Taxonomy of Clifford Cl_{3,0} subgroups: Choir and band groups

TL;DR

This paper classifies all subgroups of the basis set of Clifford algebra Cl_{3,0} into obedient 'choirs' and disobedient 'bands', introducing a novel taxonomy based on algebraic and combinatorial criteria.

Contribution

It provides a complete classification of Cl_{3,0} subgroups and introduces a new taxonomy using musical metaphors like modes, rhythms, and chords.

Findings

Identifies 21 non-isomorphic subgroups of Cl_{3,0} basis set.

Distinguishes 9 choir groups and 12 band groups.

Develops a taxonomy based on transpositions, chords, and disorder.

Abstract

We list the subgroups of the basis set of Cl_{3,0} and classify them according to three criteria for construction of universal Clifford algebras: (1) each generator squares to +1 or -1, (2) the generators within the group anticommute, and (3) the order of the resulting group is 2^{n+1}, where n is the number of nontrivial generators. Obedient groups we call choirs; disobedient groups, bands. We classify choirs by modes and bands by rhythms, based on canonical equality. Each band generator has a transposition (number of other generators it commutes with). The band's transposition signature is the band's chord. The sum of transpositions divided by twice the number of generator pair combinations is the band's beat. The band's order deviation is the band's disorder. For n less than or equal 3, we show that the Cl_{3,0} basis set has 21 non-isomorphic subgroups consisting of 9 choirs and 12 bands.