Neutrinos Again III

Carl Brannen rightly pointed out that the matrices appearing in the tribimaximal mixing papers are in fact basically the same as those that characterise MUBs and the quantum Fourier transform for the prime 3. In fact, let where $(231)$ (oops, it should be $(312)$) denotes as usual the cyclic permutation in $S_{3}$ (sometimes drawn as ribbon diagrams) and $M$ is Carl's notation. Both $(231)$ and $3M$ cube to the identity. The democratic matrix is given by $D = \frac{1}{3}[1 + (231) + (231)^{2}]$, which can be thought of as a vector $(\frac{1}{3} , \frac{1}{3} , \frac{1}{3})$. Observe that the operator $S$ from the $A_{4}$ representation obeys the rules

$D \cdot 3S = 3D$
$2D - 3S = 3I$

where $I$ is the identity. I'm beginning to wonder if those poor experimenters are ever going to detect a $\theta_{13} > 0$. Note also that the (norm square) $2 \times 2$ form of the neutrino mass matrix, which was used by Harrison et al, is expressed as which utilises the 2-circulant $3 \times 3$ matrix that happens to square to the identity. In other words, this is a $3 \times 3$ representation of the Pauli spin Fourier polynomial. Thus tribimaximal mixing is expressed as a composition of mass Fourier and spin Fourier components.