M Theory Lesson 212

The $3\times 3$ democratic matrix is useful in many ways. Here we see it acts as a cyclic shift operator for three 1-circulants that additively act like modular 3 arithmetic. But observe that $(2\cdot \text{id}{)}^2=\text{id}$ and, similarly, the square of $\left(312\right)+\left(231\right)$ is twice $\left(312\right)+\left(231\right)$. That is, we have the relations

$A^2=A$
$B^2=2A$
$C^2=\frac{1}{2}C=\text{id}$
$A+B+C=0$

Exponentiating the third relation yields the multiplicative honeycomb rule $A\cdot B\cdot C=1$. However, using ordinary matrix multiplication one obtains $ABC=2\cdot \text{id}$, so the correct normalisation factor for all four matrices is the reciprocal of the cubed root of 2, namely 1.25992. Alternatively, since 2 is really $-1$, $C^2=C$ and $B^2=-A$ and the correct normalisation is a cubed root of $-1$, that is a 6th root of unity.