Reference for representations of quaternion group

Question

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 $\mathbb{R}$.

1. One-dimensional representations that factor through $(\mathbb{Z}/2)^2$.

2. 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.

Answer 1

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

Answer 2

This example is elementary (close to an exercise), which does make it tempting just to write a couple of lines of explanation in a paper. But including such a "proof" without further comment tends to leave the impression that the ideas were just discovered. It's easy to refer to an older textbook such as I.M. Isaacs Character Theory of Finite Groups (now an AMS reprint): Exercise (2.4) combined with page 145 on the real case. The easy fact, usually just stated without comment, is that the complex linear characters of a finite group are those of the quotient by the derived group (here a Klein 4-group). But the real case needs an explicit reminder.

There's no magic way to deal with "well-known" facts (if they are indeed well-known), but here it's easy to give in to a referee request without inflating the paper. By the way, neither Serre nor Fulton-Harris pretends to be as comprehensive a reference for finite group characters as the books by Curtis-Reiner and Isaacs.

Answer 3

It seems easier simply to supply a proof of two lines. Th unique two dimensional complex representation $V$ is not a real representation, so the direct sum of $V$ and $V\simeq {\overline V}$ is defined over ${\mathbb R}$. By dimension count, you have the four characters (four one dimensional real representations), and this irreducible four dimensional one; so this gives the dimension of the group algebra over ${\mathbb R}$ (namely $8$).

Answer 4

Here is a stupid proof which, I think, would not elicit any further questions:

Let $\left[1\right]$, $\left[-1\right]$, $\left[i\right]$, $\left[-i\right]$, $\left[j\right]$, $\left[-j\right]$, $\left[k\right]$, $\left[-k\right]$ be the elements of $Q$ which correspond to $1$, $-1$, $i$, $-i$, $j$, $-j$, $k$, $-k$ under the canonical embedding $Q\subseteq \mathbb H^{\times}$. Then, the $\mathbb R$-linear map $\mathbb H\oplus \mathbb R\oplus \mathbb R\oplus \mathbb R \oplus \mathbb R$ which sends every $\left(a+bi+cj+dk,e,f,g,h\right)$ to

$\dfrac{\left[1\right]-\left[-1\right]}{2}\left(a+b\left[i\right]+c\left[j\right]+d\left[k\right]\right) $ $+ \dfrac{\left[1\right]+\left[-1\right]}{2}\left(\left(e+f+g+h\right)\left[1\right]+\left(e-f-g+h\right)\left[i\right]+\left(e+f-g-h\right)\left[j\right]+\left(e-f+g-h\right)\left[k\right]\right)$

(for all reals $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$) is easily seen to be an $\mathbb R$-algebra isomorphism. Now, the representation theory of a direct product of algebras is well-known, and so is the representation theory of fields.