How do I determine a real matrix form for a group representation?

Question

Hello mathoverflow community,

I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$ induced from the additive character $\chi$ of $\mathbb{Z}_p$ given by $\chi(g)=e^{\frac{2\pi i g}{p}}$, I obtain a complex matrix form. That is for $(a,b)\in\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$, I find matrices $R(a,b)\in\mathbb{C}^{\mathbb{Z}_p^*\times\mathbb{Z}_p^*}$ with

$ (R(a,b))_{j,k}=\begin{cases} \chi(aj^{-1})&\text{if }k=b^{-1}j,\\ 0&\text{else.} \end{cases} $

In theory, there must be an equivalent representation with real matrix form (e.g. by Schur-Frobenius test). My aim is to determine a real symmetry-adapted basis and so I would like to compute this real matrix form. Is there anything known how to do that in practise or do you see any good way to do this basis transformation?

Best regards Aron

P.S: Notations: p is a prime number. $\mathbb{Z}_p$, resp. $\mathbb{Z}_p^*$ is the additive, resp. multiplicative, group of integers modulo $p$.

Answer 1

Here's one way: Note that your semidirect product has a normal subgroup $\mathbb{Z}_{p} \mathbb{Z}_{2}$ which is dihedral with $2p$ elements ( I assume the semidirect product you want to work with is the holomorph of $\mathbb{Z}_{p}$).

It is easy to see a real faithful representation (of degree $2$) of that dihedral group ( using a rotation through $\frac{2 \pi }{p}$ and a reflection). Now you can induce that representation to the whole holomorph to get a real absolutely irreducible faithful representation of degree $p-1$.

In fact, the $p-1$ dimensional representation may be realized over $\mathbb{Q}$, but the above answers your immediate question.