MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS
11
the minimal multiplicity situations, the antiunitary operators K that occur in Type 7/corep-
resentations map a one dimensional space to itself. A simple calculation shows that such
operators satisfy K 1. Thus, in the minimal multiplicity situation, K is a conjugation,
and D{K2) = - 1 .
This structure theory of corepresentations shows that if two minimal multiplicity eigen-
values Ej[(X) and E${X) cross, then there are nine possibilities:
Typ e C Crossings The two irreducible corepresentations of G that correspond to
Ej^{X) and Eg(X) are both of Type 7, but are not unitarily equivalent to one another. Both
eigenvalues have multiplicity 1 away from the crossing.
Typ e D Crossings The two irreducible corepresentations of G that correspond to
Ej[(X) and Eg(X) are both of Type 77, but are not unitarily equivalent to one another.
Both eigenvalues have multiplicity 2 away from the crossing.
Typ e E Crossings The two irreducible corepresentations of G that correspond to
Ej[(X) and Eg(X) are both of Type 777, but are not unitarily equivalent to one another.
Both eigenvalues have multiplicity 2 away from the crossing.
Typ e F Crossings The two irreducible corepresentations of G that correspond to
EA(X) and Eg(X) are of Types 7 and 77. Away from the crossing, the eigenvalue associated
with the Type 7 corepresentation has multiplicity 1 and the other eigenvalue has multiplicity
2 away from the crossing.
Typ e G Crossings The two irreducible corepresentations of G that correspond to
Ej[(X) and Eg(X) are of Types 7 and 777. Away from the crossing, the eigenvalue associ-
ated with the Type 7 corepresentation has simple multiplicity and the other eigenvalue has
multiplicity 2.
Typ e H Crossings The two irreducible corepresentations of G that correspond to
Ej[(X) and Eg(X) are of Types 77 and 777. Both eigenvalues have multiplicity 2 away from
the crossing.
Typ e I Crossings The two irreducible corepresentations of G that correspond to
Ej{(X) and Eg(X) are both of Type 7 and are unitarily equivalent to one another. Both
eigenvalues are multiplicity 1 away from the crossing.
Typ e J Crossings The two irreducible corepresentations of G that correspond to
Ej\(X) and Eg(X) are both of Type 77 and are unitarily equivalent to one another. Both
eigenvalues are multiplicity 2 away from the crossing.
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