Harold T. Stokes and Branton J. Campbell, Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA, branton_campbell@byu.edu
There are two most commonly used tabulations for the irreducible representations (IRs) of the 230 crystallographic space groups, those of (1) Kovalev (KOV) (1993) and those of (2) Cracknell, Davies, Miller & Love (CDML) (1979) which is based on Miller & Love (1967). In order for the ISOTROPY Software Suite, which uses the CDML IR tabulation, to be more useful to those who employ the KOV IR tabulation, we have generated a mapping between complete CDML and KOV IRs for both complex irreducible representations (CIRs, see Table 1) and physically irreducible representations (PIRs, see Table 2).
In Stokes & Hatch (1988), the mapping of PIRs from CDML to KOV settings are given in Table 7 for IRs at special k points. This 1988 mapping appears to agree with our present Table 2. (We did not do a detailed comparison for the 4777 IRs involved.) However, we did find some disagreements with the mapping in the database of the ISOTROPY Software Suite; out of the 4777 IRs involved, the mapping in the present Table 2 and the mapping in the pre-2022 versions of the ISOTROPY Software Suite disagree in 18 cases (No. 39: Y1-4 and T1-4; No. 41: Y1-4; No. 46: X1-4 and S1-2). We do not know when or why these changes occurred in the database. We are, however, confident in the present mapping.
Note that none of the KOV IRs map onto the CDML ZA IRs found in space groups Nos. 195, 198, 200, 201, and 205. (See ISO-IR.)
Strictly speaking, a space-group IR is independent of space-group setting, meaning that the matrix representation of a given space-group operation and IR does not depend on the basis or origin that define the unit cell. However, the presentation of the IR is setting dependent because labels associated with space-group generators and operations are setting dependent. For example, the IR matrix corresponding to the mirror plane called −x,y,z in one setting will correspond to the mirror plane called x,−y,z in a setting rotated 90° around the z axis.
Many of the IR-label mappings shown in Table 1 are somewhat arbitrary in principle, because they can depend on a choice of a transformations which takes space-group operation from the KOV setting to the CDML setting. In general, there can be more than one way to choose such a transformation, resulting in different label mappings. We present our transformation choices in Table 3, which are consistent with the transformations given in Table 6 of Stokes and Hatch (1988). Note that for five space groups (Nos. 26, 30, 62, 63, and 64), Stokes and Hatch (1988) inadvertently listed negative-determinant transformations between the KOV and CDML space group settings, which would switch the handedness of the lattice basis. In the present Table 3, we uniformly employ setting transformations with positive determinants.
Each space-group transformation from old to new (primed)
setting can be written as a 4×4 affine matrix:
T=(TR,Tτ / 0,1)
which represents a proper rotation TR followed by an
origin shift Tτ. Various quantities are
transformed in this way as follows:
(1) affine operations: g′ = T g
T −1 or {R′|t′} =
{TR R TR−1 |
TR t + Tτ
− TR R TR−1
Tτ}
(2) atomic positions: r′ = TR
r + Tτ
(3) k vectors: k′ = k
TR−1
(4) lattice basis vectors: b′i
= ∑j (TR)ji
bj
In Kovalev (1993), the IRs of 14 space groups (Nos. 26, 30, 38, 39, 40, 41,51, 52, 54, 57, 60, 62, 63, 64) are given in both the KOV setting and the setting of the International Tables of Crystallography (IT) (2016). We follow this same practice in the present Table 1, where these settings are denoted by a "K" or "I" following the space-group symbol.
The transformations we chose for transforming space-group operations from the IT settings to the CDML settings are given in Table 3. Note that our default IT settings employ (1) monoclinic space groups with unique axis b and cell choice 1, (2) trigonal space groups with hexagonal axes, and (3) cell choice 2 whenever relevant.
Technically, the KOV and CDML tables contain little-k-group IRs from which we can calculate complete space-group CIRs. Since KOV and CDML do not always select the same defining k vector for an IR, we determine the mapping by comparing the matrix characters of the KOV and CDML CIRs.
