# Normal mode decomposition of a triangular hexagonal lattice

I was trying to understand and redo the methods used in a previous question:

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

and I face some dificulties. I don't understand the way the author find the normal modes, which is probably due to the fact that I'm not used to the formalism.

First I'm disagree concerning the group of symmetry chosen. The author is using D$$_{6}$$ but as far as I'm concern, this lattice belongs to D$$_{6h}$$.

Second, if I well understood, the author claims that by diagonalizing the matrix representation of the symmetry operations, you find the eigenvectors that he is plotting (At least for the non-degenerate part). But if I do so for $$\Gamma_{C_{6}}$$, as it is a rotation matrix, I don't find the eigenvectors he is plotting but something else, and I find eigenvalues lying on the unit circle of the complex plane (which I think is normal for a rotation matrix), but not at all the values he found. (If it can help I can upload the eigenvectors I found and their corresponding eigenvalues).

Is someone used to this formalism and could help me/us to understand the way the author found his eigenvectors and corresponding eigenvalues?