great orthogonality theorem
From Mathematics wiki
Theorem:
Consider two unitary, irreducible matrix representations and of a group . Then

Proof:
Define the matrix :
 ,
where is some unspecified matrix. Then
,  
,  
,  
, (relabeling )  
,  
. 
By the converse of Schur's lemma, either , or . If , then by Schur's lemma , where we find by taking the trace.
 ,
 . So
 ,
or, with index notation
 ,
but, since is arbitrary,
 ,
or, relabeling indices,
 .
Consequences
Choosing to be the identity representation, we get
 .
Also, the great orthogonality theorem implies an orthogonality between characters.
References
^{[1]} ^{[2]} ^{[3]} ^{[4]}
 ↑ M. Tinkham (1992). Group Theory and Quantum Mechanics. Dover Publications. ISBN 9780486432472.
 ↑ M. Hamermesh (1989). Group Theory and its Applications to Physical Problems. Dover publications. ISBN 9780486661810.
 ↑ W. Miller, Jr. (1972). Symmetry Groups and their Applications. Academic Press. ISBN 9780124974609.
 ↑ J. F. Cornwell (1997). Group Theory in Physics (Three volumes), Volume 1. Academic Press. ISBN 9780121898007.