# Dimension of the space of G-module homeomorphisms between direct sums of representations

I've been looking for a proof that if $$V$$ is an irreducible representation of a finite group $$G$$ $$\dim(\mathrm{Hom}(V,W)^G)$$ is equal to the multiplicity of $$V$$ in the decomposition of $$W$$, where $$\mathrm{Hom}(V,W)^G$$ is the invariant subspace under $$G$$ of $$\mathrm{Hom}(V,W)$$. Everything is clear except when everyone states one form or another of $$\mathrm{dim(Hom}(V,W_1\oplus W_2)^G)=\mathrm{dim(Hom(}V,W_1)^G)+\mathrm{dim(Hom(}V,W_2)^G)$$

this fact is not obvious to me, even though it might be under a layer of rust over my linear algebra.

I know that $$\dim(\mathrm{Hom}(V,W_1\oplus W_2))=\dim(\mathrm{Hom}(V,W_1))+ \dim(\mathrm{Hom}(V,W_2))$$ from $$\mathrm{Hom}(V,W)\cong V^*\otimes W$$, but how can I see that this statement translates to the restriction over the invariant subspace under $$G$$?

The idea is that $$\oplus$$ is also a direct sum as $$G$$-modules, and $$\hom (V_1,V_2)^G$$ is the same as $$\hom_G(V_1,V_2)$$, that is, morphisms of $$G$$-modules (also called morphisms of representations or intertwiners)
Therefore a morphism of $$G$$-modules $$V\to W\oplus Z$$ is the same as morphisms of $$G$$-modules $$V\to W, V\to Z$$
• Well the relation is that the space of $G$-morphisms is the space of invariants. Then as to why direct sum of $G$-modules obeys the same rules with respect to $G$-morphisms, that's just basic module theory : if you have any ring $R$ and $R$-modules $A,B,C$ then $\hom_R(A,B\oplus C) \simeq \hom_R(A,B)\oplus \hom_R(A,C)$. That's because for finitely many factors, direct sum and direct product coincide, then that isomorphism is just the statement of the universal property of a product – Max Jun 26 '19 at 12:39