# Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $$N$$, let $$\omega=e^{2i\pi/N}$$ and $$A$$, $$B$$ be the clock and shift operators:

$$A=\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \omega & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \omega^{p-1} \\ \end{matrix}\right)$$ and $$B=\left(\begin{matrix} 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & \cdots & 0 & 0 \\ \end{matrix}\right)$$ . We have

1. $$AB=\omega BA, A^N=B^N=I$$
2. $$(k,l)\mapsto A^kB^l$$ is a irreducible projective representation of $${Z}_N\times Z_N$$

Weyl theorems says that converesly, if two matrices $$U,V$$ satisfy either 1. or 2., then they must be unitarely equivalent to $$A,B$$.

I think I have a third characterization but I cannot find a way to prove it.

Assume that we have $$N$$ matrices $$U_1, U_2, ..., U_N$$ such that:

• $$U_k^N=I$$ for $$k=1...N$$
• defining $$V_l=\frac{e^{i\pi/r}}{\sqrt{N}}\sum_{k=1}^N \omega^{kl}U_k,$$ for an appropriate $$r$$, we find $$V_l^N=I$$ for all $$l=1...N$$.

Then $$U_1,U_2$$ are unitarely equivalent to $$A,B$$.

PS: for $$N=3$$, $$r=18$$.