Tables of Irreducible Representations

of the 230 Crystallographic Space Groups and Their Superspace Extensions

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).

Rational k vectors: 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 X_{1} of space group No. 90 P42_{1}2.
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:
**k**_{1}=(0,1/2,0)
and **k**_{2}=(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

0 | 0 | 1 | 0 |

0 | 0 | 0 | -1 |

0 | -1 | 0 | 0 |

1 | 0 | 0 | 0 |

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(2π**k**_{1}.**v**),
cos(2π**k**_{2}.**v**),..., and
the symbols s1,s2,... refer to
sin(2π**k**_{1}.**v**),
sin(2π**k**_{2}.**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 Δ_{1} of space group No. 90 P42_{1}2.
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 **k**_{1}=(0,β,0)
and **k**_{2}=(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) | 0 | 0 |

-sin(πβ) | cos(πβ) | 0 | 0 |

0 | 0 | cos(πβ) | sin(πβ) |

0 | 0 | -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),

0 | 0 | cos(πβ) | sin(πβ) |

0 | 0 | -sin(πβ) | cos(πβ) |

cos(πβ) | -sin(πβ) | 0 | 0 |

-sin(πβ) | -cos(πβ) | 0 | 0 |

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 Δ_{1} 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.

Irrational k vectors: (3+d)-Dimensional Superspace

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(2πδ_{1}),
cos(2πδ_{2}),..., and
the symbols s1,s2,... refer to
sin(2πδ_{1}),
sin(2πδ_{2}),...

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.

Rational k vectors: 3-Dimensional Space

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(2πi**k**_{1}.**v**),
exp(2πi**k**_{2}.**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.

Irrational k vectors: (3+d)-Dimensional Superspace