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 $p$.