10
GEORGE A. HAGEDORN
Typ e A Crossings The two irreducible representations of G that correspond to
E^(X) and Eg(X) are not unitarily equivalent to one another.
Typ e B Crossings The two irreducible representations of G that correspond to
EA(X) and E&(X) are unitarily equivalent to one another.
When H is a subgroup of index 2, standard group representation theory does not apply.
Instead of representations, the basic objects of interest are what Wigner [48] called corep-
resentations in the early days of quantum mechanics. A general theory of corepresentations
was first developed by Wigner [48]. A more modern, non-basis-dependent treatment can
be found in [33]. This general theory shows that any corepresentation can be decomposed
as a direct sum of irreducible corepresentations. Furthermore, there are three distinct types
of irreducible corepresentations which are called Types /, //, and III.
To describe these three types, we first note that G can be decomposed as G = H U /C//,
where K is an arbitrary, but fixed, antiunitary element of G. Then, if U is an irreducible
corepresentation of G, we let UJJ denote the restriction of U to H. Then the three types are
described as follows [33]:
Type I Corepresentations UJJ is an irreducible representation.
Type II Corepresentations UJJ decomposes into a direct sum of two equivalent irre-
ducible representations, Uff D ® D. Furthermore, U may be cast in the form
U(h) = (DW D(h)\ U ^ = \K ~
0
j ' a n d u(Kh) = U(JC)U(h),ioia\lheH.
Here K is an antiunitary operator that satisfies K2 = —D(K?) and K D{K~^hK) K~l =
D(h) for all heH.
Type III Corepresentations UJJ decomposes into a direct sum of two inequivalent
irreducible representations, UJJ = D C. Furthermore, U may be cast in the form
U(h) = ( D f )
c
°
f t )
) , U(a) = ( 0. ^ ( ^ " ^ . a n d U(Kh) = U(K) U(h), for all
heH. Here K : Hp —• HQ is an antiunitary operator that satisfies K D(/C-1/i/C) K~l =
C[h) for all heH.
When G / //, each distinct eigenvalue of h(X) is associated with a unique corepresen-
tation of G. From the structure theory outlined above, it is clear that minimal multiplicity
eigenvalues associated with Type /corepresentations have multiplicity 1. Minimal multiplic-
ity eigenvalues associated with Type //o r Type ///corepresentations have multiplicity 2. In
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