Explanation of Schur's Lemma Corollary

[praharmitra]

Hi, I am studying Group Theory from Georgi's book. Here is an excerpt from Chap 1, Pg 14.

He proves the following theorem (theorem 1.4):

If $$D(g)A = AD(g) \forall g \in G$$ where D is a finite dimensional irreducible representation of G, then $$A \propto I$$

He then goes on to say, the following:

A consequence of Schur's lemma is that the form of the basis states of an irreducible representation are essentially unique. We can rewrite theorem 1.4 as the statement

$$A^{-1} D(g) A = D(g) \forall g \in G \Rightarrow A \propto I$$

for any irreducible representation D. This means once the form of D is fixed, there is no further freedom to make nontrivial similarity transformations on the states. The only unitary transformation you can make is to multiply all the states by the same phase factor.

I have not understood the inference he has made at all. Can you please explain?

 

[gtoguy97]

The inference that he is making is that once the form of the irreducible representation D is fixed, then any similarity transformation (such as those represented by matrix A) that can be applied to the states of the representation must be proportional to the identity matrix. This means that the only nontrivial unitary transformation that can be applied to the states is to multiply them all by the same phase factor.