Schur’s second lemma

Schur’s second lemma describes the restrictions on a matrix that commutes with elements of two distinct irreducible representations, which may have different dimensions.

Consider an arbitrary matrix  and two irreducible representations of a group,  of dimension  and of dimension , such that

where .

Taking the conjugate transpose of eq10 and using the matrix identity , we have . As every element of a representation of a group can be expressed as a unitary matrix via a similarity transformation without any loss of generality, and as ,

Since the inverse property of a group states that  and , we can express eq10 as

Multiplying eq11 on the left by  and using eq12, we have  and therefore , which implies that  commutes with all elements of an irreducible representation of . With reference to Schur’s first lemma,

where  is a constant and  is the identity matrix.

If we multiply eq11 on the right by  and repeat the steps above, we have

Let’s consider the following cases for eq13:

Case 1:

Let the -th entry of  be . If , we can rewrite eq13 in terms of matrix entries: . If , we have , which implies that  is the zero matrix because  for all .

Combining eq13 and eq14, we have  or . This implies that  exists if . We can therefore rewrite eq10 as , which is a similarity transformation if .

Case 2:

If , the arbitrary matrix (denoted by ) is an  matrix with reference to eq10. Suppose ; we have:

If we enlarge  to form an  matrix  with the additional elements equal to zero, we have

Due to the zeroes,  . Taking the determinants,

Using the determinant identities ,  and  , we have

Since one of the columns of  is zero, . So, , which implies that  must be a zero matrix according to the results of case 1.

Finally, we can summarise Schur’s second lemma as follows:

Given an arbitrary matrix  and two irreducible representations,  of dimension  and  of dimension , where  , then

    1. if , either  or the representations are related by a similarity transformation, i.e. equivalent representations.
    2. If , .


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