Co-representations

Multiplication of Co-Representation Matrices.—We have referred above to the representations of nonunitary groups as co-representations. This distinction is made because the co-representation matrices for the group operators do not multiply in the same way as do the operators themselves.5 As will be seen below, this is a direct result of the fact that some of the operators in the group are antilinear. Consider that is the a 6 basis function of the i411 irreducible co-representation of G. The co-representation matrices D (u) and D (a) may be defined such that... 

This is, therefore, the form taken by a similarity transformation of co-representation matrices of nonunitary groups, and the two sets of matrices D and B are considered to be equivalent. It is interesting to note that if one lets V = o>E be a multiple of the unit matrix B(i)(u) =... 

D(l)(u) but B(1)(a) = to 2D<0(a) such that a common arbitrary phase factor remains in the co-representation matrices D(0(a) for the anti-unitary elements of nonunitary groups. 

Irreducible Co-Representations of Nonunitary Croups.—To determine the co-representation matrices D , let us select a set of l functions, which forms a basis for an irreducible representation A (u) of the unitary subgroup H. That is... 

Case (b) aproduces a set of functions t/rj, which is independent of the set but which also forms a basis for A (u) of H. The irreducible co-representation D of G corresponds again to a single irreducible representation of H, but has twice its dimension. In this case the dimension of A (u) is doubled. 

In order to obtain explicit forms for the co-representation matrices D1 of the nonunitary group G, in terms of the representation matrices A (u) of the unitary subgroup H, it is necessary to know how the set of functions atransforms under the operations of H. Let a0 = iffa, then... 

The explicit forms for the co-representation matrices obtained by Wigner are as follows ... 

We will not be concerned further with the explicit forms of the co-representation matrices. Instead we need ask only to which of the three cases a specific representation A (u) of the group H belongs when H is considered as a subgroup of O. The co-representation matrices can be written down immediately once this is known. The irreducible representations of H can be obtained by standard means since H is unitary. It, therefore, remains to obtain a method by which one can decide between the three cases given the group 0 and an irreducible representation of H.9 In order to do this we need the fact that the matrices / and A (u) may be assumed to be unitary,6 and that the A((u) matrices satisfy the usual orthogonality relation... 

Application, to Point Groups.—Equations (12-27) and (12-28) may be used directly to obtain the co-representations of nonunitary point groups.9 It is convenient to consider the 122 point groups in three categories. 

Co-representation matrices explicit forms, 733 multiplication of, 731 of the nonunitary group, 732 Corliss, L. M., 757 Corson, E. M., 498 Coulomb field Dirac equation in, 637 Coulomb gauge, 643,657,664 Counting functions, 165 Covariance matrix, 160 Covariant amplitude of one-particle system, 511 of one, two, etc. particle systems, 511... 

Nonlinear systems, 78 analytical methods, 349 Nonlinearities, nonanalytic, 383,389 Nonsingular matrix, 57 Nonunitary groups, 725 as co-representations, 731 representation theory, 728 structure of, 727 Nonunitary point groups, 737 No-particle state. 540,708 expectation value of current operator, 587 out, 586... 

Given the IRs T of H, all the irreducible co-representations F of G can be determined from eqs. (40)-(42). Although the equivalence of T, T and the sign of c(Z) provide a criterion for the classification of the co-representations of point groups with antiunitary operators, this will be more useftd in the form of a character test. 

Example 14.2-2 Find the co-representations of the magnetic point group Amm or C4v(C2v). Take Q = rr.A, with a the unit vector along [110]. 

The character table of 2mm (C2v) is given in Table 14.4, together with the determination of the type of co-representation of 4mm from h 1 (B2) and the projective factors... 

Table 14.6. Clebsch-Gordan coefficients for the inner direct products of irreducible co-representations.

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