A zeta function approach to the relation between the numbers of symmetry planes and axes of a polytope

TL;DR

This paper explores the connection between symmetry properties of polytopes and zeta functions, deriving geometric relations through advanced spectral analysis techniques on orbifolds and higher-dimensional manifolds.

Contribution

It extends the Cesàro-Fedorov relation to higher dimensions using the Selberg trace formula and heat-kernel coefficients, linking symmetry counts to geometric measures.

Findings

Derived the Cesàro-Fedorov relation from the Selberg trace formula.

Extended symmetry relations to higher-dimensional spheres.

Connected polynomial coefficients to geometric properties of fundamental domains.

Abstract

A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-Todd polynomial to the geometry of the fundamental domain. For the 3-sphere we show that $b_4$ is given by the ratio of the volume of the fundamental tetrahedron to its Schl\"afli reciprocal.