Dirac character

Table A2.1. Multiplication table for the Dirac characters (class sums) of S(3).

The second equality in eq. (11) follows because f>, E and the identity, which can only arise when l = k, is repeated ck times. Because the Dirac characters commute (eq. (4)), the triple product in eq. (11) is invariant under any permutation of i,j, k, so that... 

Multiplication of the Dirac characters produces a linear combination of Dirac characters (see eq. (4.2.8)), as do the operations of addition and scalar multiplication. The Dirac characters therefore satisfy the requirements of a linear associative algebra in which the elements are linear combinations of Dirac characters. Since the classes are disjoint sets, the Nc Dirac characters in a group G are linearly independent, but any set of N< I 1 vectors made up of sums of group elements is necessarily linearly dependent. We need, therefore, only a satisfactory definition of the inner product for the class algebra to form a vector space. The inner product of two Dirac characters i lj is defined as the coefficient of the identity C in the expansion of the product il[ ilj in eq. (A2.2.8),... 

This definition satisfies all the requirements of an inner product (see, for example, Cornwell (1984), p.274). Therefore, the Dirac characters form a set of orthogonal but not normalized vectors. An orthonormal basis can be defined by (c ) 2 G,, for then... 

Whereas the MRs of group elements do not necessarily commute, the MRs of the Dirac characters do commute with one another and therefore also with their adjoints. They therefore form a set of normal matrices which can all be diagonalized by similarity transformations with the same unitary matrix S. The Dirac character matrices therefore have Nc (not necessarily distinct) eigenvalues and Nc corresponding eigenvectors. 

In general, when a Dirac character multiplies a Dirac character, as in eq. (A2.2.8), it produces a linear combination of Dirac characters (vectors). But for a particular linear combination of Dirac characters... 

Table A2.2. Character table of S(3) deduced from the diagonalization of the MRs of the Dirac characters.

Table A2.4 Characteristic determinants obtained in the diagonalization of the Dirac characters for the quaternion group Q.

There is available a variant of this method of determining character tables by diagonalization of the Dirac characters which is completely unambiguous (apart from the ordering of the rows of the character table, which is arbitrary) but which involves rather more work. 

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