Expanding the Hamiltonian around a point of symmetry

[Qturtle]

Hey
I have a tight binding Hamiltonian of a BCC lattice which is a 4x4 matrix in k space (the 4 elements correspond to 4 atoms that are in a unit cell)
I want to expand it for small k's around the symmetry points P or Gamma or H.
I'm looking at a paper by J. L. Ma˜nes, PHYSICAL REVIEW B 85, 155118 (2012), where he is doing this by some kind of Unitary transformation to the Hamiltonian that changes the basis from the four Bloch functions basis to the "symmetry-adapted basis". I didn't find an explanation in the paper to the operator he uses to go from one basis to another, and i don't understand what is this symmetry adapted basis anyway.
If anyone understand this i would be very happy to know :)http://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.155118

 

[DrDu]

This is quite a common method, see, e.g.:
http://chemwiki.ucdavis.edu/Physica...al_Modes/Symmetry_Adapted_Linear_Combinations

 

[Qturtle]

hey DrDu
since i don't have a very good background on group theory (abstract math is not my strong suit.. \:), and since it looks like a more general subject, is there a more concentrated example that applies this method strictly to the problem at hand? (i.e changing the basis of a tight binding Hamiltonian)

Thanks!