6.1: Introduction to Irreducible Representations

Determining the IRs for the space group \(\mathcal G\) of a crystalline structure can be useful when analyzing electronic structure or vibrational modes because the eigenfunctions of the Hamiltonian \(H(r)\) are basis functions of the IRs. Bloch functions are basis functions for the 1-d IRs of the group of lattice translations \(\mathcal L\), which is an invariant subgroup of \(\mathcal G\). However, due to the inclusion of rotations in the space group, these IRs are no longer necessarily 1-d. Evaluating the similarity transformation of the lattice translation operation \(\left( 1 \middle| \boldsymbol{T} \right)\) with respect to \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) yields:

\[\left( R \middle| \boldsymbol{\tau}_{R} \right)^{- 1}\left( 1 \middle| \boldsymbol{T} \right)\left( R \middle| \boldsymbol{\tau}_{R} \right) = \left( R^{- 1} \middle| - R^{- 1}\boldsymbol{\tau}_{R} \right)\left( 1 \middle| \boldsymbol{T} \right)\left( R \middle| \boldsymbol{\tau}_{R} \right) = \left( 1 \middle| R^{- 1}\boldsymbol{T} \right) \nonumber \]

Since the two lattice operations \(\left( 1 \middle| \boldsymbol{T} \right)\) and \(\left( 1 \middle| R^{- 1}\boldsymbol{T} \right)\) belong to the same class, the space group has degenerate IRs. Although each lattice translation is no longer a distinct class, Bloch functions can still be used to construct the basis functions for IRs of a space group. When periodic boundary conditions are applied, space groups are finite groups and orthogonality relationships between IRs apply to construct the character table and elucidate basis functions for the IRs.

THE 1-d CHAIN of ATOMS

To demonstrate how translational and rotational symmetry combine to give the IRs of a space group, we determine the character table and IRs for the space group associated with a 1-d chain of atoms using the simplified periodic boundary condition \(\left( 1 \middle| 4a \right) \equiv \left( 1 \middle| 0 \right)\):

Figure \(\PageIndex{1}\): Atoms are located at lattice points. These sites include inversion symmetry \overline{1}. Then, the space group \mathcal G is the product of the set of 4 translation operations \(L = \left\{ \left( 1 \middle| 0 \right),\left( 1 \middle| a \right),\left( 1 \middle| 2a \right),\left( 1 \middle| - a \right) \right\}\) and the set of 2 essential symmetry operations. The space group \(\mathcal G = \mathcal L \otimes \mathcal R\) contains 8 operations.

Figure \(\PageIndex{2}\): Since \(\mathcal L\) contains 4 members, the first Brillouin zone (gray region) contains 4 allowed wave-vectors corresponding to the 4 IRs of the subgroup \(\mathcal L\):

\(k = 0:\Gamma^{(k)}\)

\(k = \frac{\pi}{2a}:\Gamma^{(k)}\)

\(k = \frac{- \pi}{2a}:\Gamma^{(k)}\)

\(k = \frac{\pi}{a}:\Gamma^{(k)}\) & \(\Gamma^{(k)}\left( 1 \middle| ma \right) = ( - i)^{m}\)

\(\Gamma^{(k)}\left( 1 \middle| ma \right) = i^{m}\)

\(\Gamma^{(k)}\left( 1 \middle| ma \right) = ( - 1)^{m}\)

To build the character table of \(\mathcal G\), the different classes must be enumerated. There are two general types of operations in this group, \(\left( 1 \middle| ma \right)\) and \(\left( \overline{1} \middle| ma \right)\), which lead to 4 expressions of similarity transformations:

\(\left( 1 \middle| na \right)^{- 1}\left( 1 \middle| ma \right)\left( 1 \middle| na \right) = \left( 1 \middle| - na \right)\left( 1 \middle| ma \right)\left( 1 \middle| na \right) = \left( 1 \middle| ma \right)\)

\(\left( \overline{1} \middle| na \right)^{- 1}\left( 1 \middle| ma \right)\left( \overline{1} \middle| na \right) = \left( \overline{1} \middle| na \right)\left( 1 \middle| ma \right)\left( \overline{1} \middle| na \right) = \left( 1 \middle| - ma \right)\)

\(\left( 1 \middle| na \right)^{- 1}\left( \overline{1} \middle| ma \right)\left( 1 \middle| na \right) = \left( 1 \middle| - na \right)\left( \overline{1} \middle| ma \right)\left( 1 \middle| na \right) = \left( \overline{1} \middle| (m - 2n)a \right)\)

\(\left( \overline{1} \middle| na \right)^{- 1}\left( \overline{1} \middle| ma \right)\left( \overline{1} \middle| na \right) = \left( \overline{1} \middle| na \right)\left( \overline{1} \middle| ma \right)\left( \overline{1} \middle| na \right) = \left( \overline{1} \middle| ( - m + 2n)a \right)\).

