# Representation theory of $M_n(\mathbb{C})$ and $M_n(\mathbb{H})$ as associative algebras over $\mathbb{R}$

I want to ask whether or not the natural representation $$\rho_1 : M_n(\mathbb{C}) \rightarrow End_{\mathbb{R}}(\mathbb{R}^{2 n})$$ and $$\rho_2 : M_n(\mathbb{H}) \rightarrow End_{\mathbb{R}}(\mathbb{R}^{4 n})$$ are irreducible. If so, how to prove that? And are their irreducible representations are unique up to isomorphism? i.e. Are $$\rho_1$$ and $$\rho_2$$ the only irreducible representations of $$M_n(\mathbb{C})$$ and $$M_n(\mathbb{H})$$ respectively?

## 1 Answer

Let $$A$$ be a $$K$$-algebra and suppose the $$K$$-vector space $$V$$ is a simple $$M_n(A)$$-module. Define $$e_{ii}$$ to be the diagonal matrix with $$1$$ in the $$ii$$th entry and $$0$$s elsewhere. Then since $$1=e_{11}+\cdots+e_{nn}$$,

$$V=\bigoplus_{i=1}^n e_{ii}V$$

as $$A$$-modules (where $$A$$ is viewed as a subalgebra of $$M_n(A)$$ and we consider restriction of scalars). Then, we can use permutation matrices to establish $$e_{ii}V\cong e_{jj}V$$ as $$A$$-modules for different $$i,j$$. Call the isomorphism class of $$A$$-modules $$[M]$$. Then $$V\cong M^n\cong A^n\otimes_A M$$ as an $$M_n(A)$$-module. To be simple, $$M$$ must be simple. If $$A$$ is a division algebra, the only simple $$A$$-module is $$A$$ itself (as all $$A$$-modules are $$A^m$$ for some $$m$$, by linear algebra over a skew field).