Fischer decomposition for osp(4|2)‐monogenics in quaternionic Clifford analysis. (7th April 2016) - Brackx, Fred; Schepper, Hennie De; Eelbode, David; Lávička, Roman; Souček, Vladimir
Abstract : 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 o s p ( 4 | 2 ) ‐monogenic polynomials, the Lie super algebra o s p ( 4 | 2 ) being the Howe dual partner to the symplectic group Sp( p ). In order to obtain Sp( p )‐irreducibility, this new concept of o s p ( 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 o s p ( 4 | 2 ) × Sp ( p ), the action of which will make the Fischer decomposition multiplicity free. Copyright © 2016 John Wiley & Sons, Ltd.