According to these expressions, the 4 translation operations form the 3 classes: \(\left\{ \left( 1 \middle| 0 \right) \right\}\), \(\left\{ \left( 1 \middle| a \right),\left( 1 \middle| - a \right) \right\}\), and \(\left\{ \left( 1 \middle| 2a \right) \right\}\); and the 4 inversion operations form 2 classes: \(\left\{ \left( \overline{1} \middle| 0 \right),\left( \overline{1} \middle| 2a \right) \right\}\) and \(\left\{ \left( \overline{1} \middle| a \right),\left( \overline{1} \middle| - a \right) \right\}\). Therefore, this small model of a space group consists of 5 IRs, which are summarized in the following character table:

	\(\left( 1 \middle| 0 \right)\)	\(\left( 1 \middle| a \right)\) \(\left( 1 \middle| - a \right)\)	\(\left( 1 \middle| 2a \right)\)	\(\left( \overline{1} \middle| 0 \right)\) \(\left( \overline{1} \middle| 2a \right)\)	\(\left( \overline{1} \middle| a \right)\) \(\left( \overline{1} \middle| -a \right)\)	Basis Functions \(\psi_{nk}(x) = A(x)e^{ikx}\)
\(Γ_g\)	1	1	1	1	1	\(k = 0,A( - x) = A(x)\)
\(Γ_u\)	1	1	1	-1	-1	\(k = 0,A( - x) = - A(x)\)
\(X_g\)	1	-1	1	1	-1	\(k = \frac{\pi}{a},A( - x) = A(x)\)
\(X_u\)	1	-1	1	-1	1	\(k = \frac{\pi}{a},A( - x) = - A(x)\)
\(Δ\)	2	0	-2	0	0	\(k = \frac{\pi}{2a} \land \frac{- \pi}{2a}\)

• Two 1-d IRs correspond to the wavevector \(k = 0\) (lattice point Γ in reciprocal space). These IRs have +1 characters for all translation operations, and their basis functions have the full periodicity of the 1-d lattice, but they differ in their response to inversion: for \(\left( \overline{1} \middle| ma \right)\psi_{n\Gamma}(x) = + \psi_{n\Gamma}(x)\), the IR is even (gerade = g); for \(\left( \overline{1} \middle| ma \right)\psi_{n\Gamma}(x) = - \psi_{n\Gamma}(x)\), the IR is odd (ungerade = u). Therefore, the labels of these IRs are Γ_g and Γ_u.
• Two 1-d IRs correspond to the wavevector \(k = \frac{\pi}{a}\) (boundary point X of the first Brillouin zone). These IRs have +1 characters for even translations and –1 characters for odd translations. Their basis functions also differ in their response to inversion. As a result, the IR labels are X_g and X_u.
• One 2-d IR corresponds to wavevectors \(k = \frac{\pi}{2a}\) and \(\frac{- \pi}{2a}\) (interior points Δ of the first Brillouin zone). Using these Bloch functions as a basis, the matrix representatives for the translation operations \(\left( 1 \middle| ma \right)\) are diagonal matrices:

\(\left( 1 \middle| 0 \right) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix};\left( 1 \middle| a \right) = \begin{pmatrix} - i & 0 \\ 0 & i \\ \end{pmatrix};\left( 1 \middle| 2a \right) = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \\ \end{pmatrix};\left( 1 \middle| - a \right) = \begin{pmatrix} i & 0 \\ 0 & - i \\ \end{pmatrix}\)

The two wavevectors and their Bloch functions are related to each other by inversion, so the matrix representatives for the inversion operations \(\left( \overline{1} \middle| ma \right)\) are matrices with 0-valued diagonal elements:

\(\left( \overline{1} \middle| 0 \right) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix};\left( \overline{1} \middle| a \right) = \begin{pmatrix} 0 & i \\ - i & 0 \\ \end{pmatrix};\left( \overline{1} \middle| 2a \right) = \begin{pmatrix} 0 & - 1 \\ - 1 & 0 \\ \end{pmatrix};\left( \overline{1} \middle| - a \right) = \begin{pmatrix} 0 & - i \\ i & 0 \\ \end{pmatrix}\)

Therefore, inversion takes \(\psi_{nk}(x)\) into \(\psi_{n( - k)}(x)\), and vice versa. As a result, this IR is not specifically labeled for inversion (Δ).

To summarize the results of the analysis of this simplified 1-d space group:

1. The rotations, \(1\) and \(\overline{1}\), give distinct sets of classes. The set \(\mathcal{L} = \left\{ \left( 1 \middle| ma \right) \right\}\) forms 3 classes due to inversion, but \(\left( \overline{1} \middle| 0 \right)\) of \(\mathcal R\) belongs to the class that includes \(\left( \overline{1} \middle| 2a \right)\). As a result, \(\mathcal L\) is an invariant subgroup of \(\mathcal G\), but \(\mathcal R\) is not. Nevertheless, \(\mathcal R\) is a group, so this space group is symmorphic.
2. Plane-wave Bloch functions can serve as basis functions for the IRs of a space group. The amplitude function adopts rotational symmetry characteristics that depend on the wavevector \(k\).
3. The wavevectors Γ (a reciprocal lattice point) and X (a first Brillouin zone boundary point) have inversion symmetry in reciprocal space. As a result, the IRs associated with these two wavevectors can each be designated as gerade (even) and ungerade (odd).
4. The IR Δ is doubly degenerate, corresponding to wavevectors \(k\) and \(- k\), which are related to each other by inversion. However, the rotational symmetry for each separate wavevector is just the identity operation. Nevertheless, the eigenfunctions of the Hamiltonian for these two wavevectors are degenerate: \(E_{n}\left( \frac{\pi}{2a} \right) = E_{n}\left( \frac{- \pi}{2a} \right)\).

