International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.10, pp. 262-268

## Section 1.10.5. Tables

T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

### 1.10.5. Tables

| top | pdf |

In this section are presented the irreducible representations of point groups of quasiperiodic structures up to rank six that do not occur as three-dimensional crystallographic point groups.

Table 1.10.5.1 gives the characters of the point groups with n = 5, 8, 10, 12, with n = 5, 8, 10, 12, and the icosahedral group I. The direct products with then follow easily. Although these direct products of a group K with do not belong to the isomorphism class of K, their irreducible representations are nevertheless given in the table for K because these irreducible representations have the same labels as those for K apart from an additional subindex u. The reresentations of the subgroup K of are the same as for K itself, those for the cosets get an additional minus sign. In the tables, the characters for the groups are separated from those for K by a horizontal rule. In addition to the characters are given the realizations of crystallographic point groups, and the irreducible components of the vector representations in direct space and internal space for these realizations. The vector representation in is called the perpendicular representation.

 Table 1.10.5.1| top | pdf | Character tables of some point groups for quasicrystals
 (a) .
n11111
Order15555
1 1 1 1 1
1
1
1
1
GeneratorsVector representationPerpendicular representation
5
 (b) .
n1225
Order1552
1 1 1 1
1 1 1
2 0
2 0
GeneratorsVector representationPerpendicular representation
52

5m

 (c) .
11111111
Order18482868
1 1 1 1 1 1 1 1
1 i
1 i i
1 i
1
1 i
1 i i
1 i
GeneratorsVector representationPerpendicular representation
8
 (d)
1222144
Order1848222
1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1
1 1 1 1
2 0 0 0
2 0 0 2 0 0
2 0 0 0
GeneratorsVector representationPerpendicular representation
822

 (e) .
n11111
Order15555
1 1 1 1 1
1
1
1
1
1 1 1 1 1
1
1
1
1
n11111
Order210101010
1 1 1 1 1
1
1
1
1
GeneratorsVector representationPerpendicular representation
10
 (f) .
n1222
Order110510
1 1 1 1
1 1 1 1
1 1
1 1
2
2
2
2
n2155
Order5222
1 1 1 1
1 1
1 1
1 1
0 0
2 0 0
0 0
2 0 0
GeneratorsVector representationPerpendicular representation

 (g) .
n111111
Order11264312
1 1 1 1 1 1
1 i
1
1 i 1 i
1
1 i
1 1 1
1
1
1 i 1
1
1
n111111
Order21234612
1 1 1 1 1 1
1
1 i
1
1 1 1
i
1
i 1 i
1
i
GeneratorsVector representationPerpendicular representation
12
 (h)
n1222
Order11264
1 1 1 1
1 1 1 1
1 1
1 1
2 1 0
2 1
2 0 0
2 2
2 1 0
n22166
Order312222
1 1 1 1 1
1 1 1
1 1 1
1 1 1
0 0
1 2 0 0
2 0 0 0
2 0 0
0 0
GeneratorsVector representationPerpendicular representation

 (i) I .
I
n112122015
Order15532
1 1 1 1 1
3 0
3 0
4 1 0
5 0 0 1
GeneratorsVector representationPerpendicular representation
532

In Table 1.10.5.2 the representation matrices for the irreducible representations in more than one dimension are given (one-dimensional representations are just the characters). For the cyclic groups there are only one-dimensional representations, for the dihedral groups there are one- and two-dimensional irreducible representations. There are four irreducible representations of I of dimension larger than one. The four- and five-dimensional ones are given as integer representations. They form crystallographic groups in 4D and 5D. The two three-dimensional representations have the same matrices. The elements, however, are connected by an outer automorphism. That means that the ith element is represented by in the representation , and by in . The element is another element . The corresponding j for each i is given in Table 1.10.5.3. Examples of physical tensors are given in Table 1.10.5.4 with their respective numbers of free parameters as determined using representations of three nD point groups.

 Table 1.10.5.2| top | pdf | Matrices of the irreducible representations of dimension corresponding to the irreps of Table 1.10.5.1
 (a)
Representation
 (b)
Representation
 (c)
Representation
 (d)
Representation
 (e) I. First column: numbering of the elements. . Horizontal rules separate conjugation classes.
No.Order
1 1
2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 5
10 5
11 5
12 5
13 5
14 5
15 5
16 5
17 5
18 5
19 5
20 5
21 5
22 5
23 5
24 5
25 5
26 3
27 3
28 3
29 3
30 3
31 3
32 3
33 3
34 3
35 3
36 3
37 3
38 3
39 3
40 3
41 3
42 3
43 3
44 3
45 3
46 2
47 2
48 2
49 2
50 2
51 2
52 2
53 2
54 2
55 2
56 2
57 2
58 2
59 2
60 2
 Table 1.10.5.3| top | pdf | The representation matrices for
 The representation matrices for are the same as for . Correspondences are given as pairs i, j: .
ijijijijijij
1 1 11 21 21 5 31 42 41 29 51 48
2 14 12 16 22 6 32 45 42 39 52 54
3 23 13 17 23 8 33 36 43 33 53 46
4 15 14 4 24 10 34 27 44 30 54 50
5 25 15 2 25 11 35 26 45 38 55 52
6 24 16 13 26 34 36 28 46 49 56 57
7 19 17 12 27 35 37 31 47 53 57 59
8 20 18 7 28 43 38 40 48 51 58 56
9 18 19 9 29 44 39 37 49 47 59 58
10 22 20 3 30 41 40 32 50 55 60 60
 Table 1.10.5.4| top | pdf | Number of free parameters for some tensors and their symmetry groups
TensorElements222(−1−11)622(1−1−1)
Metric 5 4 2
Elasticity 17 10 5
Phonon elasticity 9 5 2
Phason elasticity 3 2 2
Phonon–phason elasticity 5 3 1