How to get the basis functions for a given matrix realization of an irrep?

Hello,

As far as I understand, once GTPack provides a matrix realization for an irrep of a point group (through any method: Cornwell, Flodmark, Induction), one should be able to obtain symmetry-adapted linear combinations of e.g. spherical harmonics that transform exactly according to that matrix realization, i.e. that verify Eq. (5.47) of the GTPack book. Although I intend ultimately to treat rather general cases, I am having problems at present with a simple case, namely the two-dimensional irrep E” of group D3h. By using the command GTWignerProjectionOperator I arrive to the conclusion that the basis functions of the matrix irrep (obtained by Cornwell method) should be:

phi1 = 1/2 Y[2,-1]+1/2 Y[2,1] –> y z
phi2 = 1/2 Y[2,-1]-1/2 Y[2,1] –> x z

BUT the problem is that, when I try to check that these functions verify Eq. (5.47) they don’t !!! (at least not for all symmetry operations)

I repeat the whole procedure for irrep matrices obtained by Flodmark and Induction methods and get the same (if not worst) result.

(I like to think the symmetry operations as active so I am using GTReinstallAxes[“active”]. I do not know if this is of any relevance).

Can anybody help me?

Thanks in advance.
Alberto.