This example of the 1-d chain illustrates how the rotations of a space group create equivalent translations resulting in degenerate IRs. It also points out the enhanced symmetry of reciprocal lattice points and boundary points of the first Brillouin zone vis-à-vis general interior points. The procedure can be applied to build the character table of any space group, but determining the classes of these large-order groups can be tedious. Therefore, we desire a different approach that takes advantage of features of the space group (see slide (29)):

• Every space group \(\mathcal G\) is the product of a Bravais lattice group \(\mathcal L = \{\left( 1 \middle| \boldsymbol{T} \right)\}\) and a finite set of essential symmetry operations \(\mathcal R = \{\left( R \middle| \boldsymbol{\tau}_{R} \right)\}\), which leads to the sum of cosets:

\[\mathcal G = \mathcal L \otimes \mathcal R = \left\{ \left( 1 \middle| \boldsymbol{T} \right) \right\} \otimes \left\{ \left( 1 \middle| 0 \right),\left( R_{2} \middle| \boldsymbol{\tau}_{2} \right),\ldots,\left( R_{h} \middle| \boldsymbol{\tau}_{h} \right) \right\} \nonumber \]

\[\mathcal G = \mathcal L \otimes \mathcal R = \left\{ \left( 1 \middle| \boldsymbol{T} \right) \right\} + \left\{ \left( R_{2} \middle| \boldsymbol{\tau}_{2} + \boldsymbol{T} \right) \right\} + \cdots + \left\{ \left( R_{h} \middle| \boldsymbol{\tau}_{h} + \boldsymbol{T} \right) \right\} \nonumber \]

• The set \(\mathcal{G}_{0} = \left\{ 1,R_{2},\ldots,R_{h} \right\}\) of proper and improper rotations in the operations \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) of \(\mathcal R\) is a group, called the point group of the space group, and identifies the crystal class.
• The displacements \(\boldsymbol{\tau}_{R}\) are either \(\boldsymbol 0\) or rational fractions of lattice vectors.
• For symmorphic space groups, \(\boldsymbol{\tau}_{R} = \boldsymbol 0\) for all operations \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) and \(\mathcal R\) is a group isomorphous with \(\mathcal{G}_{0}\).
• For nonsymmorphic space groups, \(\boldsymbol{\tau}_{R} \neq \boldsymbol 0\) for some operations \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) and \(\mathcal R\) is not a group.
• \(\mathcal L\) is an invariant subgroup of \(\mathcal G\), and the factor group \(\mathcal G / \mathcal L\) is also isomorphous with \(\mathcal{G}_{0}\).
• Each coset \(\left\{ \left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol{T} \right) \right\}\) in \(\mathcal G / \mathcal L\) corresponds to \(R_{i}\) in \(\mathcal{G}_{0}\), and \(\mathcal L\) serves as the identity member.

Therefore, to determine the IRs of \(\mathcal G\), it is advantageous to start with the IRs of \(\mathcal L\).

Rotational Symmetry of Bloch Functions

Any Bloch function \(\psi_{n\boldsymbol{k}}(\boldsymbol{r})\) is a basis function for the 1-d IR \(\Gamma^{(\boldsymbol{k})}\) of the Bravais lattice group \(\mathcal L\). The representative of \(\Gamma^{(\boldsymbol{k})}\left( 1 \middle| \boldsymbol{T} \right)\) is \(e^{- i\boldsymbol{k} \cdot \boldsymbol{T}}\) and

\[\left( 1 \middle| \boldsymbol{T} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) = e^{- i\boldsymbol{k} \cdot \boldsymbol{T}}\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \nonumber \]

Now, how does the operation \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) of \(R\) affect a Bloch function \(\psi_{n\boldsymbol{k}}(\boldsymbol{r})\)? To answer this question, examine the effect of \(\left( 1 \middle| \boldsymbol{T} \right)\) on \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r})\):

\(\left( 1 \middle| \boldsymbol{T} \right)\left\{ \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \right\} = \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}\left( \left( 1 \middle| \boldsymbol{T} \right)^{- 1}\boldsymbol{r} \right) = \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r} - \boldsymbol{T})\)

\(\left( 1 \middle| \boldsymbol{T} \right)\left\{ \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \right\} = \psi_{n\boldsymbol{k}}\left( \left( R \middle| \boldsymbol{\tau}_{R} \right)^{- 1}(\boldsymbol{r} - \boldsymbol{T}) \right) = \psi_{n\boldsymbol{k}}\left( R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) - R^{- 1}\boldsymbol{T} \right)\)

\(\left( 1 \middle| \boldsymbol{T} \right)\left\{ \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \right\} = \left( 1 \middle| R^{- 1}\boldsymbol{T} \right)\psi_{n\boldsymbol{k}}\left( R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) \right) = e^{- i\boldsymbol{k} \cdot R^{- 1}\boldsymbol{T}}\psi_{n\boldsymbol{k}}\left( R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) \right)\)

At this point, apply a useful identity for the dot product of vectors \(\boldsymbol{k}\) and \(\boldsymbol{T}\) when one of them is transformed by an orthogonal matrix \(R\): \(\boldsymbol{k} \cdot R^{- 1}\boldsymbol{T} = R\boldsymbol{k} \cdot \boldsymbol{T}\), so that

\[\left( 1 \middle| \boldsymbol{T} \right)\left\{ \left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \right\} = e^{- R\boldsymbol{k} \cdot \boldsymbol{T}}\psi_{n\boldsymbol{k}}\left( R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) \right) \nonumber \]

Therefore, \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r})\) behaves like a Bloch function with wavevector \(R\boldsymbol{k}\). Also, the coordinate \(\boldsymbol{r}\) is transformed to \(R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right)\), so that

\[\left( R \middle| \boldsymbol{\tau}_{R} \right)\psi_{n\boldsymbol{k}}(\boldsymbol{r}) = \psi_{n\boldsymbol{k}}\left( R^{- 1}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) \right) \equiv \psi_{n(R\boldsymbol{k})}\left( \boldsymbol{r} - \boldsymbol{\tau}_{R} \right) \nonumber \]

As a result, the members \(\left( R \middle| \boldsymbol{\tau}_{R} \right)\) of the set of essential symmetry operations \(R\) transform every wavevector \(\boldsymbol{k}\) among various wavevectors \(R\boldsymbol{k}\) and may introduce a change of origin, i.e., phase change, to the corresponding Bloch functions.

