## Spherical harmonics, invariant theory and Maxwell's poles

##### Date

##### Authors

Dowker, J. S.

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

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).

Comment: 45 pages. JyTex; added analysis and references, minor corrections

Comment: 45 pages. JyTex; added analysis and references, minor corrections

##### Keywords

Mathematical Physics, Astrophysics, High Energy Physics - Theory, Mathematics - Algebraic Geometry