R . Wallach Symmetry , Representations , and Invariants Graduate Texts in Mathematics 255 Springer

In this appendix we shall find all the irreducible representations of the symmetry groups of the Platonic solids, by a mixture of geometric methods and algebraic methods similar to those used in Chapters 5 for representations of the classical groups. We shall also see how these representations occur naturally in the harmonic analysis of functions on the Platonic solids. This is a discrete analogue of the decomposition of functions on the 2-sphere under the action of the orthogonal group, and it will serve to illustrate the methods of Fourier analysis on groups in relatively simple but beautiful examples. As has been known from antiquity (Weyl [5], Sternberg [3, §1.8]), there are three types of three-dimensional Platonic solids: 1. the tetrahedron, with rotational symmetry group the alternating group A4, 2. the octahedron (and its dual cube), with rotational symmetry group S4, 3. the icosahedron (and its dual dodecahedron), with rotational symmetry group the alternating group A5. In considering each of type of solid, we urge the reader to draw pictures, or even better to construct a 3-dimensional model, in order to verify the properties of the symmetries of the solid. As Felix Klein writes, “we are treating of concrete matters, which may easily be conceived with the assistance of the suggested aids, but which may occasionally offer difficulties if these are neglected” ([2, §1.4]). G.1 Symmetries of the tetrahedron We begin with the simplest Platonic solid, the regular tetrahedron T. The full group of symmetries of T is S4, acting by permuting the 4 vertices, and the subgroup of orientation-preserving symmetries is the alternating group A4. The action of S4 on T is linear and arises from the standard three-dimensional representation (ρ,V ) of S4 of Section 4.4.3. To see this explicitly, we take the or-