Physical Chemistry Third Edition

1296 I Matrix Representations of Groups

Each member of anabelian group(a group in which all members commute with

each other) is also in a class by itself. Since the operators commute with each other,

A−^1 BAA−^1 ABEBB (ifA,A−^1 , andBcommute) (I-14)

The identity element is always in a class by itself, since the identity matrix commutes

with every other matrix. If a group includes the inversion operator, this operator is also

in a class by itself.

I.3 Character Tables

For some purposes it is not necessary to use the entire matrices in a representation to get

useful information. Thetraceof a square matrix is defined to be the sum of the diagonal

elements of the matrix. The set of traces of the matrices in a given representation are

called thecharactersof the representation and the list of them is called acharacter

table. We denote the three-dimensional representation of Eqs. (I-3)–(I-6) by the symbol

Γ. Its character table is

C 2 v E C 2 σyz σxz

Γ 31 1 1

Each character is listed below the symbol for the symmetry operation. The character

table for each of the three one-dimensional representations of theC 2 vgroup is just the

list of the elements of the one-by-one matrices, since there is only one term in each

trace sum. These character tables are as follows:

C 2 v E C 2 σyz σxz Function

A 1 1 111 z

B 1 1 − 1 − 11 x

B 2 1 − 11 − 1 y

The A and B symbols attached to these representations are obtained as follows:^14

One-dimensional representations are designated by A if they are symmetric to rotation

by 2π/nradians about the principaln-fold rotation axis (n2 for a 180◦rotation

in this case) and are designated by B if they are antisymmetric to this rotation. The

subscripts 1 and 2 designate whether (in this case) they are symmetric or antisymmetric

to reflection in a vertical plane. A two-dimensional representation is designated by E

(not to be confused with the identity operation), and a three-dimensional representation

is designated by T. Subscripts g and u are sometimes added to specify the symmetry

with respect to inversion (ggeradeeven; uungeradeodd). A representation

with all characters equal to 1, like the A 1 representation in this case, is called thetotally

symmetric representation.

(^14) A. W. Adamson,A Textbook of Physical Chemistry, 3rd ed., Academic Press, Orlando, FL, 1986,
p. 747ff.