EXPLICIT ORTHONORMAL FIXED BASES FOR SPACES OF FUNCTIONS THAT ARE TOTALLY SYMMETRICAL UNDER THE ROTATIONAL SYMMETRIES OF A PLATONIC SOLID

Explicit complete orthonormal fixed bases are computed for subspaces o f the space of square-integrable functions on the sphere where the sub spaces contain functions that are totally symmetric under the rotation al symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symm etry (icosahedral/dodecahedral, octahedral/cubic, tetrahedral), the ca lculation has three steps: First, a bilinear equation is derived for t he coefficients in the linear combination by equating the expansion of a symmetrized delta function in both spherical harmonics and the fixe d basis functions for the appropriate subspace. The equation is parame terized by the location (theta(0), phi(0)) of the delta function and m ust be satisfied for all locations. Second, the dependence on the delt a-function location is expressed in a Fourier (phi(0)) and a Taylor (t heta(0)) series and thereby a new system of bilinear equations is deri ved 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.