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I am wondering how to write rotations of binary icosahedral group as matrices. For example, the identity element of this group corresponds:

$R_1 = 1 + 0i + 0j + 0k$

since this group corresponds to spin3, it is a 3*3 identity matrix.

$R_2 = 0 + 1i + 0j + 0k$

where i,j,k can be thought as pauli_X, pauli_Y and pauli_Z matrices R2 corresponds $pi$ rotation about an edge aligned with an axis. How to write it as a matrix?

Same with R3:

$R_3 = 1/2 + 1/2i + 1/2j + 1/2k$

which is a $2pi/3$ rotation about one of the faces centered in an octant.

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    $\begingroup$ You are the action of the quaternion group as rotation on the sphere? Of you Check how quaternions are represented with matrices, you might get what you want $\endgroup$ Commented Jun 30, 2023 at 0:37
  • $\begingroup$ got it!! Thank you $\endgroup$
    – j.doe
    Commented Jun 30, 2023 at 0:59
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    $\begingroup$ Also the you are comes from my autocorrect, the if too… my French keyboard messes with me $\endgroup$ Commented Jun 30, 2023 at 1:32

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