To satisfy a referee, I need a reference for the following well-known fact (which is not hard to prove, but it seems silly to prove it in a paper). I can't find it in either Serre or Fulton-Harris's books on representation theory. Can anyone provide me a reference?
Let $Q$ be the $8$-element quaternion group, so we have a central extension
$$1 \longrightarrow \mathbb{Z}/2 \longrightarrow Q \longrightarrow (\mathbb{Z}/2)^2 \longrightarrow 1.$$
Then the following constitute a complete list of irreducible representations of $Q$ over $\mathbb{R}$.
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One-dimensional representations that factor through $(\mathbb{Z}/2)^2$.
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A four-dimensional representation $W$ obtained from the left action of $Q$ on the real quaternions (viewed as a $4$-dimensional real vector space).
I remark that the representation $W$ is irreducible but not absolutely irreducible — when we extend the field of scalars to $\mathbb{C}$, it breaks up into two copies of the (unique) two-dimensional irreducible complex representation.
Best Answer
Since $Q_8$ is an extraspecial group, if you want you can say that the result follows from Quillen's classification of their real representations in
Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups
Mathematische Annalen, 1971