Character table of quaternions $Q_8$

Question

I wanted to compute the character table of the quaternion group Q_8, and to this end I was reading this answer. For the degree 1 representation I used this question but for the degree 2 representation I don't understand how the hint from the first link works.

Answer 1

The hint is to map your quaternions to matrices with complex entries of the form $$\left(\begin{array}{cc}z&w\\-\bar{w}&\bar{z}\end{array}\right).$$

This will give you a representation, on $\mathbb{C}^2$, and by taking the traces of the matrices you get the characters.

It is no surprise that the identity $e\in Q_8$ maps to the identity matrix. So you are left with 3 quaternions, $i,j,k$ that you need to decide where to map to. Note the space of such matrices is 4 dimensional over the reals, parameterised by the real part of $z$, the imaginary part of $z$, the real part of $w$, and the imaginary part of $w$, from which you get a natural basis. You already sent $e$ to one of these basis vectors (corresponding to the real part of $z$). You could try mapping $i,j,k$ to the other three basis vectors.

Remember you need to check that the identities that hold between the elements of $Q_8$ also hold between the matrices you send them to.

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You also need to decide to do with $-1\in Q_8$ but there is a really obvious candidate for where to send it. Also it is fixed once you decide where to send $i,j$ or $k$.

Also you may be using the notation $Q_8=\langle x,y|y^2=x^2,xyx=y\rangle$. In this case by $i$ I mean $x$, by $j$ I mean $y$, by $k$ I mean $xy$ and by $-1$ I mean $y^2$.