# Questions

## On $(A_2,A_6)$

### sol 1

$$\begin{bmatrix} z_0^{(1)} & z_0^{(2)} \\ z_1^{(1)} & z_1^{(2)} \\ z_2^{(1)} & z_2^{(2)} \\ z_3^{(1)} & z_3^{(2)} \\ z_4^{(1)} & z_4^{(2)} \\ z_5^{(1)} & z_5^{(2)} \\ z_6^{(1)} & z_6^{(2)} \\ z_7^{(1)} & z_7^{(2)}  \end{bmatrix} = \begin{bmatrix} 1. & 1. \\ 2.24698 & 2.24698 \\ 2.80194 & 2.80194 \\ 2.24698 & 2.24698 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1.  \end{bmatrix}$$ Here $2.24698\cdots$ is a solution of $x^3-2 x^2-x+1=0$.

This solution can be obtained from $g\in SU(3)$ given by $$g=\left( \begin{array}{ccc} e^{\frac{2 i \pi }{7}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{-\frac{2 i \pi }{7}} \\  \end{array} \right)$$

### sol 2

$$\begin{bmatrix} z_0^{(1)} & z_0^{(2)} \\ z_1^{(1)} & z_1^{(2)} \\ z_2^{(1)} & z_2^{(2)} \\ z_3^{(1)} & z_3^{(2)} \\ z_4^{(1)} & z_4^{(2)} \\ z_5^{(1)} & z_5^{(2)} \\ z_6^{(1)} & z_6^{(2)} \\ z_7^{(1)} & z_7^{(2)}  \end{bmatrix} = \begin{bmatrix} 1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1.  \end{bmatrix}$$ where $1.61803\cdots$ is the Golden ratio.

This solution can be obtained from $g\in SU(3)$ given by $$g=\left( \begin{array}{ccc} e^{\frac{2 i \pi }{5}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{-\frac{2 i \pi }{5}} \\  \end{array} \right)$$