The Fueter theorem states that regular (resp. monogenic) functions in quaternionic (resp. Clifford) analysis can be constructed from holomorphic functions 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 -modules.

In this paper, we will determine the fundamental solution for the higher spin Dirac operator Qλ, which is a generalisation of the classical Rarita–Schwinger operator to more complicated irreducible (half-integer) representations for the spin group in m dimensions. This will allow us to generalise the Stokes theorem, the Cauchy–Pompeiu theorem and the Cauchy integral formula, which lie at the very heart of the function theory behind arbitrary elliptic higher spin operators.

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space R^m. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on R^m. Fix the direct sum R^m = R^p x R^q. In this paper we will study the decomposition of the space M_n(R^m;C_m) of spherical monogenics of order n under the action of Spin(p) x Spin(q). As a result we obtain a Spin(p) x Spin(q)-invariant orthonormal basis for M_n(R^m;C_m). In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(R^m;C_m).

In this paper we will present two proofs of the monogenic Fischer decomposition in two vector variables. The first one is based on the so-called “Harmonic Separation of Variables Theorem” while the second one relies on some simple dimension arguments. We also show that these decomposition are still valid under milder assumptions than the usual stable range condition. In the process, we derive explicit formula for the summands in the monogenic Fischer decomposition of harmonics.

The space of spherical monogenics \({\mathcal{M}}_k\) in \({\mathbb{R}}^m\) can be regarded as a model for the irreducible representation of Spin(m) with weight \((k + \frac{1}{2}, \frac{1}{2}, \cdots , \frac{1}{2})\). In this paper we construct an orthonormal basis for \({\mathcal{M}}_k\). To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra \(\mathfrak{sl}\)(2) on \({\mathcal{M}}_k\).

This article deals with the Hardy space related to solutions of generalized Rarita–Schwinger operators in the ball and the half space. These operators generalize the Rarita–Schwinger operator for spin -fields to the case of functions taking values in irreducible representation spaces with weight .

In this paper a generalization of the classicalRarita–Schwinger equations for spin 3/2 fields to the case of spin fieldswith values in irreducible representation spaces with weight k +1/2 isgiven. It corresponds to the study of serie of first orderconformal invariant operators, which are constructed from twisted Diracoperators. The representation character of polynomial solutions of the equations onflat space and their relations are described in details.

In this paper we consider harmonic and monogenic polynomials of simplicial type. It is proved that these polynomials provide explicit realizations of all irreducible representations of Spin ( m ).

In this paper we investigate a generalization of the classical Rarita–Schwinger equations for spin 3/2 fields to the case of functions taking values in irreducible representation spaces with weight k+1/2. These fields may be realised as functions taking values in spaces of spherical monogenics earlier considered in F. Sommen and N. Van Acker (1993, in “Clifford Algebras and Their Applications in Mathematical Physics,” Kluwer Academic, Dordrecht/Norwell, MA). In this paper we develop the main function theoretic results.

Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere. Sm−1 of . In this paper we will consider the L2 -boundary value theory for this class of functions. The main Theorem is a higher dimensional version of the Kerzman-Stein Theorem of complex analysis in the plane.

Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere Sm-1 of R m. In this paper we will given explicit formulae for the Taylor and Laurent series for this class of functions. This leads to an explicity residue theory for spherical monogenics having point singularities.

Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere S(m-1) of R(m). In this paper we will consider Runge approximation Theorems and some of their consequences for this class of functions.

This paper illustrates that the theory of monogenic functions satisfying a fixed homogeneity condition leads to a new function theory, parallel to but different from the standard theory of monogenic functions. The function theory thus obtained includes the theory of the Dirac operator on the unit sphere.

In this paper we give some basic integral formulas related to spherical monogenics of complex degree. These monogenics, locally defined on the sphere Sm-1, are eigenfunctions of the Gamma-operator corresponding to complex eigenvalues and generalise the classical spherical monogenics.

In this paper we define a product for monogenic functions derived from Fischer's decomposition. Although this product can be defined for all monogenic polynomials, it can be canonically defined only for a special subclass of polynomials. The definition of the product can be generalized to the setting of monogenic tensors.