All allowed wavevectors \(\boldsymbol{k}\) for the IRs of \(\mathcal L\) belong to the first Brillouin zone of the reciprocal lattice, a region which is the Wigner-Seitz cell for any reciprocal lattice point. Every boundary point of the first Brillouin zone is equivalent to at least one other boundary point because two boundary points are separated by a reciprocal lattice vector \(\boldsymbol K\). Now, the effect of any rotation operation \(R\) of \(\mathcal{G}_{0}\) on a wavevector \(\boldsymbol k\) has one of three possible outcomes:

1. \(R\boldsymbol k = \boldsymbol k\): \(R\) generates the same wavevector. Therefore, \(\boldsymbol k\) must lie along the symmetry element of \(R\), i.e., \(\boldsymbol k\) is parallel to the rotation axis or falls in the reflection plane (line).
2. \(R\boldsymbol k = \boldsymbol k + \boldsymbol K(\boldsymbol K \neq \boldsymbol 0,\text{a reciprocal lattice vector})\): \(R\) generates an equivalent wavevector. This outcome only occurs for wavevectors on the first Brillouin zone boundary.
3. \(R\boldsymbol k = \boldsymbol k' \neq \boldsymbol k \text{ or } \boldsymbol k + \boldsymbol K\): \(R\) generates a different wavevector. These wavevectors are inside the first Brillouin zone and do not lie along symmetry elements of \(R\).

For a given wavevector \(\boldsymbol k\) in the first Brillouin zone, the \(h\) rotations of \(\mathcal G_{0}\) generate a set of wavevectors \(\mathcal S(\boldsymbol k) = \left\{ \boldsymbol k,R_{2}\boldsymbol k,\ldots,R_{h}\boldsymbol k \right\}\), not all necesscarily distinct, called the star of \(\boldsymbol k\). The subset of \(q\) rotations \((q \leq h)\) that take \(\boldsymbol k\) to the same wavevector \(\boldsymbol k\) or an equivalent wavevector \(\boldsymbol k + \boldsymbol K\) form a group \(\mathcal G_{0}(\boldsymbol k) = \left\{ E,R_{2},\ldots,R_{q} \right\}\), called the point group of the wavevector \(\boldsymbol k\). These features are illustrated below for three different wavevectors \(\boldsymbol k = k_{1}\boldsymbol a_{1}^{*} + k_{2}\boldsymbol a_{2}^{*} \equiv \left( k_{1},k_{2} \right)\), shown as yellow dots, of the hexagonal system with \(\mathcal G_{0} = 6mm\). The members of \(\mathcal G_{0}(\boldsymbol k)\) are highlighted in blue:

@ >p(- 4) * >p(- 4) * >p(- 4) * @

() (1) \(k = (0,0) = \Gamma\) & (2) \(k = \left( k_{1},0 \right) = \Sigma\) & (3) \(k = (⅓,⅓) = \text{K}\)
() () (1) \(k = (0,0) = \Gamma\) & (2) \(k = \left( k_{1},0 \right) = \Sigma\) & (3) \(k = (⅓,⅓) = \text{K}\)
()

Figure 6.3 &

Figure 6.4 &

Figure 6.5
()

1. \(\Gamma = (0,0)\) is a reciprocal lattice point. The 6-fold rotation and all six mirror planes keep \(\Gamma\) invariant. Therefore, \(\mathcal G_{0}(\Gamma) = \mathcal G_{0} = 6mm\) with order 12, and \(\mathcal S(\Gamma)\) contains just one wavevector \(\boldsymbol k = (0,0)\).
2. \(\Sigma = \left( k_{1},0 \right)\) designates any wavevector \(k_{1}\boldsymbol a_{1}^{*}\) for \(0 < k_{1} < \frac{\pi}{a}\). Only one mirror plane keeps \(\Sigma\) invariant. \(\mathcal G_{0}(\Sigma) = m\) with order 2, and \(\mathcal S(\Sigma)\) contains six wavevectors in the first Brillouin zone:

   \(\boldsymbol k = \left( k_{1},0 \right);6\boldsymbol k = \left( 0,k_{1} \right);6^{2}\boldsymbol k = \left( {- k}_{1},k_{1} \right);6^{3}\boldsymbol k = \left( {- k}_{1},0 \right);6^{4}\boldsymbol k = \left( 0,{- k}_{1} \right);6^{5}\boldsymbol k = \left( k_{1},{- k}_{1} \right)\)

3. \(\text{K} = (⅓,⅓)\) is located at a corner of the first Brillouin zone. The 3-fold (6²-fold) rotation and three mirror planes keep \(\text{K}\) invariant or send it to an equivalent point at either \(( - ⅔,⅓)\) or \((⅓, - ⅔)\). \(\mathcal G_{0}(\text{K}) = 3m\) with order 6, and \(\mathcal S(\text{K})\) contains two wavevectors on the boundary of the first Brillouin zone: \(\boldsymbol k = (⅓,⅓);6\boldsymbol k = ( - ⅓,⅔)\).

As this example illustrates, for each wavevector \(\boldsymbol k\) of the first Brillouin zone,

Order of \(\mathcal G_{0}\) = Order of \(\mathcal G_{0}(\boldsymbol k)\) \(\times\) Number of distinct wavevectors in \(\mathcal S(\boldsymbol k)\).

