Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

Explicit complete orthonormal fixed bases are computed for subspaces of the space of square-integrable functions on the sphere where the subspaces contain functions that are totally symmetric under the rotational symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symmetry (icosahedral/dodecahedral, octahedral/cubic, tetrahedral), the calculation has three steps: First, a bilinear equation is derived for the coefficients in the linear combination by equating the expansion of a symmetrized δ function in both spherical harmonics and the fixed basis functions for the appropriate subspace. The equation is parameterized by the location (θ₀, φ₀) of the δ function and must be satisfied for all locations. Second, the dependence on the δ-function location is expressed in a Fourier (φ₀) and a Taylor (θ₀) series and thereby a new system of bilinear equations is derived by equating selected coefficients. Third, a recursive solution of the new system is derived and the recursion is solved explicitly with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.