I have been wondering how you can get the irredcible representations of the Dihedral group $\mathbb{D}_{8}$ of order 8 from its Character table. $\mathbb{D}_{8}= \left \langle a,x : a^4=x^2=e, \, xax=a^{-1} \right \rangle$

I have found a website which states the irreducible representations which I am trying to find.

Now the first 4 one-dimensional representations are easy. They can just be read off from the character table. But I don't know how to find the matrices for the 5th irreducible representation (of dimension 2). The 2 dimensional matrices for the 5th irreducible representation are:

For $a^l$

$\begin{pmatrix} \cos \frac{2l\pi}{4} & -\sin \frac{2l\pi}{4} \\ \sin \frac{2l\pi}{4} & \cos \frac{2l\pi}{4} \end{pmatrix}$

For $x$

$\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$

And for $a^l x$

$\begin{pmatrix} \cos \frac{2l\pi}{4} & \sin \frac{2l\pi}{4} \\ \sin \frac{2l\pi}{4} & -\cos \frac{2l\pi}{4} \end{pmatrix}$

My question is, how can I get these matrices by using the information in the character table? I am very new to group theory so this may be something which invoves a lot of work. I am interested to see how you would approach this. Also, is there a general method to adopt for doing this for higher orders, such as $\mathbb{D}_{12}$ and $\mathbb{D}_{16}$?

  • $\begingroup$ Perhaps it should be mentioned that the general problem of getting a representation from a character appears to be far from easy $\endgroup$ – Andreas Caranti Mar 12 '16 at 16:48
  • $\begingroup$ I feared as much. I may instead change my question to: how can i find the character table for $D_{8}$. Is there a methodical way of approaching this? It is easy to deduce that $D_{8}$ has 4 1 dimensional irreducible representations and one 2 dimensional representation. But determining the character of that 2D irreducible representation is beyond me. $\endgroup$ – user42418 Mar 12 '16 at 16:54
  • 1
    $\begingroup$ If you know about induced characters, then it is not hard to see that the $2$-dimensional character is the induction of a $1$-dimensional faithful character of the cyclic subgroup of order $4$, and that enables you to write calculate the corresponding representation. $\endgroup$ – Derek Holt Mar 12 '16 at 17:07
  • $\begingroup$ Unfortunately I haven't got any knowledge when it comes to induced characters. I think I'm first going to try to understand how to construct the character table from first principles. Could anybody offer me assistence with this? $\endgroup$ – user42418 Mar 12 '16 at 17:21
  • $\begingroup$ In this example, you have all of the characters except 1. In that case you can immediately calculate the final character from the orthogonality relations: two distinct columns of the table are mutually orthogonal. $\endgroup$ – Derek Holt Mar 12 '16 at 18:29

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