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Let $\omega=e^{2i \pi/p}$. Weyl theorems give all representations of matrix algebra span by $A,B$ such that either

  1. $AB=\omega BA, A^p=B^p=I$,

or

  1. $(k,l)\mapsto A^kB^l$ is a irreducible projective representation of ${Z}_p\times Z_p$

(the irreducible representation is the algebra generated by the clock and shift operators https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices). Of course, if $\omega$ is a different root of unity, it still works.

I'd like to know if there is an alternative result for which the "choice of the root of unity" is not done. More precisly, I'd like to know what are the alternative characterization of the algebra of block diagonal matrices $A,B$ such that $A^p=B^p=I$ with block $a_i,b_i$ satisfying $a_i b_i= \omega_i b_ia_i$, where $\omega_i$ is a $p$-root of $1$. I have a first characterization:

Let $T=A^\dagger B^\dagger AB$. We have $A^p=B^p=T^p=I$.

Does it corresponds to some theory, are they alternative characterization somewhere, does it reminds someone of something?

Thanks!

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  • $\begingroup$ In my case, I ave a familly $A_0, ... A_{p-1}$ of matrices such that $A_k^p=I, \hat{A}_l^p=\mu I$, where the $\hat{A}_l$ are the Fourrier transforms of the $A_k$ and $\mu$ a phase (it is $(1-\omega^2)/\sqrt{3}$ for $p=3$, $1$ for $p=5,13$, $-i$ for $p=7,11$,...). It implies $\forall l, \forall t+s=p, \sum_{i_1+...+i_t=-l}A_{i_1}...A_{i_t}=\frac{\mu}{n^{(s-t)/2}}\sum_{j_1+...+j_s=l}A_{j_1}^\dagger...A_{j_s}^\dagger$. For $p =3$, I found how to do (the anticommutator $\{A_0,A_1\}$ is hermitian hence $I+T+T^\dagger=0$) but I cannot find a way to generalize this... $\endgroup$
    – MarcO
    Commented Apr 23, 2019 at 15:13

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