Reduce the following reducible representations using the great orthogonality theorem.
(a) In the
(b) In the
(c) In the
(d) In the

(a)
Interpretation:
The given reducible representations are to be reduced by using the great orthogonality theorem.
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is that the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 13.59E
The linear combination for given irreducible representation of
Explanation of Solution
The given reducible representation of
This reducible representation is reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Thus, the linear combination is
The linear combination for given irreducible representation of

(b)
Interpretation:
The given reducible representations are to be reduced by using the great orthogonality theorem.
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is that the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 13.59E
The linear combination for given irreducible representation of
Explanation of Solution
The given reducible representation of
This reducible representation is reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem of the irreducible representation of
and
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Thus, the linear combination is
The linear combination for given irreducible representation of

(c)
Interpretation:
The given reducible representations are to be reduced by using the great orthogonality theorem.
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is that the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 13.59E
The linear combination for given irreducible representation of
Explanation of Solution
The given reducible representation of
This reducible representation is reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Thus, the linear combination is
The linear combination for given irreducible representation of

(d)
Interpretation:
The given reducible representations are to be reduced by using the great orthogonality theorem.
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is that the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 13.59E
The linear combination for given irreducible representation of
Explanation of Solution
The given reducible representation of
This reducible representation reduced using great orthogonality theorem as shown below.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
The great orthogonality theorem of the irreducible representation of
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Thus, the linear combination is
The linear combination for given irreducible representation of
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Chapter 13 Solutions
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