KOV and CDML define the little-k-group IR matrices Dk(t) for lattice translations t differently. In KOV, D(t)=exp(−2πi k⋅t), whereas in CDML, D(t)=exp(+2πi k⋅t). This difference in sign usually causes a KOV IR to be mapped onto the complex conjugate of the corresponding CDML IR, which only matters for IRs of type 3, for which an IR and its complex conjugate are not equivalent. An asterisk symbol (*) at the right-hand side of an IR label indicates a complex conjugate.
From an IR at +k, another IR can be generated at −k by simply reversing the sign of k in the k-dependent part of each IR matrix. When there is no operation in the space group that takes k into −k, these two IRs will be inequivalent. In the CDML system, the label for the IR at −k is obtained by modifying the label of the IR at +k as follows. If the k-vector label in the IR label has only one alphabet letter, it is appended with a second alphabet letter; for example, H becomes HA or Q becomes QA. If the k-vector label in the IR label has two alphabet letters, the second letter is replaced with another letter of the alphabet; for example DT (for Δ) becomes DU, SM (for Σ) becomes SN, LD (for Λ) becomes LE, and GP (for general point) becomes GQ. See CDML for a table of such extra k-vector labels and a more detailed explanation. We see in the very first entry in Table 1 that the KOV IR k0t1 for space group No. 1 P1 is mapped onto the CDML IR GQ1=GP1*, which is of type 3 since no space-group operation takes +k into −k, and GP1 is therefore not equivalent to GQ1=GP1*.
When we say that a certain KOV IR is mapped onto some CDML IR, we mean that they are equivalent and have the same matrix characters for equivalent space-group operations. In other words, if setting transformation T provides a one-to-one mapping of space-group operations gKOV in the KOV setting to operations gCDML in the CDML setting (i.e. if gCDML = T gKOV T −1), then characters χ of the corresponding KOV and CDML IR matrices are also identical: χCDML(gCDML) = χKOV(gKOV)
Fortunately, we do not need to test every operation in the space group. Each IR is associated with a star of k. If the KOV-to-CDML T takes a KOV k into one of the arms of the star of a CDML k, then we know that each KOV IR associated with the KOV k must be equivalent to one of the CDML IRs associated with the equivalent CDML k. Thus, we only need to consider the number of operations sufficient to distinguish different IRs for a given k by their characters. This a very small number of operations.
As an example, consider space group No. 30 Pnc2. We generate the representative operations in the CDML setting and then transform each operation into the KOV setting using the transformation −y,−1⁄4+x,z from Table 3. The operations are shown in the table below using the Seitz notation, {R|v}, which denotes a point operation R followed by a translation v. The symbol 1 denotes the identity operation, 2[001] denotes a two-fold rotation about the z axis, and −2[100] denotes a reflection through the plane perpendicular to the x axis, etc.
Consider the KOV k vector k12=(0,1⁄2,0). When we transform this k into the CDML setting, we obtain CDML k=(−1⁄2,0,0) which is equivalent to X=(1⁄2,0,0). So KOV k12 IRs are mapped onto CDML X IRs. The characters for the CDML IRs X1,2,3,4 and for the KOV IRs k12t1,2,3,4 are shown in the table below. By matching the CDML and KOV characters, we easily obtain the mapping k21t1=X1, k21t2=X2, k21t3=X4, k21t4=X3, in agreement with the entry in Table 1.
