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Molien Series for a Few Double Groups

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
.
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