Explicit Computation of Orthonormal Symmetrized Harmonics with Application to the Identity Representation of the Icosahedral Group

A novel method to explicitly compute orthonormal symmetrized harmonics is presented and the method is applied to the identity representation of the icosahedral group. Spherical viruses have icosahedral symmetry and the motivating application is the parametric representation of spherical viruses for use in inverse problems based on x-ray scattering data and cryoelectron microscopy images. The symmetrized harmonics are computed in the form of linear combinations of spherical harmonics of one order and therefore have simple rotational properties which is valuable in the electron microscopy application. The method is based on equating the expansions of a symmetrized delta function in spherical and in symmetrized harmonics from which bilinear equations for the weights in the linear combinations can be derived. The explicit character of the calculation is reflected in the fact that both explicit expressions and an efficient recursive algorithm are derived for computing the weights in the linear combinations.