The Fischer decomposition for Hodge‐de Rham systems in Euclidean spaces

The classical Fischer decomposition of spinor‐valued polynomials is a key result on solutions of the Dirac equation in the Euclidean space . As is well‐known, it can be understood as an irreducible decomposition with respect to the so‐called L‐action of the Pin group Pin(m). But, in Clifford algebra valued polynomials, we can consider also the H‐action of Pin(m). In this paper, the corresponding Fischer decomposition for the H‐action is obtained. It turns out that, in this case, basic building blocks are the spaces of homogeneous solutions to the Hodge‐de Rham system. Moreover, it is shown that the Fischer decomposition for the H‐action can be viewed even as a refinement of the classical one. Copyright © 2011 John Wiley & Sons, Ltd.