Irreducible representation over splitting field surjects to matrix ring

Statement

Suppose $G$ is a finite group and $k$ is a splitting field for $G$. Suppose $\rho :G\to GL(n,k)$ is an irreducible representation for $G$ over $k$. Then, the map extends uniquely by $k$-linearity to a $k$-linear map from the group ring to the matrix ring:

${\tilde {\rho }}:kG\to M(n,k)$

The claim is that ${\tilde {\rho }}$ is surjective.

Instead of requiring $k$ to be a splitting field, we can require only that $k$ have characteristic not dividing the order of $G$ and the representation $\rho$ be absolutely irreducible.

Related facts

• Central implies image under every irreducible representation is scalar
• Schur's lemma