There are two questions which are interrelated hence I want to mention them over here.
Given the irrep $\Gamma^{(3)}$ of group C3V, which is of 2 dim. It can be diagonalized further, into a simple diagonal matrix. We still say this set is irrep because a common matrix S does not diagonalize them simultaneously.
I want to ask if I am right on this one?
Now, if I am right on this one.
If, I create a reducible form of C3v group by
$\Gamma^{(2)}\bigoplus\Gamma^{(3)}$ then this set also cannot be diagonalized simultaneously, then why do not we call this representation as irreducible.
I would like to clear the confusion. I may be wrong somewhere, may in the calculation or understanding the concept of block diagonalization.
I have been reading from Physics textbooks and unfortunately they do not properly emphasize on this. Please consider my limited knowledge of advanced abstract mathematics while answering.