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