The reducible representation, Γ, in the table is equal to the following superposition of the irreducible representations of C_2v point group.

C2v	E	C2	σv
A1	1	1	1	1
A2	1	1	−1	−1
B1	1	−1	1	−1
B2	1	−1	−1	1
Γ	8	−2	−6	4

Concept:

• The sum of the squares of the dimensions of all the irreducible representations is equal to the order of the group.

∑di2 = h, where h = order of the group, and d = dimension

Explanation:-

• The irreducible representations can be obtained from reducible representations from the formula,

where h is the order of the group,

Xi is the character of the reducible representation,

Yi is the character of the irreducible representation,

Zi is the coefficient of the symmetry element.

• The characters of a reducible representation ΓR under the C2v point group are presented below:

• The order of the group ​is,

= 1² + 1² + 1² + 1²

= 4

• The correct coefficients of the irreducible representations A₁ will be,

= 1

• The correct coefficients of the irreducible representations A₂ will be,

​

= 2

• The correct coefficients of the irreducible representations B1 will be,

​

= 0

• The correct coefficients of the irreducible representations B2 will be,

​

= 2
Thus, the irreducible representations of the C2v point group is

​A1 ⊕ 2A2 ⊕ 0B1 ⊕ 5B2

= A1 ⊕ 2A2 ⊕ 5B2

Conclusion:-

Hence, A1 ⊕ 2A2 ⊕ 5B2 is correct.