The following is the character table for the C_2v point group

There are two functions f₁ and f₂ belonging to A₂ and B₁ representations, respectively. The correct option for the product of two functions f₁ and f₂ and the integral ∫f₁​f₂dτ is

​

Correct option is B

​The elements of the matrices which constitute the irreducible representations of a group follow ‘the great orthogonality theorem’ which is stated as follows:

​Where

i. The summation R represents the summation of various operations in a group.

ii. 

represents the matrix element R_mn of the matrix R representing Rth symmetry operation of the ith representation of a group.

iii. 

iv. h is the order of a group which is equal to the number of symmetry operations in a group.

v. I_i and I_j are the dimensions of the ith and jth representations (which respectively are equal to the order of the corresponding matrices) of a group.

vi. 

This implies that the sum of the products of the corresponding elements of matrices of the various symmetry operations belonging to two different irreducible representations is zero.

The product of the two functions A₂ and B₁ are 

The product belongs to B₂ representation.