Faithful representation group

If each group element corresponds to a different matrix, the representation is said to ht faithful. A faithful representation is a matrix group that is isomorphic to the group being represented. 

The set of components of the vector r1 in eq. (13) is the Jones symbol or Jones faithful representation of the symmetry operator R, and is usually written as (x / /) or x / z. For example, from eq. (15) the Jones symbol of the operator R (n/2 z) is (yxz) or yxz. In order to save space, particularly in tables, we will usually present Jones symbols without parentheses. A faithful representation is one which obeys the same multiplication table as the group elements (symmetry operators). 

This shows that the representation of the group on the nuclear position vectors referred to the laboratory fjxed coordinate system is not a faithful representation of the isometric group ( ) = r[Pg.25]

Although the H nuclei of the OH groups of glycol lay in the symmetry plane of the Cs tops, it is convenient to introduce nuclei in general site in a SRM associated to a particular molecule to be sure that a faithful representation of sr on the distances of that set is generated. 

Three other representations of the group C2v are the sets of four 2X2 matrices obtained by omitting a particular row and column from each of the matrices of Eq. 4.1. Only one of these gives a faithful representation. 

If a group is homomorphic onto the matrix representation then the representation is a faithful representation of the factor group. A factor group of Cjv consists of the cosets of the invariant subgroup iJ, = [Pg.234]

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 ... 

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

Figure 13.5. Schematic representation of the electronic structure of ti states in Ceo- Left eigenstates of the angular momentum in a potential with spherical symmetry states with angular momentum I = 0(s), l(p), 2(d), 3(/), 4(g), 5(h), 6(j), l(k) are shown, each with degeneracy (21+ 1). Right sequence of states in Ceo with the appropriate labels from the irreducible representations of the icosahedral group. The filled states are marked by two short vertical lines, indicating occupancy by two electrons of opposite spin. The sequence of levels reflects that obtained from detail calculations [210,211], but fteir relative spacing is not faithful and was chosen mainly to illustrate the correspondence between the spherical and icosahedral sets.

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