The number of harmonic frames of prime order

TL;DR

This paper investigates harmonic frames of prime order, providing an exact recursive formula for counting inequivalent frames and analyzing their symmetry groups.

Contribution

It introduces a recursive formula for enumerating inequivalent harmonic frames of prime order and explores their symmetry group structure.

Findings

Exact recursive formula for counting inequivalent harmonic frames

Establishment of a correspondence between frames and orbit sets

Identification of key subgroup structures in symmetry groups

Abstract

Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order harmonic frames is shown to contain a subgroup consisting of a diagonal matrix as well as a permutation matrix, each of which is dependent on the particular harmonic frame in question.