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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 $N=3$, $r=18$.

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