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 $\mathbb{C}^{6}$. The tribimaximal mixing matrix for neutrinos is then expressed in the form

$\frac{1}{\sqrt{3}} (e^{i \theta_{1}}) + \frac{\sqrt{2}}{\sqrt{3}} (e^{i \theta_{2}})$

where $\theta_{1} = - \theta_{2} = \frac{\pi}{4}$ are phases in two complex directions, $((231),(213))$ and $((123),(321))$. 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.