| CDML g | KOV g | X1 | X2 | X3 | X4 | k21t1 | k21t2 | k21t3 | k21t4 |
| {1|0,0,0} | {1|0,0,0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| {2[001]|0,0,0} | {2[001]|1⁄2,0,0} | 1 | 1 | −1 | −1 | 1 | 1 | −1 | −1 |
| {−2[100]|0,1⁄2,1⁄2} | {−2[010]|1⁄2,0,1⁄2} | 1 | −1 | 1 | −1 | 1 | −1 | −1 | 1 |
| {−2[010]|0,1⁄2,1⁄2} | {−2[100]|1,0,1⁄2} | 1 | −1 | −1 | 1 | 1 | −1 | 1 | −1 |
As a more complicated example, consider the KOV k vector k4=(μ,1⁄2,ν) for the same space group No. 30 Pnc2. We evaluate this k using arbitrary parameter values μ=1⁄5 and ν=1⁄7 and obtain the KOV k=(1⁄5,1⁄2,1⁄7). We use parameters like 1⁄5 and 1⁄7 to ensure that the evaluated k doesn't accidentally have more symmetry than k4. When we transform this k into the CDML setting, we obtain CDML k=(−1⁄2,1⁄5,1⁄7) which is equivalent to L=(1⁄2,α,β) with α=1⁄5 and β=1⁄7. So KOV k4 IRs are mapped onto CDML L IRs.
The characters for the CDML IRs L1,2 and for the KOV IRs k4t1,2 are shown for selected operations g in the table below. We see that none of the KOV IRs have the same characters as any of the CDML IRs. However the characters for the KOV IR k4t1 are equal to the complex conjugate of the characters for the CDML IR L1, and the characters for the KOV IR k4t2 are equal to the complex conjugate of the characters for the CDML IR L2, so we obtain the mapping k4t1=L1*, k4t2=L2*, in agreement with the entry in Table 1. Note that the irrational form of 1.457+i 0.702 is [cos(12π/35)+cos(2π/35)]+i [sin(12π/35)-sin(2π/35)].
| CDML g | KOV g | L1 | L2 | k4t1 | k4t2 |
| {1|0,0,0} | {1|0,0,0} | 2 | 2 | 2 | 2 |
| {−2[100]|0,1⁄2,1⁄2} | {−2[010]|1⁄2,0,1⁄2} | 1.457+i 0.702 | 1.457−i 0.702 | −1.457−i 0.702 | −1.457+i 0.702 |
For space groups with centered lattices, computer calculations are more conveniently carried out in a setting using a primitive lattice. For one thing, all lattice vectors are simply triplets of integers. Both KOV and CDML provide transformations to settings with primitive lattices. We denote these settings as pKOV, pCDML, and pIT (for the 14 space groups where KOV provides an IT setting).
As an example, consider space group No. 39 Abm2 for the IT setting in KOV. In the table below, we show the transformations between different space-group settings. Then we show the representative operations, primitive lattice translations, and the general k vector in each setting. We see in the CDML and IT columns that the CDML setting is C2mb, and the IT setting is Abm2.
| pCDML | CDML | IT | pIT | ||||
| pCDML to CDML 1⁄2x+1⁄2y,−1⁄2x+1⁄2y,z |
CDML to IT z,y,−x |
IT to pIT y−z,y+z,x | |||||
| CDML to pCDML x−y,x+y,z |
IT to CDML −z,y,x |
pIT to IT z,1⁄2x+1⁄2y,−1⁄2x+1⁄2y |
|||||
| g=x,y,z | x,y,z={1|0,0,0} | x,y,z={1|0,0,0} | x,y,z | ||||
| g=y,x,−z | x,−y,−z={2[100]|0,0,0} | −x,−y,z={2[001]|0,0,0} | −y,−x,−z | ||||
| g=1⁄2+y,1⁄2+x,z | 1⁄2+x,−y,z={−2[010]|1⁄2,0,0} | x,−y,−1⁄2+z={−2[010]|0,0,−1⁄2} | 1⁄2−y,−1⁄2−x,z | ||||
| g=1⁄2+x,1⁄2+y,−z | 1⁄2+x,y,−z={−2[001]|1⁄2,0,0} | −x,y,−1⁄2+z={−2[100]|0,0,−1⁄2} | 1⁄2+x,−1⁄2+y,−z | ||||
| t=(1,0,0) | (1⁄2,−1⁄2,0) | (0,−1⁄2,−1⁄2) | (0,−1,0) | ||||
| t=(0,1,0) | (1⁄2,1⁄2,0) | (0,1⁄2,−1⁄2) | (1,0,0) | ||||
| t=(0,0,1) | (0,0,1) | (1,0,0) | (0,0,1) | ||||
| k=(μ,ν,ω) | (μ+ν,−μ+ν,ω) | (ω,−μ+ν,−μ−ν) | (ν,−μ,ω) | ||||
For calculations involving physical properties, we often want to use real numbers. The physically irreducible representations (PIRs) map space-group operations onto real matrices and are irreducible with respect to real numbers. We use PIRs throughout the software in the ISOTROPY Software Suite.
