I would like to find all the real harmonic homogeneous polynomials of each degree in three variables that are invariant (or invariant up to a sign change) under the action of each of the point groups that act on Euclidean 3-space.
A harmonic polynomial is simply a polynomial such that the Laplacian vanishes.
For example let $z=x_1+\mathrm{i}x_2$ then the Real and Imaginary parts of the polynomials $z^m$ and $\bar{z}^m$ are invariant under the action of the group generated by a rotation about the $e_3$ axis through $2\pi/q$ where $q$ and $m$ are natural numbers and $q$ divides $m$.
I can construct other examples but I assume that since the representation theory of point groups has been studied extensively, specifically for chemistry, this might be something that has already been done. Perhaps under another name.
I am aware of Sheldon Axler's Mathematica code for harmonic functions, and can calculate a basis for harmonic polynomials of some degree, so I could in principle calculate the invariant polynomials by brute force. However I am looking for something that provides more insight in to the relations between the invariant harmonic polynomials and the group.
A complete answer is contained in the paper Burnett Meyer, On the Symmetries of Spherical Harmonics, Canadian Journal of Mathematics, Volume 6,1954 , pp. 135-157, thanks to Gábor Megyesi for pointing out the paper. Meyer finds a generating function such that the coefficients of its series expansion are the dimensions of the spaces of invariant harmonic polynomials for each degree. He then goes on to construct bases for these spaces for each of the point groups for each degree. As these are explicit one can test if each of these basis elements is invariant or invariant up to a sign change.