Fischer Decomposition for osp(4|2)-monogenics in Quaternion Clifford Analysis

TL;DR

This paper introduces a new concept of (4|2)-monogenic polynomials in quaternionic Clifford analysis, enabling a multiplicity-free Fischer decomposition of spinor-valued polynomials under the dual pair (4|2)  and Sp(p).

Contribution

It defines (4|2)-monogenicity using quaternionic Dirac operators, scalar Euler operator, and Clifford algebra operator P, refining quaternionic monogenicity for irreducibility.

Findings

Established (4|2)-monogenicity as a refinement of quaternionic monogenicity.

Constructed a multiplicity-free Fischer decomposition for spinor-valued polynomials.

Connected the decomposition to the Howe dual pair (4|2)  and Sp(p).

Abstract

Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp$( p)$. These Fischer decompositions involve spaces of homogeneous, so-called $\mathfrak{osp}(4|2)$-monogenic polynomials, the Lie superalgebra $\mathfrak{osp}(4|2)$ being the Howe dual partner to the symplectic group Sp$( p)$. In order to obtain Sp$( p)$-irreducibility this new concept of $\mathfrak{osp}(4|2)$-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator $\mathbb{E}$ underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator $P$ underlying the decomposition of spinor space into symplectic cells. These operators $\mathbb{E}$ and $P$, and their hermitian conjugates, arise naturally when constructing the Howe dual pair $\mathfrak{osp}(4|2) \times$ Sp$( p)$, the action of which will make the Fischer decomposition multiplicityfree.