Roman Lávička

Gelfand–Tsetlin bases for spherical monogenics in dimension 3

The main aim of this paper is to recall the notion of the Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3 having the Appell property and we compare them with those constructed by the first and the second author recently (by a direct analytic approach).

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‐Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis.

Gel'fand–Tsetlin bases of orthogonal polynomials in Hermitean Clifford analysis

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centered around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators. Copyright

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].

Fischer Decompositions of Kernels of Hermitean Dirac Operators

In this note we describe explicitly irreducible decompositions of kernels of the Hermitean Dirac Operators. In [6], it is shown that these decompositions are essential for a construction of orthogonal (or even Gelfand‐Tsetlin) bases of homogeneous Hermitean monogenic polynomials.

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 Mobius transformations.

The Fischer Decomposition for the H -action and Its Applications

Recently the Fischer decomposition for the H-action of the Pin group on Clifford algebra-valued polynomials has been obtained. We apply this tool to get various decompositions of special monogenic and inframonogenic polynomials in terms of two-sided monogenic ones.

Generalized Appell Property for the Riesz System in Dimension 3

In this note, we give an explicit description of orthogonal (even the Gelfand‐Tsetlin) bases for the spaces of homogeneous solutions of the Riesz system in R3 using hypergeometric series. Moreover, we show that the basis elements in dimension 3 possess the Appell property not only with respect to one variable but even with respect to all variables.

On the Structure of Monogenic Multi‐Vector Valued Polynomials

Let F be a smooth s‐vector valued function in the Euclidean space Rm satisfying the equation ∂F = 0 where ∂ is the Dirac operator in Rm. In [3], it was shown that there exists an s‐vector valued harmonic function H in Rm such that F = ∂H∂. In this note, we construct such a function H quite explicitly in the case when F is an s‐vector valued homogeneous polynomial and, as an easy consequence, when the function F is defined on a ball in Rm. Moreover, we decompose harmonic and one‐sided monogenic (multi‐vector valued) polynomials only in terms of two‐sided monogenic ones.

Fischer decomposition for osp(4|2)‐monogenics in quaternionic Clifford analysis

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 osp(4|2)-monogenic polynomials, the Lie super algebra 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 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 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 E and P, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair osp(4|2)×Sp(p), the action of which will make the Fischer decomposition multiplicity free. Copyright