A type-1 CIR can always be transformed to real matrices. For type-2 and type-3 CIRs, we form the PIR from the direct sum of the CIR matrices and their complex conjugates, which can then be transformed to real matrices.
A type-2 CIR is equivalent to their own complex conjugate, so we obtain the PIR from the direct sum of the CIR matrix with itself. We denote these PIRs with a symbol containing the CIR twice. For example, in space group No. 18 P22_22_1, the KOV CIR k17t1 and the equivalent CDML CIR Q1 are both of type 2. We write the KOV PIR symbol as k17t1t1 and the CDML PIR symbol as Q1Q1. Observe that we repeat the k-vector symbol Q in the CDML PIR symbol, but we repeat the IR specifier t1 rather than the k-vector symbol k17 in the KOV PIR symbol.
A type-3 CIR is not equivalent to own own complex conjugate, so that we obtain the PIR from the direct sum of the CIR with its inequivalent complex conjugate. We write the PIR symbol using both CIR symbols. For example, in space group No. 18 P22_12_1, the inequivalent KOV CIRs, k14t1 and k14t2, are of type 3 and are complex conjugates of each other, as are the corresponding CDML CIRs, P2 and P1, respectively. We write the KOV PIR symbol as k14t1t2 and the CDML PIR symbol as P1P2. Note that the order of the CIR symbols in the PIR symbol doesn't matter. The direct sum of CIRs 2 and 1 is equivalent to the direct sum of CIRs 1 and 2. We choose to write the PIR symbols with the CIR symbols in numerical order, which in this case is 1 followed by 2.
As another example, in space group No. 23 I222, the KOV CIR k16t1 and the corresponding CDML CIR W1 are of type 3. Their complex conjugates are also of type 3, but are constructed at −k rather than +k. CDML denotes this complex-conjugate CIR as WA1, and we therefore write the CDML PIR symbol as W1WA1. KOV does not have a separate symbol for an CIR evaluated at −k, so we write the KOV PIR symbol as k14t1t1*, indicating the direct sum of k14t1 with its complex conjugate k14t1*. Though we did not do so in Stokes and Hatch (1988), or in pre-2022 versions of the ISOTROPY Software Suite, we now include the "*" at the end of such a symbol to clearly distinguish between type-2 and type-3 PIRs.
Using the information in Table 1, we have generated a mapping of CDML PIRs onto KOV PIRs which is presented in Table 2.
We thank Andrew Wills at the University College of London for sharing with us computer readable files which provided information about the KOV IRs, making it possible to generate the KOV IR matrices necessary for this work.
A. P. Cracknell, B. L. Davies, S. C. Miller, and W. F. Love, Kronecker Product Tables, Vol. 1 (Plenum, New York, 1979).
International Tables for Crystallography (2016), Vol. A: Space Groups, edited by M. I. Aroyo.
O. V. Kovalev, Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Corepresentations, edited by H. T. Stokes and D. M. Hatch (Gordon and Breach, Amsterdam, 1993).
S. C. Miller and W. F. Love, Tables of Irreducible Representations of Space Groups and Co-Representations of Magnetic Space Groups (Pruett, Boulder, 1967).
H. T. Stokes and D. M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups (World Scientific, Singapore, 1988).