Abstract:Polynomials on $\Bbb R^n$ with values in an irreducible $Spin_n$-module form a natural representation space for the group $Spin_n$. These representations are completely reducible. In the paper, we give a complete description of their decompositions into irreducible components for polynomials with values in a certain range of irreducible modules. The results are used to describe the structure of kernels of conformally invariant elliptic first order systems acting on maps on $\Bbb R^n$ with values in these modules.
Keywords: conformally invariant differential operators, generalized (higher-spin) Dirac operators, representations of spin-groups, Littlewood-Richardson rule
AMS Subject Classification: 53A30, 53A55, 32A50, 43A65