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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?

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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).

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