Real basis functions of polyhedral groups

The basis of the identity representation of a polyhedral group is able to describe functions with symmetries of a platonic solid, i.e., 3-D objects which geometrically obey the cubic symmetries. However, to describe the dynamic of assembles of heterogeneous 3-D structures, a situation that each object lacks the symmetries but obeys the symmetries on a level of statistics, the basis of all representations of a group is required. While those 3-D objects are often transformed to real functions on $L_2$ space, it is desirable to generate a complete basis on real space. This paper deduces the existence of a basis on real space for each polyhedral group, and introduces a novel approach to explicitly compute these real basis functions, of which properties are further explored.