Abstract. 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.

A fifth order conformally invariant higher spin operator on the sphere AIP Conf.Calculus structure on the Lie conformal algebra complex and the variational complex This paper deals with a systematic construction of higher spin operators, defined as conformally invariant differential operators acting on functions on flat space R m with values in an arbitrary half-integer irreducible representation for the spin group. To be more precise, the higher spin version of the Dirac operator and associated twistor operators will be constructed as generators of a transvector algebra, hereby generalising the well-known fact that the classical Dirac operator on R m and its symbol generate the orthosymplectic Lie superalgebra osp(1, 2). To do so, we will use the extremal projection operator and its relation to transvector algebras. In the second part of the article, the conformal invariance of the constructed higher spin operators will be proven explicitly. C 2014 AIP Publishing LLC. [http://dx.

In a series of recent papers, we have introduced higher spin Dirac operators, which are generalisations of the classical Dirac operator. Whereas the latter acts on spinorvalued functions, the former act on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe how the polynomial kernel spaces of such operators decompose in irreducible representations of the spin group. We will hereby make use of results from representation theory.

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In this paper we work in the 'split' discrete Clifford analysis setting, i.e. the mdimensional function theory concerning null-functions of the discrete Dirac operator ∂, defined on the grid Z m , involving both forward and backward differences. This Dirac operator factorizes the (discrete) Star-Laplacian (∆ * = ∂ 2 ). We show how the space H k of discrete k-homogeneous spherical harmonics, which is a reducible so(m, C)-representation, may explicitly be decomposed into 2 2m isomorphic copies of irreducible so(m, C)-representations with highest weight (k, 0, . . . , 0). The key element is the introduction of 2 2m idempotents, dividing the discrete Clifford algebra in 2 2m subalgebras of dimension k+m−1 k − k+m−3 k .