### Neutrinos Again

A paper by Altarelli and Feruglio on tribimaximal mixing was mentioned in talks at Neutrino 08, so I thought I should take a look at it. They begin by describing the 3 dimensional representation of $A_{4}$ which has generators $S$ and $T$ satisfying $(ST)^{3} = S^{2} = 1$, just like the modular group, and also the relation $T^{3} = 1$. Then, letting $\omega$ be the cubed root of unity, one has for $T$ the matrix

1 0 0

0 $\omega^{2}$ 0

0 0 $\omega$

and for $3 S$ the circulant matrix

-1 2 2

2 -1 2

2 2 -1

The 12 elements of $A_{4}$ are given by all possible combinations of these generators. Gee, it already sounds a bit like M Theory. I can't imagine what they want with all the fairy fields and SUSY's mumbo jumbo, although in short shrift the tribimaximal mixing matrix appears.

1 0 0

0 $\omega^{2}$ 0

0 0 $\omega$

and for $3 S$ the circulant matrix

-1 2 2

2 -1 2

2 2 -1

The 12 elements of $A_{4}$ are given by all possible combinations of these generators. Gee, it already sounds a bit like M Theory. I can't imagine what they want with all the fairy fields and SUSY's mumbo jumbo, although in short shrift the tribimaximal mixing matrix appears.

## 5 Comments:

Hi Kea,

Feruglio is a colleague in Padova... I'll give a look at the paper!

Funny how one is blissfully oblivious at times of what colleagues working in the adjoining office do.

Cheers,

T.

... and thanks for the link ;')

Cheers,

T.

Cool, Tommaso! Check out his nice slides.

This reminds me of qutrit MUBs, i.e. see matrix M.

And your S, is closely related to 1+J+J^2.

Why yes, Carl, indeed. I'm wondering why I can't find any papers that explain this connection .... dammit, does that mean I have to write another paper?

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