MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

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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.