Recall that Ma's finite group $A_4$ is a subgroup of the permutation group $S_4$ on four letters. Thus it labels $12$ of the $24$ vertices of the permutohedron polytope in three dimensions, and contains the Klein four group (using $2143, 3412$ and $4321$). The reduction of the permutohedron to the associahedron, which itself exhibits triality, suggests a selection of $7$ vertices out of the $14$, which happens to be the number of points (or lines) in the Fano plane.

The neutrino literature tends to associate these finite groups directly with larger continuous symmetry groups, but this really misses the fundamental nature of arithmetic structure in emergent geometry. When the neutrino set is doubled to include mirror neutrinos, we can presumably characterise mixing with the full set of vertices on operad polytopes. Similarly, Fourier circulants can arise from a Fano star, or double tetractys. Since the tetractys indexes a much larger (secondary) polytope, based on the central point, the polytopes provide more information than the basic finite groups.

6 years ago

Note that Ma's work relates $A_4$ and $S_4$ directly to the important question of the $\theta_{13}$ deviation from exact tribimaximal mixing. With $S_4$ supersymmetry, we can better analyse the $\theta_{13}$ behaviour, which is essentially non zero due to the existence of mirror neutrinos.

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