Intertwiners, the spaces ${\mathrm{Hom}}_G(V,W)$ and ${\mathrm{End}}_G(V)$, equivalent representations, and faithful representations

Let $\rho :G\to \mathrm{GL}(V)$ and $\sigma :G\to \mathrm{GL}(W)$ be representations of $G$ over the same field $k$.

A linear map $f:V\to W$ (Linear map between vector spaces over the same field) is an intertwiner, or $G$-equivariant map, when $f(\rho (g)v)=\sigma (g)f(v)(g\in G,v\in V)\mathrm{.}$ Equivalently, $f\rho (g)=\sigma (g)f(g\in G)\mathrm{.}$

Write ${\mathrm{Hom}}_G(V,W)$ for the set of all intertwiners. It is a subset of the vector space $\mathcal{L}(V,W)$ of all linear maps (The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication). When $V=W$, write ${\mathrm{End}}_G(V):={\mathrm{Hom}}_G(V,V)\mathrm{.}$

The two representations are equivalent if there is an invertible intertwiner $V\to W$.

A representation $\rho$ is faithful when $\rho (g)={\mathrm{id}}_V$ implies $g=e$.

• Faithfulness says that the action remembers every group element.

• The equality $f\rho (g)=\sigma (g)f$ is the representation-language form of a module homomorphism condition.