Schur's lemma

Schur's lemma is a deceptively short statement that does enormous work: it says irreducible representations are extremely rigid. There are almost no G-equivariant maps between them — only the obvious zero map, unless two irreducibles are secretly the same, and the only equivariant maps from an irreducible to itself are the boring scalars. This rigidity is what makes characters separate representations and what diagonalizes everything in sight.

Statement: let V and W be irreducible representations of G and let f: V -> W be a G-equivariant map (an intertwiner, meaning f rho_V(g) = rho_W(g) f for all g). Then f is either zero or an isomorphism. In particular, if V and W are not isomorphic, the only intertwiner is 0. Moreover, if the ground field is algebraically closed, every intertwiner f: V -> V is a scalar multiple of the identity, because f has an eigenvalue lambda and f - lambda*I is a non-invertible intertwiner, hence zero.

The algebraically closed hypothesis in the second part matters: over R the rotation-by-theta representation of a cyclic group is irreducible, yet its endomorphism ring is the complex numbers C, not just R — there is a nonscalar intertwiner (rotation by 90 degrees). So 'endomorphisms are scalars' holds verbatim only over algebraically closed fields; in general the endomorphism ring is a division algebra over k.