Both sets \(\mathcal G_{0}(\boldsymbol k)\) and \(\mathcal S(\boldsymbol k)\) are important to determine the IRs of \(\mathcal G\). Each distinct wavevector in \(\mathcal S(\boldsymbol k)\) has the same abstract point group, but \(\mathcal G_{0}(\boldsymbol k)\) is formed by different operations of \(\mathcal G_{0}\).

Group of the Wavevector \(\boldsymbol k\)

Each operation of the point group of the space group \(\mathcal G_{0}\) is associated with one member of the set of essential symmetry operations \(R\). Likewise, each operation of \(\mathcal G_{0}(\boldsymbol k)\) is associated with one member of the subset \(R(\boldsymbol k) = \left\{ \left( 1 \middle| \boldsymbol 0 \right),\ldots,\left( R_{q} \middle| \boldsymbol{\tau}_{q} \right) \right\}\), in which every rotation \(R\) takes \(\boldsymbol k\) to \(\boldsymbol k\) or \(\boldsymbol k + \boldsymbol K\). The product \(\mathcal L \otimes \mathcal R(\boldsymbol k) = \mathcal G(\boldsymbol k)\) is the group of the wavevector \(\boldsymbol k\), also called the little group, which leads to the sum of cosets:

\(\mathcal G(\boldsymbol k) = \mathcal L \otimes \mathcal R(\boldsymbol k) = \left\{ \left( 1 \middle| \boldsymbol T \right) \right\} \otimes \left\{ \left( 1 \middle| \boldsymbol 0 \right),\left( R_{2} \middle| \boldsymbol{\tau}_{2} \right),\ldots,\left( R_{q} \middle| \boldsymbol{\tau}_{q} \right) \right\}\)

\(\mathcal G(\boldsymbol k) = \mathcal L \otimes \mathcal R(\boldsymbol k) = \left\{ \left( 1 \middle| \boldsymbol T \right) \right\} + \left\{ \left( R_{2} \middle| \boldsymbol{\tau}_{2} + \boldsymbol T \right) \right\} + \cdots + \left\{ \left( R_{q} \middle| \boldsymbol{\tau}_{q} + \boldsymbol T \right) \right\}\).

\(\mathcal G(\boldsymbol k)\) is a subgroup of the space group \(\mathcal G\) and the corresponding factor group \(\frac{\mathcal G(\boldsymbol k)}{\mathcal L}\) is isomorphous with \(\mathcal G_{0}(\boldsymbol k)\), which is a subgroup of \(\mathcal G_{0}\). \(\mathcal L\) is the identity member of this factor group, but now each representative for \(\left( 1 \middle| \boldsymbol T \right)\) is \(e^{- i\boldsymbol k \cdot \boldsymbol T}E\), where \(E\) is the identity matrix.

Let \(D^{(\boldsymbol k\mu)}\) be an IR of \(\mathcal G(\boldsymbol k)\). Then,

\(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol T \right) = \Gamma^{(\boldsymbol k)}\left( 1 \middle| \boldsymbol T \right)D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right) = e^{- i\boldsymbol k \cdot \boldsymbol T}D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)\) and

\(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)D^{(\boldsymbol k\mu)}\left( R_{j} \middle| \boldsymbol{\tau}_{j} \right) = D^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \middle| \boldsymbol{\tau}_{i} + R_{i}\boldsymbol{\tau}_{j} \right)\).

Now, define the corresponding loaded representation \({\hat{D}}^{(\boldsymbol k\mu)}\) of \(\mathcal G(\boldsymbol k)\) as follows:

\({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right) \equiv e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)\), so that \(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right) = e^{- i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right)\).

According to this definition, the single matrix \({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right)\) represents every member of the coset \(\left\{ \left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol T \right) \right\}\) in the coset expansion of \(\mathcal G(\boldsymbol k)\). Since each coset is a distinct member of the factor group \(\mathcal G(\boldsymbol k) / \mathcal L\), examine the product of two loaded representatives:

\({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k\mu)}\left( R_{j} \right) = e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{j}}D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)D^{(\boldsymbol k\mu)}\left( R_{j} \middle| \boldsymbol{\tau}_{j} \right) = e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{j}}D^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \middle| \boldsymbol{\tau}_{i} + R_{i}\boldsymbol{\tau}_{j} \right)\)

\({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k\mu)}\left( R_{j} \right) = e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{j}}e^{- i\boldsymbol k \cdot \left( \boldsymbol{\tau}_{i} + R_{i}\boldsymbol{\tau}_{j} \right)}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \right) = e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{j}}e^{- i\boldsymbol k \cdot R_{i}\boldsymbol{\tau}_{j}}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \right)\)

\({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k\mu)}\left( R_{j} \right) = e^{i\boldsymbol k \cdot \boldsymbol{\tau}_{j}}e^{- iR_{i}^{- 1}\boldsymbol k \cdot \boldsymbol{\tau}_{j}}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \right)\).

Since \(R_{i}\) belongs to \(\mathcal G_{0}(\boldsymbol k)\), so does the inverse \(R_{i}^{- 1}\). If \(R_{i}^{- 1}\boldsymbol k \equiv \boldsymbol k + \boldsymbol K_{i}\), then the product becomes

\({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k\mu)}\left( R_{j} \right) = e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \right)\).

