Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$.

Define the $p$ fourrier transform of those matrices $B_0, \dotsc, B_{p-1}$ where $$B_j=1/\sqrt{p}\sum_{i=0}^{p-1}\omega^{ij}A_i.$$

If the $B_j$ are such that $\forall j, B_j^p=I$, what can we say? Can we characterize the $A_i$ in some way?

For $p=2$, it is easy to prove that $A_0$, $A_1$ anticommute, which implies that the two are codiagonal with blocks of size 2, which are $\left(\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right)$ for $A_0$ and $\left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right)$ for $A_1$.

For a general $p$, I expect that it will be something like: the $A_i$ will be codiagonal with blocks of size $p$, which are $\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \omega & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \omega^{p-1} \\ \end{matrix}\right)$ for $A_0$, $\left(\begin{matrix} 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \\ \end{matrix}\right)$ for $A_1$, …, plus a possibility to exchange the roles of the matrices.