On polynomial solutions of generalized Moisil-Théodoresco systems and Hodge-de Rham systems

The aim of the paper is to study relations between polynomial solutions of generalized Moisil-Theodoresco (GMT) systems and polynomial solutions of Hodge-de Rham systems and, using these relations, to describe polynomial solutions of GMT systems. We decompose the space of homogeneous solutions of GMT system of a given homogeneity into irreducible pieces under the action of the group O(m) and we characterize individual pieces by their highest weights and we compute their dimensions

Finely differentiable monogenic functions

summary:Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in (Lávička, R., A generalisation of monogenic functions to fine domains, preprint.) like a higher dimensional analogue of finely holomorphic functions

Gel'fand-Tsetlin procedure for the construction of orthogonal bases in Hermitean Clifford analysis

In this note, we describe the Gel’fand-Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy-Kowalewski extension theorem and the Fischer decomposition in Hermitean Clifford analysis

Fischer decompositions of kernels of Hermitean Dirac operators

In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [3], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand-Tsetlin) bases of homogeneous Hermitean monogenic polynomials

Orthogonal bases of Hermitean monogenic polynomials : an explicit construction in complex dimension 2

In this contribution we construct an orthogonal basis of Hermitean monogenic polynomials for the specific case of two complex variables. The approach combines group representation theory, see [5], with a Fischer decomposition for the kernels of each of the considered Dirac operators, see [4], and a Cauchy-Kovalevskaya extension principle, see [3]

Reversible maps in the group of quaternionic Möbius transformations

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations