Properties of the characters

(i) The character system is the same for all equivalent representations. To prove this, we need to show that TrM⁰¼ Tr S M S
¹¼ Tr M, and to prove this result we need to show first that TrAB ¼ Tr BA:

Tr AB ¼P

Equation (3) shows that the character system is invariant under a similarity transformation and therefore is the same for all equivalent representations. If for some S2 G, S R S
¹¼ T, then R and T are in the same class in G. And since the MRs obey the same multiplication table as the group elements, it follows that all members of the same class have the same character. This holds too for a direct sum of IRs.

Example 4.4-1 From Table4.1the characters of two representations of C3vare C_3v E C^þ₃ C<sub>3

d
e
f</sub>

₁ 1 1 1 1 1 1

₃ 2
1
1 0 0 0

(ii) The sum of the squares of the characters is equal to the order of the group. In eq.

(4.3.1), set q¼ p, s ¼ r, and sum over p, r, to yield

Equation (5) provides a simple test as to whether or not a representation is reducible.

Example 4.4-2 Is the 2-D representation
3of C3vreducible?

ð
3Þ ¼ f2
1
1 0 0 0g, P

T

j3ðT Þj²¼ 4 þ 1 þ 1 þ 0 þ 0 þ 0 ¼ 6 ¼ g,

so it is irreducible. The 3 3 representation in Table4.1is clearly reducible because of its block-diagonal structure, and, as expected,

4.4 The characters of a representation 75

P

T

jðT Þj²¼ 3²þ 2ð0Þ²þ 3ð1Þ²¼ 12 66¼ g:

Generally, we would take advantage of the fact that all members of the same class have the same character and so perform the sums in eqs. (4), (5), and (6) over classes rather than over group elements.

(iii) First orthogonality theorem for the characters. Performing the sum over classes

(4) P^Nc

where Ncis the number of classes and ckis the number of elements in the kth class, ck

Equation (7) states that the vectors with components ffiffiffiffiffiffiffiffiffiffi ck=g

p ⁱðckÞ, ffiffiffiffiffiffiffiffiffiffi ck=g

p ^jðckÞ are orthonormal. If we set up a table of characters in which the columns are labeled by the elements in that class and the rows by the representations – the so-called character table of the group (see Table4.2) – then we see that eq. (7) states that the rows of the character table are orthonormal. The normalization factors ffiffiffiffiffiffiffiffiffiffi

ck=g

p are omitted from the character table (see Table4.2) so that when checking for orthogonality or normalization we use eq. (7) in the form

g
¹P^Nc

k¼1

ck ⁱðckÞ^^jðckÞ ¼ ij: (8)

It is customary to include ckin the column headings along with the symbol for the elements in ck(e.g. 3
vin Table4.3). Since E is always in a class by itself, E¼ c1is placed first in the list of classes and c1¼ 1 is omitted. The first representation is always the totally symmetric representation
1.

Example 4.4-3 Using the partial character table for C3v in Table 4.3, show that the character systems {1} and {3} satisfy the orthonormality condition for the rows.

g
¹P

k

ck j1ðckÞj²¼ ð1=6Þ½1ð1Þ²þ 2ð1Þ²þ 3ð1Þ² ¼ 1;

Table 4.2. General form of the character table for a group G.

gkis a symbol for the type of element in the class ck( e.g. C2,

_v); ckis the number of elements in the kth class; g1is E, c1is 1, and
¹is the totally symmetric representation.

G

g
¹P

In how many ways can these vectors be chosen? We may choose the character ⁱ(ck) from any of the NrIRs. Therefore the number of mutually orthogonal vectors is the number of IRs, Nrand this must be Ncthe dimension of the space. In fact, we shall see shortly that the number of IRs is equal to the number of classes.

(iv) Second orthogonality theorem for the characters. Set up a matrixQ and its adjoint Q^y in which the elements of Q are the characters as in Table 4.2 but now including normalization factors, so that typical elements are

Q_ik¼ ffiffiffiffiffiffiffiffiffiffi

Equation (13) describes the orthogonality of the columns of the character table. It states that vectors with components ffiffiffiffiffiffiffiffiffiffi

ck=g

p ⁱðck) in an Nr-dimensional space are orthonormal.

Since these vectors may be chosen in Ncways (one from each of the Ncclasses),

(13) N_c N_r: (14)

Table 4.3. Partial character table for C3vobtained from the matrices of the IRs
1and
3in Table4.1.

4.4 The characters of a representation 77

But in eq. (7) the vectors with components ffiffiffiffiffiffiffiffiffiffi ck=g

p ⁱðckÞ may be chosen in Nrways (one from each of Nrrepresentations), and so

(7) N_r Nc: (15)

(14), (15) N_r¼ Nc: (16)

The number of representations Nris equal to Nc, the number of classes. In a more practical form for testing orthogonality

(13) P^Nr

i¼1

ⁱðckÞ^ⁱðc_lÞ ¼ ðg=ckÞkl: (17)

These orthogonality relations in eqs.(8) and (17), and also eq.(16), are very useful in setting up character tables.

Example 4.4-4 In C_3vthere are three classes and therefore three IRs. We have established that
1and
3are both IRs, and, usingP

i

l²_i ¼ g, we find 1 þ l²₂þ 4 ¼ 6, so that l2¼ 1. The character table for C3vis therefore as given in Table4.4(a).

From the orthogonality of the rows,

1ð1Þð1Þ þ 2ð1Þ2ðC3Þ þ 3ð1Þ2ð
Þ ¼ 0,

.Exercise 4.4-1 Check the orthogonality of the columns in the character table for C_3vwhich was completed in Example4.4-4.

(v) Reduction of a representation. For
to be a reducible representation, it must be equivalent to a representation in which each matrix
(T ) of T has the same block-diagonal structure. Suppose that the jth IR occurs c^jtimes in
; then

ðT Þ ¼P

jc^j_jðT Þ: (18)

Multiplying by i(T)^*and summing over T yields

(18), (4) P

Normally we would choose to do the sum over classes rather than over group elements.

Equation (20) is an extremely useful relation, and is used frequently in many practical applications of group theory.

(vi) The celebrated theorem. The number of times the ith IR occurs in a certain reducible representation called the regular representation
^r is equal to the dimension of the representation, li. To set up the matrices of
^rarrange the columns of the multiplication table so that only E appears on the diagonal. Then
^r(T ) is obtained by replacing T by 1 and every other element by zero (Jansen and Boon (1967)).

Example 4.4-5 Find the regular representation for the group C₃. C3¼ fE C^þ₃ C₃
g.

Interchanging the second and third columns of Table4.4(b) gives Table4.4(c).

Therefore, the matrices of the regular representation are

^rðEÞ
^rðC₃^þÞ
^rðC
₃Þ

The group C3is Abelian and has three classes; there are therefore three IRs and each IR occurs once in
^r. (But note that the matrices of
^rare not block-diagonal.)

Table 4.4(a) Character table for C3v.

C

4.4 The characters of a representation 79