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

Abstract

A derivation of the Cesàro-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 Schläfli reciprocal.

Plain TeX, 26 pages (eqn. (86) corrected)

References in corpus (2)

• Vacuum energy on orbifold factors of spheres
• Effective action in spherical domains

Cited by in corpus (1)

• p-form spectra and Casimir energies on spherical tesselations