The harmonic transvector algebra in two vector variables

TL;DR

This paper extends harmonic analysis by describing polynomial decompositions in two vector variables using a new transvector algebra, generalizing Howe duality and enabling explicit projections and integrals over Stiefel manifolds.

Contribution

It introduces a novel transvector algebra as a dual partner for the orthogonal group, generalizing Howe duality to decompose polynomials in two variables.

Findings

Derived explicit projection operators for irreducible components.

Provided formulas for integrating polynomials over Stiefel manifolds.

Extended harmonic analysis tools to two vector variables.

Abstract

The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical Howe duality. The results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated Stiefel manifold.