### M Theory Lesson 272

Today at morning tea we talked, amongst other things, about neutrino symmetries.

Recall that the symmetry group $A_4$ can be used to describe the tribimaximal mixing matrix, $T$, for neutrinos. We have probably neglected to point out before that $A_4$ is secretly a $2$-group, where the $2$ refers to categorical $2$-arrows. That is, $A_4$ is the group $C_{3}$ acting on the group $C_{2} \times C_{2}$. There is another order 12 group, which we will call $G$, where $C_{2} \times C_{2}$ is replaced by the only other four element group, $C_{4}$.

This reminds us of the Fourier decomposition $T = F_{3} F_{2}$. The group $C_{2}$ is generated by the Pauli swap, $\sigma_{X}$, and $C_{3}$ may be generated by the basic three dimensional permutation $(231)$. Any discrete Fourier expansion is expressed as an element of the group algebra for one of these groups. Note that these groups are also generated by the circulant operators $R_{d}$, and in dimension two $F_{2}$ also gives $C_{2}$. Then $T$ may be defined directly in terms of $A_4$ generators as $T = R_{3}F_{2}$.

Now since $R_{2}$ does not generate $C_2$, but $R_{2}^{2}$ generates $C_{4}$, we can consider the alternative mixing matrix $S = R_{3} R_{2}$ as a $G$ (non local) version of the mixing matrix. This was the matrix that vaguely resembled a root of the CKM matrix.

Aside: A new paper by Harrison et al discusses the nearness of the CKM matrix to unitarity.

Recall that the symmetry group $A_4$ can be used to describe the tribimaximal mixing matrix, $T$, for neutrinos. We have probably neglected to point out before that $A_4$ is secretly a $2$-group, where the $2$ refers to categorical $2$-arrows. That is, $A_4$ is the group $C_{3}$ acting on the group $C_{2} \times C_{2}$. There is another order 12 group, which we will call $G$, where $C_{2} \times C_{2}$ is replaced by the only other four element group, $C_{4}$.

This reminds us of the Fourier decomposition $T = F_{3} F_{2}$. The group $C_{2}$ is generated by the Pauli swap, $\sigma_{X}$, and $C_{3}$ may be generated by the basic three dimensional permutation $(231)$. Any discrete Fourier expansion is expressed as an element of the group algebra for one of these groups. Note that these groups are also generated by the circulant operators $R_{d}$, and in dimension two $F_{2}$ also gives $C_{2}$. Then $T$ may be defined directly in terms of $A_4$ generators as $T = R_{3}F_{2}$.

Now since $R_{2}$ does not generate $C_2$, but $R_{2}^{2}$ generates $C_{4}$, we can consider the alternative mixing matrix $S = R_{3} R_{2}$ as a $G$ (non local) version of the mixing matrix. This was the matrix that vaguely resembled a root of the CKM matrix.

Aside: A new paper by Harrison et al discusses the nearness of the CKM matrix to unitarity.

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