Polynomial Invariants and Harmonic Functions Related to Exceptional Regular Polytopes

Abstract

We compute certain polynomial invariants for the finite reflection groups of the types 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.

Keywords:

• Polynomial invariants
• harmonic functions
• exceptional regular polytopes
• mean value property
• finite reflection groups