Great orthogonality theorem

The great orthogonality theorem establishes the orthogonal relation between entries of matrices of irreducible representations of a group. Mathematically, it is expressed as


    1. refers to the matrix entry in -th row and -th column of the -th matrix of the -th irreducible representation.
    2. is the order of the group .
    3. is the dimension of the irreducible representation.

The proof of eq14 involves analysing two cases and then combining the results. Consider the matrix

where  is an  matrix associated with representation ,  is and  matrix associated with representation  and  is an arbitrary matrix with  rows and  columns.

Multiplying eq15 on the left by some matrix  associated with representation ,

Using the matrix identity ,



Proof the matrix identity .


and so .


According to the closure property of a group,  and  and thus

Case 1: .

Eq15 and eq16 becomes  and  respectively (we have changed the dummy index from  to in eq16). As every element of a representation can undergo a similarity transformation to a unitary matrix, we shall assume  and its inverse are unitary matrices. According to Schur’s first lemma, eq16 implies that  and eq15 becomes

is a similarity transformation, where  is similar to some other arbitrary matrix. Since the traces of similar matrices are the same,

where  is the identity matrix’s dimension, which is equal to the dimension of the matrix .

Substitute eq18 in eq17, we have , or in terms of matrix entries,

The RHS of the above equation is a finite summation of the product of three scalars and their order can be changed. So,

With ,

Since  is an arbitrary matrix, the above equation must satisfy any. This is only possible if

Since  is unitary,

Case 2: and  is not equivalent to .

According to Schur’s second lemma, eq15 becomes , or in terms of matrix entries,

Since  is an arbitrary matrix, the above equation must satisfy any . This is only possible if

Since  is unitary,

Combining eq19 and eq20, we have the expression for the great orthogonality theorem:

which can also be expressed as

because the RHS vanishes if , which renders the subscript  unnecessary for  .


What about the case where  and  is equivalent to ?


In this case,  can undergo a similarity transformation to become , which in turn can undergo a similarity transformation to become elements of a unitary representation. This implies that both  and  can be expressed as the same unitary representation because if  is similar to  and  is similar to , then  is similar to . In other words, we have  (where is  unitary), which is case 1.


Let’s rewrite eq20b as

Eq21 has the form of the inner product of two vectors, where  and  are  components of vectors  and  respectively in a -dimensional vector space. We can regard the components of the vectors as a function of three indices ,  and  such that the two vectors are orthogonal to each other when . This orthogonal relation of matrix entries is why eq21 is called the great orthogonality theorem.



Verify eq21 for

i) ,
ii)  with ,  and
iii) , with .




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