Fischer decomposition for the symplectic group
\textit{F. Brackx} et al. [Math. Methods Appl. Sci. 39, No. 16, 4874--4891 (2016; Zbl 1361.30085)] conjectured a form for the Fischer decomposition for the spinor-valued polynomials defined on Euclidean space of four-fold dimension in terms of spaces \(S^r_{a,b}\) of bi-homogeneous \(\text{osp}(4|2)\)-monogenic polynomials with values in the symplectic cell \(S^r_r\). The current paper formulates and proves a corrected version of this Fischer decomposition which holds in ALL cases. Also shown, using the spirit of \textit{R. Howe} [Trans. Am. Math. Soc. 313, No. 2, 539--570 (1989; Zbl 0674.15021)], is the \(\mathrm{Sp}(p)\)-irreducibility of the spaces \(S^r_{a,b}\). As preparatory for the latter deduction a special section presenting refinements of Clifford analysis through the concept of symmetry reduction is included in the paper.
scientific article; zbMATH DE number 6802346
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