Therefore, the loaded representation \({\hat{D}}^{(\boldsymbol k\mu)}\) of \(\mathcal G(\boldsymbol k)\) is an example of a projective representation in which the numerical factor \(e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}}\) depends on the two operations in the product and the order of their multiplication. If \(e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}}\) always equals 1, then all products follow the normal definition of a group, and the loaded representation \({\hat{D}}^{(\boldsymbol k\mu)}\) is an IR \(D_{0}^{(\boldsymbol k\mu)}\) of the point group \(\mathcal G_{0}(\boldsymbol k)\). If \(e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}}\) does not equal 1 for some products, then additional analysis is required to establish the IRs of \(\mathcal G(\boldsymbol k)\).

The factor \(e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}} = 1\) when \(\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j} = 0\), which occurs for \(\boldsymbol{\tau}_{j} = \boldsymbol 0\), \(\boldsymbol K_{i} = \boldsymbol 0\), and \(\boldsymbol K_{i}\ \bot\ \boldsymbol{\tau}_{j}\). To apply these restrictions, it is advantageous to consider symmorphic and nonsymmorphic space groups separately:

SYMMORPHIC SPACE GROUPS:

\(\boldsymbol{\tau}_{j} = \boldsymbol 0\) for all \(\left( R_{j} \middle| \boldsymbol{\tau}_{j} \right)\) and \(\mathcal R(\boldsymbol k) = \mathcal G_{0}(\boldsymbol k)\). As a result, \(\mathcal G(\boldsymbol k)\) can always be factored into the product of two groups, \(\mathcal L\) and \(\mathcal R(\boldsymbol k)\), and the IRs for \(\mathcal G(\boldsymbol k)\) are the products of the IRs for \(\mathcal L\) and \(\mathcal G_{0}(\boldsymbol k)\):

\(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol T \right) = \Gamma^{(\boldsymbol k)}\left( 1 \middle| \boldsymbol T \right)D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol 0 \right) = e^{- i\boldsymbol k \cdot \boldsymbol T}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right) = e^{- i\boldsymbol k \cdot \boldsymbol T}D_{0}^{(\boldsymbol k\mu)}\left( R_{i} \right)\).

This outcome will be illustrated for the \(\pi\)-bands of graphene in slides (56–61).

NONSYMMORPHIC SPACE GROUPS:

\(\boldsymbol{\tau}_{j} \neq \boldsymbol 0\) for some \(\left( R_{j} \middle| \boldsymbol{\tau}_{j} \right)\). For all wavevectors \(\boldsymbol k\) inside the first Brillouin zone, \(\boldsymbol K_{i} = \boldsymbol 0\) for every \(R_{i}\) in \(\mathcal G_{0}(\boldsymbol k)\) and the IRs for \(\mathcal G(\boldsymbol k)\) are the products of the IRs for \(\mathcal G_{0}(\boldsymbol k)\) and \(\mathcal L\), just as for symmorphic space groups, but with an additional phase factor arising from the nonzero displacements \(\boldsymbol{\tau}_{j}\):

\(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol T \right) = \Gamma^{(\boldsymbol k)}\left( 1 \middle| \boldsymbol T \right)D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right) = e^{- i\boldsymbol k \cdot \boldsymbol T}e^{- i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}D_{0}^{(\boldsymbol k\mu)}\left( R_{i} \right)\).

For certain wavevectors on the first Brillouin zone boundary, \(\boldsymbol K_{i} \neq \boldsymbol 0\) or \(\boldsymbol K_{i}\) is not perpendicular to \(\boldsymbol{\tau}_{j}\), which means \(e^{- i\boldsymbol K_{i} \cdot \boldsymbol{\tau}_{j}} \neq 1\) and some products \({\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k\mu)}\left( R_{j} \right) \neq {\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i}R_{j} \right)\). This outcome leads to only degenerate IRs for \(\mathcal G_{0}(\boldsymbol k)\) and \(\mathcal G(\boldsymbol k)\). To determine the characters of these IRs, further analysis is required:

\(D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol T \right) = \Gamma^{(\boldsymbol k)}\left( 1 \middle| \boldsymbol T \right)D^{(\boldsymbol k\mu)}\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right) = e^{- i\boldsymbol k \cdot \boldsymbol T}e^{- i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}{\hat{D}}^{(\boldsymbol k\mu)}\left( R_{i} \right)\).

Determining the IRs of \(\mathcal G(\boldsymbol k)\) for various wavevectors normally concludes this analysis, because these IRs are important for classifying states in electronic band structures and phonon dispersion curves. However, to evaluate the IRs of the entire space group \(\mathcal G\), one final step remains that makes use of the star of \(\boldsymbol k\).

Star of \(k\)

The groups \(\mathcal G(\boldsymbol k)\) for each member of \(\mathcal S(\boldsymbol k)\) are isomorphous with each other, which means they have identical IRs although the respective basis functions may be different. Consequently, there is a region within the first Brillouin zone, called the irreducible wedge, from which all wavevectors of the first Brillouin zone can be generated by the rotations of the point group of the space group \(\mathcal G_{0}\). For example, the irreducible wedges for the five 2-d Bravais lattice systems are illustrated below:

@ >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * @

()

OBLIQUE

&

RECTANGULAR

&

CENTERED
RECTANGULAR

&

TETRAGONAL

&

HEXAGONAL /
TRIGONAL

() ()

OBLIQUE

&

RECTANGULAR

&

CENTERED
RECTANGULAR

&

TETRAGONAL

&

HEXAGONAL /
TRIGONAL

()

Figure 6.6 &

Figure 6.7 &

Figure 6.8 &

Figure 6.9 &

Figure 6.10
Γ = (0, 0)

X = (½, 0)

Y = (0, ½)

C = (½, –½) & Γ = (0, 0)

X = (½, 0)

Y = (0, ½)

S = (½, ½) & Γ = (0, 0)

X = (k₁, k₁)

Y = (–½, ½)

S = (0, ½) & Γ = (0, 0)

X = (½, 0)

M = (½, ½) & Γ = (0, 0)

M = (½, 0)

K = (⅓, ⅓)
()

As a result of the rotational symmetry of reciprocal space, the eigenvalues of the electronic or vibrational Hamiltonian \(H(\boldsymbol k)\) for some wavevector \(\boldsymbol k\) of the irreducible wedge will be the same for every wavevector in \(\mathcal S(\boldsymbol k)\). Therefore, computations focus on wavevectors belonging just to the irreducible wedge, and the number of members in \(\mathcal S(\boldsymbol k)\) leads to degeneracies of the IRs.

