Homomorphism of linear representations

Definition

Abstract formulation

Suppose $G$ is a group, $V_1,V_2$ are vector spaces over a field $k$, and ${\rho }_1:G\to GL(V_1)$, ${\rho }_2:G\to GL(V_2)$ are linear representations of $G$. A homomorphism of representations from ${\rho }_1$ to ${\rho }_2$ is a $k$-linear map $f:V_1\to V_2$ such that, for all $g\in G$:

$f\circ {\rho }_1(g)={\rho }_2(g)\circ f$

Matrix formulation

Suppose $G$ is a group, $k$ is a field, and ${\rho }_1:G\to GL(m,k)$ and ${\rho }_2:G\to GL(n,k)$ are representations of $G$ over $k$. A homomorphism from ${\rho }_1$ to ${\rho }_2$ is a $n\times m$ matrix $F$ with the property that for all $g\in G$:

$F{\rho }_1(g)={\rho }_2(g)F$

where the multiplication on both sides is matrix multiplication.