  The classical Fischer decomposition of spinor-valued polynomials is a key
result on solutions of the Dirac equation in the Euclidean space R^m. 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, on 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.