By using the members of \(\mathcal S(\boldsymbol k)\) as a basis, each member of \(\mathcal G_{0}\) is a permutation matrix, which has “0” or “1” as matrix elements. The dimension of these matrices equals the number of distinct members of \(\mathcal S(\boldsymbol k)\). If \(R_{j}\boldsymbol k_{i} = \boldsymbol k_{j}\) or \(\boldsymbol k_{j} + \boldsymbol K\), then the matrix element of the \(i\)^th row and \(j\)^th column is “1”; all other matrix elements in the same row and column are “0”. The complete IRs for the space group \(\mathcal G\) are then constructed by placing the IRs \(D^{(\boldsymbol k)}\) of \(\mathcal G(\boldsymbol k)\) in these permutation matrices where matrix elements equal “1”. In this way, every IR for the space group can be worked out. However, the complete matrices are seldom used and knowing their characters is often sufficient.

To summarize how IRs for a space group \(G\) are determined:

• Select a wavevector \(\boldsymbol k\) belonging to the irreducible wedge of the first Brillouin zone. This wavevector determines the Bloch functions \(\psi_{n\boldsymbol k}(\boldsymbol r)\).
• Determine the group of the wavevector \(\boldsymbol k\), \(\mathcal G(\boldsymbol k)\), and its corresponding point group, \(\mathcal G_{0}(\boldsymbol k)\), from the set of essential symmetry operations \(R\).
• If the space group is symmorphic, then every operation \(\left( R_{i} \middle| \boldsymbol T \right)\) takes the form \(\left( 1 \middle| \boldsymbol T \right)\left(R_{i} \middle| \boldsymbol 0 \right)\), and the IRs for \(\mathcal G(\boldsymbol k)\) are \(e^{- i\boldsymbol k \cdot \boldsymbol T}\) times the IRs of \(R_{i}\) for \(\mathcal G_{0}(\boldsymbol k)\).
• If the space group is nonsymmorphic, then operations take the form \(\left( R_{i} \middle| \boldsymbol{\tau}_{i} + \boldsymbol T \right)\) and the IRs for \(\mathcal G(\boldsymbol k)\) are \(e^{- i\boldsymbol k \cdot \boldsymbol T}e^{- i\boldsymbol k \cdot \boldsymbol{\tau}_{i}}\) times the IRs of \(R_{i}\) determined from the loaded representations of \(\mathcal G_{0}(\boldsymbol k)\). For all wavevectors \(\boldsymbol k\) in the interior region of the first Brillouin zone, the loaded representations for \(\mathcal G_{0}(\boldsymbol k)\) correspond to the IRs for \(\mathcal G_{0}(\boldsymbol k)\). For wavevectors on the boundary of the first Brillouin zone, then the products \({\hat{D}}^{(\boldsymbol k)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k)}\left( R_{j} \right)\) must be evaluated. If \({\hat{D}}^{(\boldsymbol k)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k)}\left( R_{j} \right) = {\hat{D}}^{(\boldsymbol k)}\left( R_{i}R_{j} \right)\) for all members of \(\mathcal G_{0}(\boldsymbol k)\), then the loaded representations for \(\mathcal G_{0}(\boldsymbol k)\) correspond to the IRs for \(\mathcal G_{0}(\boldsymbol k)\). If \({\hat{D}}^{(\boldsymbol k)}\left( R_{i} \right){\hat{D}}^{(\boldsymbol k)}\left( R_{j} \right) \neq {\hat{D}}^{(\boldsymbol k)}\left( R_{i}R_{j} \right)\) for some members of \(\mathcal G_{0}(\boldsymbol k)\), then the loaded representations for \(\mathcal G_{0}(\boldsymbol k)\) are exclusively degenerate.
• To complete the IRs for \(\mathcal G\), construct the permutation matrices for each member of \(\mathcal S(\boldsymbol k)\) and insert the relevant IRs of \(\mathcal G(\boldsymbol k)\) into these matrices where “1” is located.

To examine the difference between symmorphic and nonsymmorphic space groups, we compare the irreducible representations of the monoclinic space groups \(P2\) and \(P2_{1}\) along the \(\boldsymbol b^*\) direction of the first Brillouin zone. The space groups are expressed as the product of their lattice subgroups and the set of essential symmetry operations; this latter set contains 2 members. From the choices of wavevectors, \(\boldsymbol k = u\boldsymbol a^* + v\boldsymbol b^* + w\boldsymbol c^* = v\boldsymbol b^*\), and \(\boldsymbol k \cdot \boldsymbol T_{mnp} = 2\pi nv(0 \leq v \leq ½)\).

\[P2 = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right) \right\} \times \left\{\left( 1 \middle| \boldsymbol 0\right), \left( 2_{010} \middle| \boldsymbol 0 \right) \right\} \nonumber \]

\(\boldsymbol k = \boldsymbol 0\ (\Gamma):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = 0\)

\[\mathcal G(\Gamma) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \boldsymbol T_{mnp} \right) \right\} \nonumber \]

()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() ()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() \(\Gamma^{(1)} = a\)	1	1
\(\Gamma^{(2)} = b\)	1	–1
()

The IRs are either symmetric or antisymmetric with respect to the 2-fold rotation. For \(\Gamma\), there are no phase changes for any lattice translation. The star of \(\Gamma\), \(\mathcal S(\Gamma)\), contains just a single wavevector.

