# Why not just use the x,y,z as basis functions instead of linear/quadratic functions of these unit vectors?

I am going through Applications of Group Theory to the Physics of Solids by M. S. Dresselhaus.

In chapter 4 he states

This is with respect to the P(3) group.

I understand basis functions must transform like the given representation e.g for the identity representation $$\Gamma_1$$ all operators leave any unit function unchanged hence you could use any of $$x,y,z,z^2$$ separately as basis functions because they are left unchanged in this representation.

For a 2D representation $$\Gamma_2$$ often functions like $$(xz,yz)$$ and $$(xy,x^2-y^2)$$ are also stated as basis functions. Why not simply use $$(x,y)$$ for the 2D case?

Do we list all other basic functions in character tables for completeness? Or is it because in certain cases basis functions like $$(xy,x^2-y^2)$$ may be more useful than $$(x,y)$$ [I am thinking that the former basis function has the symmetry of a d orbital according to Harris and Bertolucci hence it may be more useful to consider in the context of looking at atoms/molecules]

Each d subshell has five d-orbitals, and often these are written as $$\mathrm{d}_{xy}$$, $$\mathrm{d}_{yz}$$, $$\mathrm{d}_{xz}$$, $$\mathrm{d}_{z^2}$$, and $$\mathrm{d}_{x^2 - y^2}$$. These transform in the same way as the respective Cartesian products in the subscripts, so for example, the $$\mathrm{d}_{xy}$$ orbital transforms in the same way as $$xy$$. (More technically, the mathematical form of the $$\mathrm{d}_{xy}$$ orbital is $$xy$$ multiplied by some spherically symmetric term, which doesn't affect the symmetry.)
(The $$x, y, z$$ linear functions are useful for lots of things (determining symmetries of translational modes, etc.), but they're also there because of the p-orbitals, which transform in the same way as $$(x, y, z)$$.)