The harmonic transvector algebra in 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.

Fischer decomposition for polynomials on superspace

“…Here we use the fact that dim F a+ m 2 1 k = dim F a . On the other hand, using Lemma 4.1, we get (7) [ where ∧ j (C k ) is the j-th antisymmetric power of the defining representation C k for GL(k) and dim ∧ j (C k ) = k j , see [14]. In addition, for each ℓ ∈ N 0 , we have proved that π(x J M ℓ ) ≃ M ℓ .…”

“…The scalar case in the stable range is well known (see [12]). Recently, in the scalar case, an alternative approach leading to explicit formulae for different projections in the decomposition was developed in [7] for two variables.…”

The orthogonal branching problem for symplectic monogenics

“…To arrive at an explicit construction of the embedding operators turning the isomorphism of the Klimyk decomposition into an equality, we start from the Lie algebra g = sp(4), hereby following the approach from [6]. This Lie algebra can be realised as the algebra generated by rotationally invariant operators in two vector variables in R m .…”

“…These embeddings are provided by a set of rotationally invariant operators which appeared in the literature as the generators of the transvector algebra Z(sp(4), so(4)). This quadratic algebra was used by De Bie et al in [6] as a new dual partner for the orthogonal group, hence obtaining formulas for the integral over a Stiefel manifold. See also the paper of Zhelobenko [22] for a general introduction to the theory of transvector algebras, and section 4 for the necessary background in this paper.…”

The higher spin Laplace operator in several vector variables

“…Using equation (14) and the fact that N (x) = xx = |x| 2 , this implies that the action of the Kelvin inversion on functions is given by…”