In this section are presented the irreducible representations of point groups of quasiperiodic structures up to rank six that do not occur as threedimensional 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.
     
n  1  1  1  1  1 
Order  1  5  5  5  5 

1 
1 
1 
1 
1 

1 





1 





1 





1 




 Generators  Vector representation  Perpendicular representation 
5 



    
n  1  2  2  5 
Order  1  5  5  2 

1 
1 
1 
1 

1 
1 
1 


2 


0 

2 


0 
 Generators  Vector representation  Perpendicular representation 
52 







5m 











        
 1  1  1  1  1  1  1  1 
Order  1  8  4  8  2  8  6  8 

1 
1 
1 
1 
1 
1 
1 
1 

1 

i 






1 
i 



i 



1 





i 


1 








1 

i 






1 


i 



i 

1 





i 

 Generators  Vector representation  Perpendicular representation 
8 











       
 1  2  2  2  1  4  4 
Order  1  8  4  8  2  2  2 

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 
 Generators  Vector representation  Perpendicular representation 
822 



























     
n  1  1  1  1  1 
Order  1  5  5  5  5 

1 
1 
1 
1 
1 

1 





1 





1 





1 





1 
1 
1 
1 
1 

1 





1 





1 





1 




     
n  1  1  1  1  1 
Order  2  10  10  10  10 

1 
1 
1 
1 
1 

1 





1 





1 





1 


































 Generators  Vector representation  Perpendicular representation 
10 















    
n  1  2  2  2 
Order  1  10  5  10 

1 
1 
1 
1 

1 
1 
1 
1 

1 

1 


1 

1 


2 




2 




2 




2 



    
n  2  1  5  5 
Order  5  2  2  2 

1 
1 
1 
1 

1 
1 



1 

1 


1 


1 



0 
0 


2 
0 
0 



0 
0 


2 
0 
0 
 Generators  Vector representation  Perpendicular representation 




































      
n  1  1  1  1  1  1 
Order  1  12  6  4  3  12 

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 





      
n  1  1  1  1  1  1 
Order  2  12  3  4  6  12 

1 
1 
1 
1 
1 
1 








1 








1 
i 



1 













1 

1 

1 





i 



1 







i 
1 


i 

1 









i 


 Generators  Vector representation  Perpendicular representation 
12 











    
n  1  2  2  2 
Order  1  12  6  4 

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 
     
n  2  2  1  6  6 
Order  3  12  2  2  2 

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 
 Generators  Vector representation  Perpendicular representation 




























I      
n  1  12  12  20  15 
Order  1  5  5  3  2 

1 
1 
1 
1 
1 

3 


0 


3 


0 


4 


1 
0 

5 
0 
0 

1 
 Generators  Vector representation  Perpendicular representation 
532 












In Table 1.10.5.2 the representation matrices for the irreducible representations in more than one dimension are given (onedimensional representations are just the characters). For the cyclic groups there are only onedimensional representations, for the dihedral groups there are one and twodimensional irreducible representations. There are four irreducible representations of I of dimension larger than one. The four and fivedimensional ones are given as integer representations. They form crystallographic groups in 4D and 5D. The two threedimensional 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.
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 




i  j  i  j  i  j  i  j  i  j  i  j 
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 

Tensor  Elements  222(−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 
