Question

In the groups specified, determine the direct product representations and reduce them when possible. Remember that the characters belonging to each operation in the direct product of two representations is just the product of the input characters.

a) A2 x B1 in C2v

b) E x E in C4v

c) E x E in C3v

Answer 1

a) refer character tables of point group C2V

A2*B1 : 1        -1              -1             1

this is a reducible representation.To reduce it you need to apply the following reduction formula-

N=1/h r^x i^x n^x

N=number of times a particular irreducible representation appear in the representaion being reduced,

h=total number of operations in the group or order of the group)(=sum of squares of the chracters of any irreducible representation like A1,A2,B1,B2)

for A1,1^2+1^2+1^2+1^2=4=h

r^x =reducible representation for a particular class of operation,x

i^x =irreducible representation for a particular class of operation,x

n^x =number of operations in the class

the summation is taken over all classes

N(A1)=1/4[1*1 +    1* -1 +1* -1 +1* 1]=0

N(A2)=1/4[1*1 +    1* -1 +-1* -1 +-1* 1]=0

N(B1)=1/4[1*1 +    -1* -1 +1* -1 +-1* 1]=0

N(B2)=1/4[1*1 +    -1* -1 +-1* -1 +1* 1]=1

It has only one irreducible representation B2