Physical Chemistry Third Edition

1298 I Matrix Representations of Groups

There are several theorems and facts that give useful information about the irre-

ducible representations of groups. They can be used to understand and apply character

tables.^16 We state a few of them without proof:

1. The totally symmetric representation (the one-dimensional representation with all
   characters equal to 1) occurs with all groups.


2. If there is more than one operation in a class, all of the operations in a class will
   have identical characters in any representation, and can be lumped together in the
   same column of a character table, as is done in Table A.26 of Appendix A.


3. The number of irreducible representations equals the number of classes. Therefore,
   in a complete character table containing the characters of all irreducible representa-
   tions, the number of rows will equal the number of columns.


4. The sum of the square of the dimensions of the irreducible representations is equal
   to the order of the group (the number of operators in the group). If the dimension
   of irreducible representation numberiis calledliand if the order of the group ish,
   then
   ∑

i

l^2 ih (I-15)

where the sum is over all irreducible representations of the group. If we denote the

character of a given operation,O, in the representation numberibyχi(O), then

Eq. (I-15) can be written

∑

i

[χi(E)]^2 h (I-16)

since the character of the identity operator equals the dimension of the representation.

5. The sum of the squares of the characters in any irreducible representation is equal
   to the order of the group.
   ∑

O

[χ(O)]^2 h (I-17)

where the sum is over the members (operations) of the group.

6. Two irreducible representations of a group are orthogonal to each other. This means
   that if you take the product of the characters of a given operation in the two repre-
   sentations and then sum all such products, they will add to zero. Ifiandjdenote
   two irreducible representations of a group,
   ∑

O

χi(O)χj(O) 0 (I-18)

where the sum is over all members of the group.

8. The number of times that an irreducible representation occurs in a reducible repre-
   sentation is given by

Number

1

h

∑

O

χi(O)χj(O) (I-19)

wherehis the order of the group and whereistands for an irreducible representation

andjstands for a reducible representation, and where the sum is over all members

of the group.

(^16) G. L. Meissler and D. A. Tarr,Inorganic Chemistry, Prentice Hall, Englewood Cliffs, NJ, 1991, p. 104.