Storing method eigenfunctions of `AffineTransform` strips eigenvalue information.

[Alphaharrius]

Description

The current method to store the eigenfunction for a given AffineTransform is done via Dict{Tuple, BasisFunction}, for which the tuple stores the combination of function order and the rationalized eigenvalue. Yet we know that rationalization induces information loss since for a fixed denominator restricts how to look up eigenvalue from start, and it cannot be reversed.

Suggested fix

Store the eigenfunctions in the form of generator of Pair{Complex, BasisFunction} and since the order information is stored within the BasisFunction, we can regenerate the lookup table on the fly based on the precision we desire.

[Alphaharrius]

The solution is to create the lookup table structure PhaseTable that also includes the actual eigenvalue from a given phase computed from some inputs (which often is not so accurate). Apart from this change we also included an API extension to the FockMap which helps to generate a transformation of outspace or inspace to handle the symmetrizing problem with unusual unit-cell (that does not contained within the positive parallelogram of the real space).

Yet this fix unfortunately removed the feature of adding custom eigenfunction instead of using the computed version.

[Alphaharrius]

This fix also included a revamped implementation of *(::AffineTransform, ::RegionState) which supports manual symmetrization if there are no unitary transformation found that can correctly diagonalize the given set of isometry to U(1) irreducible representations (which mean the single-particle states given by the isometries are not symmetric in any perspective). The manual symmetrization process will only be performed on quasi-symmetrical isometries (which the U(1) phases of the isometry under diagonalization is approximately equal to the actual phases of the symmetry up to some tolerance).