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Tables of Irreducible Representations
of the 230 Crystallographic Space Groups and Their Superspace Extensions

2011 Version

Harold T. Stokes and Branton J. Campbell, Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA, branton_campbell@byu.edu

For further details, see H. T. Stokes, B. J. Campbell, and R. Cordes, "Tabulation of Irreducible Representations of the Crystallographic Space Groups and Their Superspace Extensions." Acta Cryst. A 69, 388-395 (2013).

Table 1. Physically Irreducible Representations,
Rational k vectors: 3-Dimensional Space

When group-representation methods are used for calculating real quantities, like atomic displacements, it is convenient to use IR matrices containing real numbers. For type-2 and type-3 IRs, which are unavoidably complex, we can obtain real matrices by taking the direct sum of the IR matrix and its complex conjugate followed by an appropriate similarity transform. Such a "physically irreducible" representation (PIR) is irreducible with respect to real numbers. In this table, we present all of the PIRs of the 230 crystallographic space groups in 3-dimensional space.

The first line for each IR contains the k vectors in the star of k and -k, the dimension nk of the IR of the little group of k, and the IR type. Consider an example: IR X1 of space group No. 90 P4212. We see from the entry, "90 P42_12 X1 (0,1/2,0),(1/2,0,0); nk=2; type=1", that there are 2 k vectors in the star of k and -k: k1=(0,1/2,0) and k2=(1/2,0,0).

Each IR matrix is displayed on a single line. Rows are separated by a slash. If there are more than one k vector in the star of k and -k, then the matrix is presented in block form, with one row or column block per k vector. Each block is enclosed in parentheses and preceded by an integer indicating which column that block goes into. As an example, consider the matrix, (2(1,0/0,-1)/1(0,-1/1,0)). The first row of blocks contains the matrix (1,0/0,-1) in the second column of blocks. The second row of blocks contains the matrix (0,-1/1,0) in the first column of blocks. The fully displayed matrix is

0010
000-1
0-100
1000

The second line for each IR contains the form of the IR matrix T(v) of a pure translation v. The symbols c1,c2,... refer to cos(2πk1.v), cos(2πk2.v),..., and the symbols s1,s2,... refer to sin(2πk1.v), sin(2πk2.v),...

The following lines for each IR contain the space-group operators g and the corresponding matrices D(g). The point operator part R of each space group operator g is also given. For R, we use a notation which uses the conventional symbol for R followed by the axis direction, if any. For example, 4[001] refers to a 90-degree (4-fold) rotation about the [001] axis. 4[00-1] refers to a 90-degree rotation about the [00-1] axis, which is the same as a -90-degree rotation about the [001] axis. -2[100] refers to a reflection (2-fold rotation followed by an inversion) through a plane perpendicular to the [100] direction. This notation helps you quickly find particular operators. As an example, the entry,
4[001] -y+1/2,x+1/2,z (2(1,0/0,-1)/1(0,-1/1,0)),
gives the matrix for a 90-degree rotation about [001] followed by a translation (1/2,1/2,0). A representative operator is given for each point operator in the space group.

For non-special k vectors, free parameters α,β,γ are given. Consider an example: IR Δ1 of space group No. 90 P4212. We see from the entry, "90 P42_12 DT1 (0,b,0),(b,0,0)", that the 2 k vectors in the star of k and -k are k1=(0,β,0) and k2=(β,0,0), where the free parameter β can take on any non-special value.

The IR matrices for non-special k vectors are written as a product of two matrices: one which depends only on the translation part of the operator and one which depends only on the point operation part of the operator. For example, consider the following entry:
4[001] -y+1/2,x+1/2,z T(1/2,1/2,0)*(2(1,0/0,1)/1(1,0/0,-1)).
The translation part is given by the entry,
T(v)=(1(c1,s1/-s1,c1)/2(c2,s2/-s2,c2)),
which, when evaluated at v=(1/2,1/2,0), becomes

cos(πβ)sin(πβ) 00
-sin(πβ)cos(πβ) 00
00 cos(πβ)sin(πβ)
00 -sin(πβ)cos(πβ)

From the product of this matrix and the matrix for the point operation part, (2(1,0/0,1)/1(1,0/0,-1)), we obtain for D(g),

00 cos(πβ)sin(πβ)
00 -sin(πβ)cos(πβ)
cos(πβ)-sin(πβ) 00
-sin(πβ)-cos(πβ) 00

Note that in the above example, the translation in T(1/2,1/2,0) was simply the translation part of the operator g. This is not always the case. Consider an example: for IR Δ1 in space group No. 85 P4/n. We find an entry,
2[001] -x+1/2,-y+1/2,z T(-1,0,0)*(1(1,0/0,-1)/2(1,0/0,-1)).
We see that the translation in T(-1,0,0) is not (1/2,1/2,0), the translation part of the operator g. These differences arise whenever the setting in the International Tables has a different origin from that in the little-group IR tables of Miller and Love.

Table 2. Physically Irreducible Representations,
Irrational k vectors: (3+d)-Dimensional Superspace

For non-special k vectors, we can extend a 3-dimensional space group to (3+d)-dimensional superspace, where d is the number of k vectors in the star of k and -k. The coordinates in this superspace are x1,x2,...,x3+d. The translation part of an operator in the superspace group has (3+d) components, vx,vy,vz,δ1, δ2,...,δd. The components v=vx,vy,vz are defined in 3-dimensional space, and the components δ=δ1,δ2,...,δd are phase shifts along each of the modulation vectors. Table 2 is very similar to Table 1. Here we only note the differences.

The second line for each IR contains the form of the IR matrix Q(δ) of a phase shift δ. The symbols c1,c2,... refer to cos(2πδ1), cos(2πδ2),..., and the symbols s1,s2,... refer to sin(2πδ1), sin(2πδ2),...

These IR matrices do not depend on the 3-dimensional translation v in 3-dimensional space. The IRs map pure translations onto the unit matrix: D(v)=1. Therefore, for each representative operator (one for each point operation in the space group), we present only a single matrix. Note also that we have found forms for these matrices that do not depend on the free parameters in the k vector.

Table 3. Complex Irreducible Representations,
Rational k vectors: 3-Dimensional Space

This table gives the IR matrices in complex form. For a type-2 or type-3 IR, the complex irreducible representation has half the dimension of the corresponding PIR in Table 1. Table 3 is very similar to Table 1. Here we only note the differences.

The first line for each IR contains k vectors in the star of k instead of the star of k and -k. If -k is in the star of k and is not equivalent to k, then there will be twice as many k vectors in the star of k as in the star of k and -k.

The second line for each IR contains the form of the IR matrix T(v) of a pure translation v. The symbols e1,e2,... refer to exp(2πik1.v), exp(2πik2.v),... Note that in the complex form, the IR matrix T(v) is diagonal.

Complex numbers are denoted by enclosing its real and imaginary parts in parentheses. For example (1,0)=1 and (0,1)=i.

Table 4. Complex Irreducible Representations,
Irrational k vectors: (3+d)-Dimensional Superspace

This table presents the IR matrices in complex form. See the explanations for Tables 1-3.

Table 5. Basis functions

This is a replacement of Table 4 in Isotropy Subgroups of the 230 Crystallographic Space Groups (Stokes and Hatch, 1988) using the 2011 version of the IR matrices. This table gives basis functions of the IRs which transform like components of polar vectors (x,y,z), axial vectors (Sx,Sy,Sz) and second-rank tensors (x2,y2,z2,yz,xz,xy). An orthogonal right-handed coordinate system is used for all crystal classes. For hexagonal crystal classes, the x axis is chosen to be in the direction of the hexagonal a axis.