 1973. URL: https://doi.org/10.1107/S0567739473001129, doi:10.1107/S0567739473001129.