Total Rarita-Schwinger operators in Clifford analysis

Rarita-Schwinger operators in Clifford analysis can be realized as first-order differential operators acting on functions f(x, u) taking values in the vector space of homogeneous monogenic polynomials. In this paper, the Scasimir operator for the orthosymplectic Lie superalgebra will be used to construct an invariant operator which acts on the full space of functions in two vector variables and therefore has more invariance properties. Also the fundamental solution for this operator will be constructed

Taylor series in Hermitean Clifford analysis

In this paper, we consider the Taylor decomposition for h-monogenic functions in Hermitean Clifford analysis. The latter is to be considered as a refinement of the classical orthogonal function theory, in which the structure group underlying the equations is reduced from so(2m) to the unitary Lie algebra u(m)

The higher spin Laplace operator in several vector variables

In this paper, an explicit expression is obtained for the conformally invariant higher spin Laplace operator $\mathcal{D}_{\lambda}$, which acts on functions taking values in an arbitrary (finite-dimensional) irreducible representation for the orthogonal group with integer valued highest weight. Once an explicit expression is obtained, a special kind of (polynomial) solutions of this operator is determined

On a special type of solutions of arbitrary higher spin Dirac operators

In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(x) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized by a half-integer highest weight. Also a special class of solutions of these operators is constructed, and the connection between these solutions and transvector algebras is explained

Gegenbauer polynomials and the Fueter theorem

The Fueter theorem states that regular (resp. monogenic) functions in quaternionic (resp. Clifford) analysis can be constructed from holomorphic functions f(z) in the complex plane, hereby using a combination of a formal substitution and the action of an appropriate power of the Laplace operator. In this paper we interpret this theorem on the level of representation theory, as an intertwining map between certain sl(2)-modules

Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables

The higher spin Dirac operator Q_{k,l} acting on functions taking values in an irreducible representation space for Spin(m) with highest weight (k+1/2,l+1/2,1/2,...,1/2), with k, l in N and k>= l, is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied

A Hermitian refinement of symplectic Clifford analysis

In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure $\mathbb{J}$ on the canonical symplectic manifold $(\mathbb {R}^{2n},\omega_0)$. This gives rise to two symplectic Dirac operators $D_s$ and $D_t$ (in the sense of Habermann), leading to a $\mathfrak{u}(n)$-invariant system of equations on $\mathbb{R}^{2n}$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of polynomials on $\mathbb {R}^{2n}$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra

Polynomial solutions for arbitrary higher spin dirac operators

In a series of recent papers, we have introduced higher spin Dirac operators, which are far-reaching generalisations of the classical Dirac operator. Whereas the latter acts on spinor-valued functions, the former acts on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe a general procedure to decompose the polynomial kernel spaces for these operators in irreducible summands for the regular action of the spin group. We will do this in an inductive way, making use of twisted higher spin operators