# Character table of quaternions $Q_8$

Mathematics Asked by roi_saumon on August 5, 2020

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.

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 $$ein Q_8$$ 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.

You also need to decide to do with $$-1in Q_8$$ 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 $$Q_8=langle x,y|y^2=x^2,xyx=yrangle$$. In this case by $$i$$ I mean $$x$$, by $$j$$ I mean $$y$$, by $$k$$ I mean $$xy$$ and by $$-1$$ I mean $$y^2$$.

Correct answer by tkf on August 5, 2020