Schur’s lemma and irreducibility

Let an intertwiner connect two nonzero irreducible representations over the same arbitrary field. Its kernel is invariant in the source and its image is invariant in the target. If the map is nonzero, irreducibility forces the kernel to be zero and the image to be the entire target. Hence the map is an isomorphism. This argument does not require the group to be finite or the characteristic to avoid its order; the conclusion follows directly from invariant subspaces.

For an irreducible representation, every nonzero equivariant endomorphism thus has an inverse, and that inverse is equivariant too. The endomorphism algebra is a division algebra. This does not yet identify it with the field of scalars. Over the complex numbers and in finite dimension, every endomorphism has an eigenvalue. Subtract that eigenvalue times the identity; the resulting intertwiner has nonzero kernel and therefore must vanish. This extra eigenvalue step is where algebraic closure and finite dimensionality enter.

Consider the real plane with a cyclic generator acting by rotation through 120 degrees. It has no invariant real line, because its rotation matrix has no real eigenvalue. The action is therefore irreducible over the real numbers. Nevertheless rotation through 90 degrees commutes with it and is not a real scalar matrix. The commutant consists of operators behaving like multiplication by complex numbers, demonstrating that the division-algebra conclusion can be stronger than scalar multiplication over the chosen base field.

After complexifying that same plane, the 120-degree rotation has the two distinct cube roots of unity as eigenvalues. The corresponding complex lines are invariant, so the complexified action is reducible. There is no contradiction: irreducibility is always relative to a field. An operator that has no real eigenvalue can acquire eigenvectors after enlarging the scalars. Whenever an irreducibility claim moves between real and complex geometry, test whether scalar extension changes the invariant subspaces before applying the complex form of Schur’s lemma.

A scalar commutant need not force irreducibility for an arbitrary representation of an arbitrary algebra. In our finite-group complex setting, however, Maschke supplies an invariant complement to every invariant subspace. A proper nonzero invariant subspace would then produce an equivariant projection that is neither zero nor identity and therefore cannot be scalar. Thus a scalar commutant is also an irreducibility criterion here. The logical dependency is important: the converse uses complete reducibility, not only Schur’s lemma.

More generally, if a complex semisimple representation contains inequivalent irreducibles with multiplicities, its commutant is a direct sum of full matrix algebras on the multiplicity spaces. Equivariant maps cannot mix inequivalent irreducibles, while repeated equivalent copies can mix freely. A multiplicity of two contributes four dimensions to the endomorphism algebra, rather than two. This observation anticipates the character norm, which will count precisely the sum of squared multiplicities without requiring an explicit decomposition basis.