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Let $G$ be a group and say $V$ is an irreducible representation over $\mathbb{R}$. Then $End_G(V) = End_{\mathbb{R}[G]}(V)$ must be a division algebra, since $V$ is a simple $\mathbb{R}[G]$-module. But division algebras over $\mathbb{R}$ are just $\mathbb{R},\mathbb{C},\mathbb{H}$. Apparently, the following is true:

  • $End(V)\cong\mathbb{R} \Leftrightarrow V\otimes_\mathbb{R}\mathbb{C}$ is an irreducible complex representation.
  • $End(V)\cong\mathbb{C} \Leftrightarrow V\otimes_\mathbb{R}\mathbb{C}$ is a direct sum of two non-isomorphic complex representations.
  • $End(V)\cong\mathbb{H} \Leftrightarrow V\otimes_\mathbb{R}\mathbb{C}$ is a direct sum of two isomorphic complex representations.

Could someone explain why these are true, or provide further literature on irreducible real representations? Thanks!

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    $\begingroup$ This is discussed in section 13.2 of Serre's Linear Representations of Finite Groups. $\endgroup$
    – Angina Seng
    Commented Sep 12, 2018 at 17:10

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