For every atom that is not shifted under C₄ and σ symmetry operations, the characters are, respectively,

​Concept:

• Cn represents the axis of rotation symmetry of order n, and molecule doesn't change after rotation by \(\frac{360^0}{n}\)
• \(\sigma \) is the plane of symmetry. Any molecule possessing \(\sigma\)symmetry, either shows reflection or bisection when plane symmetry operation is applied. 
• The number of unshifted atoms gives the character of that symmetry which is being applied, in the irreducible representation.
• Character is calculated by adding the diagonal terms of matrix formed for that symmetry operation. 

• Cn represents the axis of rotation symmetry of order n, and molecule doesn't change after rotation by \(\frac{360^0}{n}\)
• \(\sigma \) is the plane of symmetry. Any molecule possessing \(\sigma\)symmetry, either shows reflection or bisection when plane symmetry operation is applied.

Explanation:

Character for \(C_4\) : 

Matrix for character of \(C_n\) operation is given by;

 \(\begin{vmatrix} cos\theta \;\;\;sin\theta \;\;0\\-sin\theta \;cos\theta \;0\\0\;\;\;\;\;\;\;0\;\;\;\;\;\;\;1 \end{vmatrix}\) 

This gives, \(\chi_{C_n}=2cos\theta +1 \)

for \(C_4\) symmetry, \(\theta \)= 90°

 \(\chi _{C_4}=2cos90 ^0+1 \)

\(\chi _{C_4}=1\)

Character for \(\sigma \): 

For \(\sigma \) symmetry, character = 1

Conclusion:

Therefore, the character for each unshifted atom in C₄ and \(\sigma \) in irreducible representation is 1 and 1 respectively.