Diagonalization 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. 

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. 

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

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