Polynomial invariants and harmonic functions related to exceptional regular polytopes

Abstract

We compute certain polynomial invariants for the finite reflection groups of the types {\small $H_3$, $H_4$ and $F_4$}. Using this result, we explicitly determine the solution space of functions satisfying a mean value property related to the exceptional regular polytopes, namely, the icosahedron and dodecahedron in three dimensions and the 24-cell, 600-cell, and 120-cell in four dimensions.