An explicit construction of the unitarily invariant quaternionic polynomial spaces on the sphere

TL;DR

This paper provides an explicit constructive method for decomposing quaternionic polynomials on the sphere into irreducible components, facilitating applications in geometric configurations and spherical designs.

Contribution

It introduces a low-dimensional subspace approach with explicit formulas for zonal polynomials, advancing the constructive understanding of quaternionic polynomial spaces.

Findings

Explicit formulas for zonal polynomials

Orthogonal decomposition into irreducibles

Descriptions of symmetries and dimensions

Abstract

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several authors. Typically, these abstract decompositions into ``quaternionic spherical harmonics'' specify the irreducible representations involved and their multiplicities. The elementary constructive approach taken here gives an orthogonal direct sum of irreducibles, which can be described by some low-dimensional subspaces, to which commuting linear operators $L$ and $R$ are applied. These operators map harmonic polynomials to harmonic polynomials, and zonal polynomials to zonal polynomials. We give explicit formulas for the relevant ``zonal polynomials'' and describe the symmetries, dimensions, and ``complexity'' of the subspaces involved. Possible applications include the construction and analysis of desirable sets of points in quaternionic space, such as equiangular lines, lattices and spherical designs (cubature rules).