For our example we consider the point group D_4h. We install D_4h using GTInstallGroup. The character Table and the representation matrices are obtained using GTGetIreps.

We choose the irreducible representation B_1g as well as the irreducible representation E_g.

To apply projection operators, we need to specify a function, which we choose to be f(x, y, z) = x²–y² + x y + x z + z y. The character projection operator is applied using GTCharProjectionOperator. Here we can either choose a specific irreducible representation by providing its characters, or it can be applied for all irreducible representations by providing the full character system. As can be verified, the part x y transforms as B_2g, x²-y² transforms as B_1g, and x z + y z transforms as E_g.

E_g is a two-dimensional representation and hence we can find two linearly independent basis functions spanning the basis of this representation. By applying the full projection operator GTProjectionOperator for the different diagonal components of the representation matrices, we can project out two such functions.