# Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity

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.

• The hypothesis might have to change from $\forall j, B_j^p=I$ to $\forall j, B_j^p=e^{i...}I$ – MarcO Apr 12 '19 at 11:39