M Theory Lesson 237

The tribimaximal mixing matrix $T$, which is easily expressed in circulant form, is Fourier transformed to a block matrix which has exact entries

$\frac{2 \sqrt{2} - 1}{2 \sqrt{6}} \pm \frac{\sqrt{3}}{2 \sqrt{6}} i$
$\frac{\sqrt{3} + \sqrt{6}}{2 \sqrt{6}} \pm \frac{1 - \sqrt{2}}{2 \sqrt{6}} i$
$\frac{1 + \sqrt{2}}{\sqrt{6}} + \frac{1 - \sqrt{2}}{\sqrt{6}} i$

Numerically, this transform takes the form Observe that the norm of both the $GL(1)$ component and the $GL(2)$ component is 1, as in the case of the CKM transform. This reduces the parameterisation to four real parameters. The two dimensional component $A$ does not belong to $SU(2)$, but $A^{\dagger} A$ is a braided circulant, namely where the imaginary part should be exactly $2/3$, but my rounding was lazy. The real part is exactly $\sqrt{3}^{-1}$, and the determinant of $A^{\dagger} A$ is $2/9$ (a number that M theorists will recognise).