Spherical harmonics, invariant theory and Maxwell's poles

TL;DR

This paper explores the deep connections between harmonic polynomials, invariant theory, and Maxwell's poles, highlighting classical theorems and their modern interpretations within group theory and quantum mechanics.

Contribution

It clarifies the relationship between harmonic polynomials and invariant theory, proposes renaming Sylvester's theorem, and links classical constructs to SU(3) and quantum mechanics.

Findings

Homogeneous harmonic polynomials correspond to apolar ternary forms.

Sylvester's theorem is suggested to be renamed the Clebsch-Sylvester theorem.

Connections between invariant theory constructs and angular momentum in quantum mechanics are expanded.

Abstract

I discuss the relation between harmonic polynomials and invariant theory and show that homogeneous, harmonic polynomials correspond to ternary forms that are apolar to a base conic (the absolute). The calculation of Schlesinger that replaces such a form by a polarised binary form is reviewed. It is suggested that Sylvester's theorem on the uniqueness of Maxwell's pole expression for harmonics is renamed the Clebsch-Sylvester theorem. The relation between certain constructs in invariant theory and angular momentum theory is enlarged upon and I resurrect the Joos--Weinberg matrices. Hilbert's projection operators are considered and their generalisations by Story and Elliott are related to similar, more recent constructions in group theory and quantum mechanics, the ternary case being equivalent to SU(3).