# Explanation of Schur's Lemma Corollary

1. Jul 27, 2010

### 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?