Symmetries should provide a "canonical form" for group elements
The canonical form should probably be (t, r)
where t
is a realspace vector in a cartesian coordinate frame, and r
is an integer orthogonal matrix in this same frame.
This form is needed, for example, for SiteFamily
s to know how to calculate the unitary representation of a group element for the on-site Hilbert spaces.
For example, a SiteFamily
that has a spin 1/2 onsite Hilbert space where the quantization axis is in the z direction will have a different representation than one where the quantization axis is along the x-axis. In the first case any rotations in the x-y plane will map to the 2x2 identity matrix, whereas in the second case they will map to some combination of Pauli matrices.
This form is also needed, for example, to distinguish between 2 cyclic groups that rotate around different axes in realspace.