Similarity transformations yield irreducible representations, \u0393 , which lead to the useful tool in group theory \u2013 the character table. The general strategy for determining \u0393 is as follows: A, B and C are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). There is some similarity transform operator such that<\/p>\n
where v uniquely produces block-diagonalized matrices, which are matrices possessing square arrays along the diagonal and zeros outside the blocks<\/p>\n
Matrices A, B, and C are reducible. Sub-matrices A , B and C obey the same multiplication properties as A, B and C. If application of the similarity transform does not further block-diagonalize A\u2019, B\u2019 and C\u2019, then the blocks are irreducible representations. The character is the sum of the diagonal elements of \u0393 .<\/p>\n
As an example, let\u2019s continue with our exemplary group: E, C , C , \u03c3 , \u03c3 \u2019, \u03c3 \u201d by defining an arbitrary basis \u2026 a triangle<\/p>\n
The basis set is described by the triangles vertices, points A, B and C. The transformation properties of these points under the symmetry operations of the group are:<\/p>\n
These matrices are not block-diagonalized, however a suitable similarity transformation will accomplish the task,<\/p>\n
Applying the similarity transformation with C as the example,<\/p>\n
if v \u22c5 C * \u22c5 v is applied again, the matrix is not block diagonalized any further. The same diagonal sum is obtained *though offdiagonal elements may change). In this case, C * is an irreducible representation, \u0393 .<\/p>\n
The similarity transformation applied to other reducible representations yields:<\/p>\n
Thus a 3 \u00d7 3 reducible representation, \u0393 , has been decomposed under a similarity transformation into a 1 (1 \u00d7 1) and 1 (2 \u00d7 2) block-diagonalized irreducible representations, \u0393i. The traces (i.e. sum of diagonal matrix elements) of the \u0393 \u2019s under each operation yield the characters (indicated by \u03c7) of the representation. Taking the traces of each of the blocks:<\/p>\n
This collection of characters for a given irreducible representation, under the operations of a group is called a character table. As this example shows, from a completely arbitrary basis and a similarity transform, a character table is born.<\/p>\n
The triangular basis set does not uncover all \u0393 of the group defined by {E, C , C , \u03c3 , \u03c3 \u2019, \u03c3 \u2019\u2019}. A triangle represents Cartesian coordinate space (x,y,z) for which the \u0393 s were determined. May choose other basis functions in an attempt to uncover other \u0393 s. For instance, consider a rotation about the z-axis,<\/p>\n
The transformation properties of this basis function, R , under the operations of the group (will choose only 1 operation from each class, since characters of operators in a class are identical):<\/p>\n
Note, these transformation properties give rise to a \u0393 that is not contained in a triangular basis. A new (1 x 1) basis is obtained, \u0393 , which describes the transform properties for R . A summary of the \u0393 for the group defined by E, C , C , \u03c3 , \u03c3 \u2019, \u03c3 \u201d is:<\/p>\n
Is this character table complete? Irreducible representations and their characters obey certain algebraic relationships. From these 5 rules, we can ascertain whether this is a complete character table for these 6 symmetry operations.<\/p>\n","protected":false},"excerpt":{"rendered":"