Platonic orthonormal wavelets (Q1372771)

Aspects of the theory of wavelets on a finite group are considered. Let \(G\) be such a group and let \((\sigma, V)\) be an irreducible unitary representation of \(G\). Let \(H\) be a subgroup of \(G\). An \(H\)-wavelet in \(V\) is a unit vector \(v\in V\) whose stabilizer in \(H\) is trivial, and such that \(\sigma(H)v\) is an orthonormal basis of \(V\). The main concern of the paper is the group \(G\) of rotational symmetries of a Platonic solid \(S\), a polyhedron in \(\mathbb{R}^3\) whose faces are regular polygons. Theorem. Let \(G\) be the group of rotational symmetries of a Platonic solid. Then, up to conjugation, there is only one subgroup \(H\subseteq G\) for which the \(H\)-wavelets exist. In the special cases when \(S\) is a tetrahedron, a cube, an octahedron, a dodecahedron or an icosahedron, the \(H\)-wavelets are evaluated explicitly.