An explicit construction of the unitarily
invariant quaternionic polynomial spaces on the sphere
Mozhgan Mohammadpour and Shayne Waldron
Abstract:
The decomposition of the polynomials on the quaternionic unit sphere in H^d 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).
Keywords:
irreducible representations of the quaternionic unitary group, symplec-
tic group, quaternionic polynomials, spherical harmonic polynomials, zonal functions,
projective spherical t-designs, finite tight frames, quaternionic equiangular lines
Math Review Classification:
Primary 15B33, 20G20, 33C55, 42C15;
Secondary 917B10, 42-08, 46S05, 51M2017B10, 42-08, 46S05, 51M204A12.
Length: 65 Pages
Last Updated: 21 May 2024
Availability: