# Selection Rules criteria using Group Theory

When we are applying Group Theory to see whether the matrix element $$\langle i|H'|f\rangle$$ vanishes, we look at how the matrix element transforms. It can be shown that the matrix elements transforms as follows $$\Gamma = \Gamma_i\otimes\Gamma_j\otimes\Gamma_f$$ where $$\Gamma_i$$ is the irreducible representation of the state $$i$$, $$\Gamma_j$$ is the reducible (can be irreducible) representation of the perturbed Hamiltonian and $$\Gamma_f$$ is the irreducible representation of the state $$f$$. It is said that the matrix element vanishes if $$\Gamma$$ does not contain the identity representation.

What is the intuition behind it? I have been told that we require the matrix element to transform as a scalar (essentially be invariant) under the symmetry operations which would mean that the matrix element corresponds to the identity representation. However, I don't find this answer satisfactory as we only require $$\Gamma$$ to contain the identity representation. $$\Gamma$$ can also contain different irreducible representations which means that it can also transform other than the identity representation. What am I missing ?

This matrix element $$\langle i\vert H'\vert f\rangle$$ is evaluated as a mapping from your Hilbert space to the field of complex numbers.