Schur's lemma

The lemma of Schur, named after Isay Schur, describes the homomorphisms between simple modules. It states that every such homomorphism except the Nullhomomorphismus is an isomorphism.

The lemma of Schur in the module- theoretic version is ( was a ring with 1):

There are simple links modules. Then:

In the representation theoretic version of Schur 's lemma is ( be a finite group, a body ):

There are irreducible representations of. Then:

The second statement is also true in the reverse, so that a precisely then is a skew field if the representation is irreducible.

Because of the connection between representations of KG- modules over and both versions say the same.

Special case: matrix representations

Here, the evidence reduced to elementary linear algebra. There are invertible matrices, invertible matrices, and it is a matrix. True of the matrix products

Then is the core of an invariant subspace for the representation, because follows. Because of the irreducibility of may just be the zero vector space or the whole vector space. In the first case is inverted and provides a similarity between the representation matrices and transformation. In the second case is the zero matrix.

For practical purposes ( tabulation ), the matrices of an irreducible representation occasionally be standardized. For example, serve in the rotation group, the common eigenvectors of rotations about a selected axis as a standard basis. In such cases, the matrices of irreducible representations and either or inequivalent identical. Thus, the following addition to the Schur's lemma is relevant:

From all follows, that is, is a complex multiple of the unit matrix.

Proof: Let a ( complex ) eigenvalue of, and be a corresponding eigenvector. With the assumed equation also applies

Therefore, the core of an invariant subspace of the representation and may be due to irreducibility only the null space or the whole room. Since the eigenvector belongs to the core, leaving only the second option. So true.