Standard Model 🪩 Symmetries

complex numbers C are written as z = a + bi

The conjugate is z̃ = a - bi

U(1)

[ C ]

So thats 2 values in 1 cell. To get this as a real number matrix imagine a simple circle rotation / sin/cos

⎡  cos, -sin  ⎤
⎣  sin,  cos  ⎦

Thats akin to

⎡  0, -1 ⎤
⎣  1,  0 ⎦

And the reason for that is that this rotates a point [x, y] with matmul

x' =  cos(x)  +  sin(y)
y' = -sin(x)  +  cos(y)

and if you square [0, -1 | 1, 0 ] you get

⎡  0, -1  ⎤ × ⎡  0, -1  ⎤ = ⎡ -1,  0  ⎤
⎣  1,  0  ⎦   ⎣  1,  0  ⎦   ⎣  0, -1  ⎦

which is the the -1 identity matrix

I never encountered these until now (i mean technically i do since birth i guess)

That's now 8 values in 4 cells

⎡ C, C ⎤
⎣ C, C ⎦

These are used with quaternions like

1 = ⎡  1  0  ⎤
    ⎣  0  1  ⎦

i = ⎡  i  0  ⎤
    ⎣  0 -i  ⎦

j = ⎡  0  1  ⎤
    ⎣ -1  0  ⎦

k = ⎡  0  i  ⎤
    ⎣  i  0  ⎦

so every matrix is

A = ⎡  x  y  ⎤
    ⎣ -ỹ  x̃  ⎦