Unfaithful representation

Problem 6-2. Of the eight examples of representations listed in this section, which are faithful For each unfaithful representation, which subgroup of the original group does it faithfully represent ... 

The top line identifies the group and contains headings for the columns associated with each covering operation. The symbols Ai, A2, Bi and B2 are the names of the four symmetry species corresponding to the irreducible representations. The lines of numbers to the right of the symmetry species designations are the traces of the matrices of the representations. In this case, the characters themselves represent the group, albeit unfaithfully. This is not always true. 

The representation of the Cjy group that we have presented is said to have the coordinates x, y, and z as its basis. Other representations can be obtained by using other functions as a basis and determining how the symmetry operators change these functions. The matrices in a representation do not have to have any physical interpretation, but they must multiply in the same way as do the symmetry operators, must be square, and all must have the same number of rows and columns. In some representations, called unfaithful or homomorphic, there are fewer matrices than there are symmetry operators, so that one matrix occurs in the places in the multiplication table where two or more symmetry operators occur. 

Because of the way in which block-diagonal matrices multiply, the 2 by 2 bloeks in the matrices in Eq. (9.76) if taken alone form another representation of the Csy group. When a reducible representation is written with its matrices in block-diagonal form, the block submatrices form irreducible representations, and the reducible representation is said to be the direct sum of the irreducible representations. Both the representations obtained from the submatrices are irreducible. The 1 by 1 blocks form an unfaithful or homomorphic representation, in which every operator is represented by the 1 by 1 identity matrix. This one-dimensional representation is called the totally symmetric representation. In this particular case we could not get three one-dimensional representations, because the Cs(z) operator mixes the x and y coordinates of a particle, preventing the matrices from being diagonal. 

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