Classification of homomorphisms of matrix algebras

Question

Is there a classification of homomorphisms of matrix algebras $$Mat_{k\times k}(\mathbb{C})\to Mat_{n\times n}(\mathbb{C})$$ say up to conjugacy?

Answer 1

Yes. I assume you want them to be $\mathbb{C}$-linear. If $F$ is any field then the matrix ring $M_k(F)$ is semisimple and $F^k$ is the unique simple module, meaning every module is a direct sum of copies of $F^k$. It follows that a homomorphism $M_k(F) \to M_n(F)$ of $F$-algebras (equivalently, an $n$-dimensional module over $M_k(F)$) exists iff $k \mid n$ and is unique up to conjugacy, and up to conjugacy it is given by diagonal blocks

$$X \mapsto \left[ \begin{array}{cccc} X & 0 & \cdots & 0 \\ 0 & X & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & X \end{array} \right].$$