\(\boldsymbol k = v\boldsymbol b^*(\Delta):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = 2\pi nv\)

\[\mathcal G(\Delta) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \boldsymbol T_{mnp} \right) \right\} \nonumber \]

()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() ()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() \(\Delta^{(1)}\)	\(e^{-2 \pi inv}\)	\(-e^{-2 \pi inv}\)
\(\Delta^{(2)}\)	\(e^{-2 \pi inv}\)	\(-e^{-2 \pi inv}\)
()

The two complex IRs are symmetric and antisymmetric with respect to the 2-fold rotation. \(\mathcal S(\Delta)\) contains two wavevectors, \(\pm v\boldsymbol b^*\).

\(\boldsymbol k = ½\boldsymbol b^* (Z):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = \pi n\)

\[\mathcal G(Z) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \boldsymbol T_{mnp} \right) \right\} \nonumber \]

()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() ()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left( 2_{010} \middle| \boldsymbol T_{mnp} \right)\)
() \(\Zeta^{(1)}\)	\((-1)^{n}\)	\(-(-1)^{n}\)
\(\Zeta^{(2)}\)	\((-1)^{n}\)	\(-(-1)^{n}\)
()

The two real IRs are nondegenerate. \(\mathcal S(Z)\) contains just one wavevector.

\[P2_{1} = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right) \right\} \times \left\{\left( 1 \middle| \boldsymbol 0\right), \left( 2_{010} \middle| \boldsymbol b/2 \right) \right\} \nonumber \]

\(\boldsymbol k = \boldsymbol 0(\Gamma):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = 0\)

\[\mathcal G(\Gamma) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \frac{\boldsymbol b}{2} + \boldsymbol T_{mnp} \right) \right\} \nonumber \]

@ >p(- 4) * >p(- 4) * >p(- 4) * @

() & \(\left( 1 \middle| \boldsymbol T_{mnp} \right)\) &

\(\left( 2_{010} \middle| \boldsymbol b/2 + \boldsymbol T_{mnp} \right)\)

() () & \(\left( 1 \middle| \boldsymbol T_{mnp} \right)\) &

\(\left( 2_{010} \middle| \boldsymbol b/2 + \boldsymbol T_{mnp} \right)\)

() \(\Gamma^{(1)} = a\) & 1 & 1
\(\Gamma^{(2)} = a\) & 1 & –1
()

The IRs are either symmetric or antisymmetric with respect to the 2-fold rotation. For \(\Gamma\), there are no phase changes for any lattice translation. The star of \(\Gamma\), \(\mathcal S(\Gamma)\), contains just a single wavevector.

\(\boldsymbol k = v\boldsymbol b^*(\Delta):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = 2\pi nv\)

\[\mathcal G(\Delta) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \frac{\boldsymbol b}{2} + \boldsymbol T_{mnp} \right) \right\} \nonumber \]

@ >p(- 4) * >p(- 4) * >p(- 4) * @

() & \(\left( 1 \middle| \boldsymbol T_{mnp} \right)\) &

\(\left( 2_{010} \middle| \boldsymbol b/2 + \boldsymbol T_{mnp} \right)\)

() () & \(\left( 1 \middle| \boldsymbol T_{mnp} \right)\) &

\(\left( 2_{010} \middle| \boldsymbol b/2 + \boldsymbol T_{mnp} \right)\)

() \(\Delta^{(1)}\) & \(e^{-2 \pi inv}\) & \(-e^{-2 \pi inv} e^{- \pi iv}\)
\(\Delta^{(2)}\) & \(e^{-2 \pi inv}\) & \(-e^{-2 \pi inv} e^{- \pi iv}\)
()

The two complex IRs are symmetric and antisymmetric with respect to the 2-fold rotation. \(\mathcal S(\Delta)\) contains two wavevectors, \(\pm v\boldsymbol b^*\).

\(\boldsymbol k = ½\boldsymbol b^*(Z):\ \boldsymbol k \cdot \boldsymbol T_{mnp} = \pi n\)

\[\mathcal G(Z) = \left\{ \left( 1 \middle| \boldsymbol T_{mnp} \right),\left( 2_{010} \middle| \frac{\boldsymbol b}{2} + \boldsymbol T_{mnp} \right) \right\} \nonumber \]

()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left ({2} _ {010} \middle| {\boldsymbol b} / {2} + {\boldsymbol T} _ {mnp} \right )\)
() ()	\(\left( 1 \middle| \boldsymbol T_{mnp} \right)\)	\(\left ({2} _ {010} \middle| {\boldsymbol b} / {2} + {\boldsymbol T} _ {mnp} \right )\)
() \(\Zeta^{(1)}\)	\((-1)^n\)	\(-i(-1)^n\)
\(\Zeta^{(2)}\)	\((-1)^n\)	\(-i(-1)^n\)
()

The two IRs are complex conjugates of each other and, therefore, degenerate. \(\mathcal S(Z)\) has just one wavevector.

To see the significance of determining the irreducible representations of space groups for electronic band structures and phonon dispersion curves, we examine and analyze the symmetry characteristics of the electronic \(\pi\)-bands of graphene.