Homomorphisms and Endomorphisms of Group Representations

Definition: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. A Group Representation Homomorphism (or simply Morphism) between these group representations is a linear map $\varphi : V_1 \to V_2$ that is $G$-equivariant, that is, for all $g \in G$ and for all $v \in V_1$ we have that $\varphi([\rho_1(g)](v)) = [\rho_2(g)](\varphi(v))$. The space of all homomorphisms of $(V_1, \rho_1)$ to $(V_2, \rho_2) $] is denoted by [[$ \mathrm{Hom}^G(V_1, V_2)$.

The following proposition tells us that if $(V_1, \rho_1)$ and $(V_2, \rho_2)$ are group representations of $G$ and if $\varphi \in \mathrm{Hom}^G(V_1, V_2)$ then $\ker \varphi$ is a subrepresentation of $V_1$ and $\mathrm{range} \varphi$ is a subrepresentation of $V_2$.

Proposition 1: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. If $\varphi \in \mathrm{Hom}^G(V_1, V_2)$ then $\ker \varphi$ is a subrepresentation of $V_1$ and $\mathrm{range} \varphi$ is a subrepresentation of $V_2$.

• Proof: Clearly $\ker \varphi$ is a subspace of $V_1$. Furthermore, for all $g \in G$ and all $v \in \ker \varphi$ we have that:

(1)

\begin{align} \quad \varphi([\rho_1(g)](v)) = [\rho_2(g)](\varphi(v)) = [\rho_2(g)](0) = 0 \end{align}

• So if $v \in \ker \varphi$ we see that $[\rho_1(g)](v) \in \ker \varphi$. So $\ker \varphi$ is $G$-invariant and thus $\ker \varphi$ is a subrepresentation of $V_1$.

• Similarly, clearly $\mathrm{range} \varphi$ is a subspace of $V_2$. Furthermore, for all $g \in G$ and all $w \in \mathrm{range} \varphi$ we have that there exists a $v \in V_1$ such tht $\varphi(v) = w$. So:

(2)

\begin{align} \quad [\rho_2(g)](w) = [\rho_2(g)](\varphi(v)) = \varphi([\rho_1(g)](v)) \end{align}

• So if $w \in \mathrm{range} (\varphi)$ we see that $\rho_2(g) \in \mathrm{range} \varphi$. So $\mathrm{range} \varphi$ is $G$-invarint and thus $\mathrm{range} \varphi$ is a subrepresentation of $V_2$. $\blacksquare$

Definition: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$. An Group Representation Endomorphism is a group representation homomorphism from $(V, \rho)$ to $(V, \rho)$. The space of all group endomorphisms on $(V, \rho)$ is denoted by $\mathrm{End}^G(V)$.