Neutrinos Again VII

Recall that Carl's magic form for the experimentally verified tribimaximal mixing involves a sum of a 1-circulant and a 2-circulant. Such objects naturally live in a group algebra for the permutation group $S_3$. That is, let the 6 elements of $S_3$ (three 1-circulants and three 2-circulants) represent unit basis vectors for a six dimensional vector space, nominally ${ℂ}^6$. The tribimaximal mixing matrix for neutrinos is then expressed in the form

$\frac{1}{\sqrt{3}}\left(e^{i{\theta }_1}\right)+\frac{\sqrt{2}}{\sqrt{3}}\left(e^{i{\theta }_2}\right)$

where ${\theta }_1=-{\theta }_2=\frac{\pi }{4}$ are phases in two complex directions, $\left(\right(231),(213\left)\right)$ and $\left(\right(123),(321\left)\right)$. Given the simplicity of the coefficients, a restriction of the number field would be feasible here. M theorists will also recognise the dimension of twistor space.