I am curious about the irreducible representations $\rho: D\_{2p} \rightarrow GL\_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two questions, especially the first question:


Given a prime number $p$, what are the faithful irreducible representations of the dihedral group $D\_{2p}$? What are the irreducible representations of the dihedral group $D\_{2p}$?


For $p=5$, do we have irreducible faithful representations of dimensions less than $4$? How can we write down the $4\times4$ rational matrices corresponding to $D\_{10}$?



Faithful irreducible representations of the dihedral group $D_{2p}$, of dimension at most $p-1$
Over $\mathbb{Q}$, $\mathbb{Z}/p$ has two irreducible representations: the trivial one, and one of dimension $p-1$. Indeed, the group algebra is $\mathbb{Q}[t]/(t^p-1) = \mathbb{Q} \times \mathbb{Q}[t]/(1+t+\ldots+t^{p-1}) = \mathbb{Q} \times \mathbb{Q}(\zeta\_p)$ by the irreducibility of the cyclotomic polynomial.


A faithful representation of $D\_{2p}$ must have non-trivial restriction to $\mathbb{Z}/p$, so must contain the non-trivial representation of this group, so must have dimension $\geq p-1$.


To write an action by $(p-1 \times p-1)$-matrices concretely, take $V = \mathbb{Q}(\zeta\_p)$ with evident basis $1,\zeta\_p,\ldots,\zeta\_p^{p-2}$, let $\mathbb{Z}/p \subset D\_{2p}$ act by multiplication by $\zeta\_p$, and let $\mathbb{Z}/2 \subset D\_{2p}$ act by complex conjugation.


