WOLFRAM|DEMONSTRATIONS PROJECT

Molien Series for a Few Double Groups

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Trigonal
Octahedral
Icosahedral
U(3) ⊃ 2O:
T
2
U(3) ⊃ 2O:
T
2
A
0
A
3
E
2
T
1
T
2
E
1/2
E
5/2
G
3/2
Total
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
3
2
1
0
1
0
1
0
0
0
6
3
1
0
0
1
2
0
0
0
10
4
2
0
2
1
2
0
0
0
15
5
1
0
1
2
4
0
0
0
21
6
3
1
3
2
4
0
0
0
28
7
2
0
2
4
6
0
0
0
36
8
4
1
5
4
6
0
0
0
45
9
3
1
3
6
9
0
0
0
55
10
5
2
7
6
9
0
0
0
66
11
4
1
5
9
12
0
0
0
78
12
7
3
9
9
12
0
0
0
91
13
5
2
7
12
16
0
0
0
105
14
8
4
12
12
16
0
0
0
120
15
7
3
9
16
20
0
0
0
136
16
10
5
15
16
20
0
0
0
153
17
8
4
12
20
25
0
0
0
171
18
12
7
18
20
25
0
0
0
190
19
10
5
15
25
30
0
0
0
210
20
14
8
22
25
30
0
0
0
231
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
The Molien equation [1, 2] determines symmetry correlation tables [3]. The tables presented here describe correlations between representations of the unitary group and various finite subgroups: double trigonal
2
D
3
, double octahedral
2O
, and double icosahedral
2
A
5
. In the history of science, double groups were introduced by Klein [4] and subsequently applied to physics by Bethe [5]. Both German works are available in English.
The tables
U(2)⊃2G:
E
1/2
, some of which already appear in Bethe's 1929 article (cf. [5], tables 2 and 12), are of particular interest to physics. These tables are identical to rotational correlation tables, usually computed by multiplying group characters [5, 6, 7]. Rotational tables explain patterns of degeneracy observed in the rotational spectra of symmetrical molecules [6]. According to the fundamental importance of
SU(2)~SO(3)
in quantum mechanics, we derive representation naming conventions from the tables
U(2)⊃2G:
E
1/2
.