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

Abstrakt

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.

Klíčová slova

• Fischer decomposition
• Hodge-de Rham systems