Representation Theory of Finite Groups - Morphisms Between Representations

Morphisms Between Representations

Given two representations ρ: G → GL(n,C) and τ: G → GL(m,C) a morphism between ρ and τ is a linear map T : Cn → Cm so that for all g in G we have the following commuting relation: T ° ρ(g) = τ(g) ° T.

According to Schur's lemma, a non-zero morphism between two irreducible complex representations is invertible, and moreover, is given in matrix form as a scalar multiple of the identity matrix.

This result holds as the complex numbers are algebraically closed. For a counterexample over the real numbers, consider the two dimensional irreducible real representation of the cyclic group C₄ = 〈x〉 given by:

Then the matrix defines an automorphism of ρ, which is clearly not a scalar multiple of the identity matrix.